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XVIII Latin American Algebra Colloquium
Sao Pedro, SP, Brazil, August 3rd – 8th, 2009
Held at Hotel Fonte Colina Verde, Sao Pedro, SP, Brazil.
http://www.ime.usp.br/∼[email protected]
Organized by:• Universidade de Sao Paulo• Instituto Nacional de Ciencia e Tecnologia de Matematica (INCTMat)
Sponsored by• International Mathematical Union (IMU)
With the support of• FAPESP• CNPq• CAPES
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Scientific Committee
• E. Aljadeff (Technion, Israel)• N. Andruskiewitsch (U. Nac. de Cordoba, Argentina)• Y. Bahturin (Memorial U. of Newfoundland, Canada)• F.U. Coelho (USP, Brazil)• E. Esteves (IMPA, Brazil)• V. Ferreira (USP, Brazil)• W. Ferrer (U. de la Republica, Uruguay)• M. Ferrero (UFRGS, Brazil)• V. Futorny (USP, Brazil)• A. Garcia (IMPA, Brazil)• A. Giambruno (U. de Palermo, Italy)• V. Kharchenko (UNAM, Mexico)• A. Labra (U. de Chile, Chile)• E.N. Marcos (USP, Brazil)• S. Natale (U. Nac. Cordoba, Argentina)• D. Panario (Carleton U., Canada)• M.I. Platzeck (U. de Bahia Blanca, Argentina)• C. Polcino Milies (USP, Brazil)• M.J. Redondo (U. Nac. del Sur, Argentina)• V. Serganova (U.C. Berkeley, USA)• I. Shestakov (USP, Brazil)• S.K. Sehgal (U. of Alberta, Canada)• S. Sidki (UnB, Brazil)• J. Tirao (U. Nac. de Cordoba, Argentina)• I. Vainsencher (UFMG, Brazil)
Organizing Committee
• R.A. Ferraz• V. Ferreira• V. Futorny• G. Chalom• L.S.I. Murakami• C. Polcino Milies (chairman)
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Session organizers
• Number Theory, Algebraic Geometry and Commutative Algebra: I. Vaisencher and E. Esteves• Group Theory: S. Sidki and J. Tirao• Ring Theory: M. Ferrero and A. Giambruno• Representation of Algebras: F.U. Coelho and M.I. Platzeck• Homological Methods in Algebra: E.N. Marcos and M.J. Redondo• Hopf Algebras: S. Natale and V. Ferreira• Non-Associative Algebra: I. Shestakov and A. Labra• Lie Theory and Applications in Mathematical Physics: V. Futorny and V. Serganova• Finite Fields, Coding Theory and Cryptography: D. Panario and A. Garcia
Plenary Speakers
• Nicolas Andruskiewitsch (Universidad Nacional de Cordoba, Argentina)• Severino Collier Coutinho (UFRJ, Brazil)• Jose Antonio de la Pena (UNAM, Mexico)• Surender K. Jain (Ohio University, USA)• Christian Kassel (CNRS & Universite de Strasbourg, France)• Vera Serganova (University of California, Berkeley, USA)• Said Sidki (UnB, Brazil)
Mini-courses
• Quantum groups and Hopf algebrasGaston Andres Garcia (Universidad Nacional de Cordoba, Argentina)
• Cluster algebras and cluster categoriesRalf Schiffler (University of Connecticut, USA)
• An introduction to central simple algebras and the Brauer groupEduardo Tengan (ICMC-USP, Brazil)
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Session Talks
Number Theory, Algebraic Geometry and Commutative Algebra
• Michela Artebani
• Angel Carocca
• Juliana Coelho
• Hemar Godinho
• Victor Gonzalez Aguilera
• Marcos Jardim
• Takao Kato
• Antonio Laface
• Victor Gonzalo Lopez-Neumann
• Renato Vidal Martins
• Margarida Melo
• Frank Neumann
• Akira Ohbuchi
• Alvaro Rittatore
• Anita Rojas
• Cecılia Salgado
• Filippo Viviani
Group Theory
• Sheila Chagas
• Claudia Egea
• Gabriel Minian
• Ricardo Nunes de Oliveira
• Noraı Romeu Rocco
• Olga Patricia Salazar-Diaz
• Pavel Zalesskii
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Ring Theory
• Dirceu Baggio
• Rosali Brusamarello
• John Clark
• Wagner Cortes
• Michael Dokuchaev
• Raul Antonio Ferraz
• Miguel Ferrero
• Antonio Giambruno
• Jairo Zacarias Goncalves
• Claus Haetinger
• Thierry Petit Lobao
• Virginia Rodrigues
• Alveri Sant’Ana
• Clotilzio Santos
• Ednei A. Santulo Jr
• Mazi Shirvani
• Viviane Silva
• Antonio Calixto Souza Filho
• Paula Murgel Veloso
Representation of Algebras
• Edson Ribeiro Alvares
• Ibrahim Assem
• Diane Castonguay
• Claudia Chaio
• Maria Andrea Gatica
• Patrick Le Meur
• Daniel Rivera
• Gordana Todorov
• Sonia Trepode
• Rosana Retsos Signorelli Vargas
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Homological Methods in Algebra
• Eli Aljadeff
• Viktor Bekkert
• Leandro Cagliero
• Matias L. del Hoyo
• Marco Farinati
• Estanislao Herscovich
• Kiyoshi Igusa
• Eduardo N. Marcos
• Gabriel Minian
• Maria Julia Redondo
• Natalia Abad Santos
Hopf Algebras
• Marcelo Muniz S. Alves
• Fernando Araujo Borges
• Alexei Davydov
• Walter Ferrer Santos
• Agustın Garcıa Iglesias
• Barbara Pogorelsky
• Cristian Vay
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Non-Associative Algebra
• Antonio Behn
• Maria de Lourdes Giuliani
• Iryna Kashuba
• Alexandr Kornev
• Plamen Koshlukov
• Alicia Labra
• Luiz Antonio Peresi
• Carlos Rojas
• Ivan Shestakov
• Maria Trushina
• Raul Velasquez
• Luis Alberto Wills-Toro
Lie Theory and Applications in Mathematical Physics
• Carina Boyallian
• Andre Bueno
• Thomas Bunke
• Eduardo Hoefel
• Olivier Mathieu
• Adriano Moura
• Roldao da Rocha
• Juan Tirao
• Francesco Toppan
• Jorge Vargas
• Milen Yakimov
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Finite Fields, Coding Theory and Cryptography
• Ricardo Alfaro
• Alp Bassa
• Herivelto Borges
• Peter Beelen
• Maria Bras-Amoros
• Cicero Carvalho
• Francis Castro
• Luciane Quoos Conte
• Ricardo Dahab
• Sudhir Ghorpade
• Masaaki Homma
• Guillermo Matera
• C. Moreno
• Enric Nart
• Leo Storme
• Horacio Tapia Recillas
• Fernando Torres
• Qiang (Steven) Wang
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Posters
• Jaime Edmundo Apaza Rodriguez
• German Benitez Monsalve
• Paula Andrea Cadavid
• Joyce Caetano
• Yohny Ferney Calderon Henao
• Gladys Chalom
• Mariana Cornelissen
• Jose Antonio O. Freitas
• Marines Guerreiro
• Sandra Mara Alves Jorge
• Tiago Macedo
• Thiago Castilho de Mello
• Agustin Moreno Canadas
• Beatriz Motta
• Joacir Lucas de Oliveira
• Rafael Peixoto
• Fernanda de Andrade Pereira
• Luiz Henrique Pereira
• Anderson L.P. Porto
• Joao Eloir Strapasson
• Marcio Andre Traesel
• Flavia Ferreira Ramos Zapata
• Theo Zapata
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Contents
Self-dual codes over Fq[u]/(ut)Ricardo Alfaro 21
On Moore’s conjecture: overview and some new resultsEli Aljadeff 21
From trisections in module categories to quasi-directed componentsEdson Ribeiro Alvares 21
On partial actions of Hopf algebrasMarcelo Muniz Silva Alves 22
On pointed Hopf algebras with non-abelian groupNicolas Andruskiewitsch 22
Codigos de controle da paridade e codigos de GoppaJaime Edmundo Apaza Rodriguez 23
Spinor class fields for lattices in function fieldsLuis Arenas-Carmona 23
On Cox rings of K3 surfacesMichela Artebani 24
Cluster-tilted algebras without clustersIbrahim Assem 24
Partial actions of ordered groupoids on ringsDirceu Baggio 25
Questions on towers of function fields motivated by applications in multi-party computationand secret sharingAlp Bassa 25
Weight of codewords from evaluation codesPeter Beelen 25
Train algebras which satisfy the additional identity (x2 − ω(x)x)2 = 0Antonio Behn 26
Derived tame local and two-point algebrasViktor Bekkert 26
Almost split sequences for Hopf AlgebrasFernando Araujo Borges 27
The multi-Frobenius non-classical curvesHerivelto Borges 27
On the classification of irreducible modules over finite simple Lie conformal superalgebrasCarina Boyallian 27
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On numerical semigroups and algebraic codesMaria Bras-Amoros 28
Incidence algebrasRosali Brusamarello 29
Free field realizations of the elliptic affine Lie algebrasAndre Bueno 29
Classification of irreducible non-dense modules for A(2)2
Thomas Bunke 30
Interaction of Koszul and Ringel dualitiesPaula Andrea Cadavid Salazar 30
Codigos geometricos de GoppaJoyce dos Santos Caetano 31
On the cohomology ring of truncated quiver algebrasLeandro Cagliero 31
Una introduccion a la teorıa de representaciones de algebrasYohny Ferney Calderon Henao 31
Jacobians with complex multiplicationAngel Carocca 32
Weierstrass semigroups at several points and generalized Hamming weightsCicero Carvalho 32
Freely connected algebraDiane Castonguay 32
Divisibility of the Number of Solutions of Polynomials with Prescribed Leaders MonomialsFrancis Castro 33
Conjugacy separability and commensurabilitySheila Campos Chagas 33
Grados de morfismos irreducibles y tipo de representacion de un algebraClaudia Chaio 33
Idempotents for some cyclic codesGladys Chalom 34
The Weyr form for matrices and various varieties of commuting matricesJohn Clark 34
Abel maps for curves of compact typeJuliana Coelho 34
Further examples of maximal curvesLuciane Quoos Conte 35
Classifying indecomposable RA LoopsMariana Cornelissen 35
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On partial skew Armendariz ringsWagner Cortes 36
TbaSeverino Collier Coutinho 36
Fast finite field arithmetic for cryptographic applicationsRicardo Dahab 36
Modular invariants for group-theoretical modular dataAlexei Davydov 36
TbaJose Antonio de la Pena 37
On the loop space of a 2-categoryMatias L. del Hoyo 37
Invariantes de algebras de grupo parciaisM. Dokuchaev 38
Parametrization of representations of braid groupsClaudia Egea 39
Solvable Lie bialgebrasMarco Farinati 40
Units in U(ZCp)Raul Ferraz 40
Relative invariant theoryWalter Ferrer 40
Partial actions on semiprime rings: GlobalizationMiguel Ferrero 40
Polynomial identities for graded tensor products of algebrasJose Antonio O. Freitas 41
Finite dimensional pointed Hopf algebras over S4
Agustın Garcıa Iglesias 41
Quantum groups and Hopf algebrasGaston Andres Garcıa 42
Representation type and Hochschild cohomology of toupie algebrasMaria Andrea Gatica 42
Grassmann codes and their relativesSudhir R. Ghorpade 43
Graded polynomial identites and exponential growthAntonio Giambruno 43
Right-division in Moufang loopsMaria de Lourdes Merlini Giuliani 43
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p-adic diagonal equationsHemar Godinho 44
Bass cyclic units as factors in a free product in integral group ring unitsJ. Z. Goncalves and Angel del Rio 44
On smooth cubic projective hypersurfacesVictor Gonzalez Aguilera 44
Idempotents in abelian group algebrasMarines Guerreiro 45On left Jordan centralizersClaus Haetinger 45
Representations of Yang-Mills algebrasEstanislao Herscovich 45
On the coalgebra description of OCHAEduardo Hoefel 45
On the number of points of a plane curve over a finite fieldMasaaki Homma 46
Exceptional sequences, braid groups and clustersKiyoshi Igusa 47
Group algebras satisfying certain homological properties - A surveySurender K. Jain 47
Decomposability criterion for coherent sheavesMarcos Jardim 47
On graded central polynomials of the graded algebra M2(E)Sandra Mara Alves Jorge 48
Representation type of the universal enveloping algebras of alternative algebrasIryna Kashuba 48
Homology (of) Hopf algebrasChristian Kassel 48
Bielliptic Weierstrass pointsTakao Kato 49
As algebras de Maltsev livresAlexandr Kornev 49
Graded identities in Lie algebrasPlamen Koshlukov 50
A class of locally nilpotent commutative algebrasAlicia Labra 51
Cox rings of rational surfacesAntonio Laface 52
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Coverings of laura algebrasPatrick Le Meur 52
A generalization of the Jacobson radicalThierry Petit Lobao 53
A generalisation of the dual Kummer surfaceVictor Gonzalo Lopez-Neumann 53
Chevalley groups and hyperalgebrasTiago Macedo 53
2− d-Koszul algebrasEduardo N. Marcos 53
Max Noether theorem for singular curvesRenato Vidal Martins 54
TbaGuillermo Matera 54
On modular unipotent representation of GL(n,K)Olivier Mathieu 54
Compactified Picard stacks over the moduli space of curves with marked pointsMargarida Melo 54
On the center of the relatively free algebra of Ma,b and the ring of generic matricesThiago Castilho de Mello 55
The poset of p-subgroups of a finite groupGabriel Minian 55
The homology of reduced lattices and some combinatorial duality theoremsGabriel Minian 56
Methods of poset representation theory in steganographyAgustın Moreno Canadas 57
TbaC. Moreno 57
Plane arcs from plane curvesBeatriz Motta 58
Tensor products, characters, and blocks of finite-dimensional representations of quantumaffine algebras at roots of unityAdriano Moura 58
Genus 3 curves: a world to exploreEnric Nart 59
Etale homotopy types of moduli stacksFrank Neumann 59
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On a 4-gonal curve of genus 9Akira Ohbuchi 59
Leavitt path algebrasJoacir Lucas de Oliveira 60
On commutativity and finiteness in groupsRicardo Nunes de Oliveira 61
Low complexity normal bases in finite fieldsDaniel Panario 61
Near orders on higher dimensional varietiesRafael Peixoto 62
The Kirillov-Reshetikhin modules associated to E6
Fernanda de Andrade Pereira 62The Gerstenhaber bracket in the Hamiltonian formalismLuiz Henrique Pereira 63
Polynomial identities of ternary algebrasLuiz A. Peresi 63
Right coideal subalgebras in the quantum Borel algebra of type G2
Barbara Pogorelsky 63
Lie nilpotence of skew symmetric elements in group ringsCesar Polcino Milies 64
p-adic representation of cyclic groups and pro-p groupsAnderson L.P. Porto 64
Hereditary abelian categoriesMaria Julia Redondo 64
The endomophisms monoid of a homogeneous vector bundleAlvaro Rittatore 64
Serre relations for Lie algebras associated to quasi-Cartan matrix of type Bn and CnAntonio Daniel Rivera 65
On non-abelian tensor powers of a groupNoraı R. Rocco 65
Hecke algebras and quantum Clifford algebrasRoldao da Rocha 65
Covering coalgebras and dual non-singularityVirginia Rodrigues 66
Prym-Tyurin varieties via Hecke algebrasAnita Rojas 66
Trace forms and ideals on commutative algebras satisfying an identity of degree fourCarlos Rojas-Bruna 66
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Subgroups embedded in R. Thompson’s group VOlga Patricia Salazar-Dıaz 67
On the rank of the fibres of rational elliptic surfacesCecılia Salgado 67
Galois correspondence for α-partial Galois Azumaya extensionsAlveri A. Sant’Ana 67
Somas de quadrados e rormas hermitianasClotilzio Moreira dos Santos 68
Homologıa de Hochschild y homologıa cıclica de algebras de dimension finitaNatalia Abad Santos 68
Classification of involutions on incidence algebrasEdnei A. Santulo Jr. 69
Cluster algebras and cluster categoriesRalf Schiffler 69
On Kac Wakimoto conjecture about dimension of simple representation of Lie superalgebraVera Serganova 69
Construcao de algebras primas degeneradas atraves de algebra de GrassmannIvan Shestakov 69
Free unitary and symmetric pairs in group rings of characteristic 2Mazi Shirvani 70
State-closed groupsSaid Sidki 70
On Z2-graded identities and central polynomials of the Grassmann algebraViviane Ribeiro Tomaz da Silva 70
The hyperbolic propertyA. C. Souza Filho 71
Galois geometries and coding theory: two interacting research areasLeo Storme 71
Projection lattices pointsJoao Eloir Strapasson 72
On the quasi-cyclicity and linearity of the Gray image of a code over a Galois ringHoracio Tapia-Recillas 72
An introduction to central simple algebras and the Brauer groupEduardo Tengan 72
The algebra of differential operators associated to a weight matrixJuan Tirao 73
TbaGordana Todorov 73
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On the representations of the 1D N-extended SuperalgebraFrancesco Toppan 73
Castle curves and codesFernando Torres 73
Generalized octonionic structures on S7 and Clifford algebrasMarcio Andre Traesel 74
On the representation dimension of some classes of tame algebrasSonia Trepode 74
Irreducible bimodules over alternative superalgebrasMaria Trushina 74
Multiplicity formulae for admissible restriction of discrete series representationsJorge Vargas 75
Some algebras with directed gluingsRosana Retsos Signorelli Vargas 75
Hopf algebra of dimension 16Cristian Vay 76
About quasi-Jordan algebras generated by dialgebras and their relation with Leibniz algebrasRaul Velasquez 76
Free subgroups of U(ZG) generated by alternating unitsPaula Murgel Veloso 77
Deformations of restricted simple Lie algebras in positive characteristicFilippo Viviani 77
On a group of permutation polynomialsQiang (Steven) Wang 78
Some classification results on nonassociative graded algebrasLuis Alberto Wills-Toro 78
Spectra of quantum Schubert cells and quantum flag varietiesMilen Yakimov 78
Genus for groupsPavel Zalesskii 78
Grupos agindo sobre arvores e automatosFlavia Ferreira Ramos Zapata 79
Pro-finite limit groupsTheo Zapata 79
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Abstracts
Self-dual codes over Fq[u]/(ut)Ricardo AlfaroUniversity of Michigan-Flint, USA
Two characterizations of self-dual codes over Fq[u]/(ut) are determined in terms of linearcodes over Fq. An algorithm to produce such self-dual codes is also established.
On Moore’s conjecture: overview and some new resultsEli AljadeffTechnion-Israel Institute of Technology, Israel
A theorem of J.P. Serre (1969) says that if Γ is a torsion free group and H is a subgroup offinite index then they have the same cohomological dimension.
In 1976 J. Moore posed a conjecture which is a far reaching generalization of Serre’s theorem:If Γ is torsion free and H is a subgroup of finite index then the projective dimension of anyZΓ-module M coincides with the projective dimension of M as a module over ZH. In particular(and in fact equivalently) any ZΓ-module M which is projective over ZH is projective also overZΓ.
The conjecture is known for large families of groups, as solvable groups, linear groups andsome special groups (as the Thompson’s group). In the lecture I will present some old results(joint work with Cornick, Ginosar and Kropholler) together with some recent results (joint workwith Udi Meir).
From trisections in module categories to quasi-directed componentsEdson Ribeiro AlvaresUFPR, Brazil
Joint with Assem, I., Coelho, F.U., Pena, M.I, Trepode, S.We define a special type of trisections in a module category, namely the compact trisections
which characterise quasi-directed components. We apply this notion to the study of laura alge-bras and we use it to define a class of algebras with predictable Auslander-Reiten components.
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On partial actions of Hopf algebrasMarcelo Muniz Silva AlvesUFPR, Brazil
Partial group actions on algebras were first studied in the context of operator algebras,but soon enough became an independent topic of interest in ring theory. Partial actions ofHopf algebras were introduced as a natural generalization of partial actions of groups (whichcorrespond, in a certain way, to partial actions of the Hopf algebra kG). In this talk it willbe shown that important results regarding partial group actions also hold in the Hopf case, asthe existence of an enveloping H-module algebra B for a partial H-module algebra A, and of astrict Morita context between the partial smash product A#H and B#H. Results concerningpartial coactions of a Hopf algebra H and will also be presented, as well as some results on theextension AH ⊂ A, where AH is the subalgebra of (partial) invariants of A.
On pointed Hopf algebras with non-abelian groupNicolas AndruskiewitschUniversidad Nacional de Cordoba, Argentina
I will survey recent developments on the classification of finite-dimensional pointed Hopfalgebras with non-abelian group. Concretely, I will describe two main topics:
i) How to attach a generalized root system, or equivalently a Weyl groupoid, to a completelyreducible Yetter-Drinfeld module W . Reference: [AHS]. This is done by looking at the Nicholsalgebra B(W ); I will try to avoid technicalities and concentrate on some applications.
ii) How to show that for a given finite non-abelian group G, there are no finite-dimensionalpointed Hopf algebras with group G. So far, we know that this happens for the alternatinggroups An, n > 6 or n = 5; for some sporadic groups (some of the Mathieu, some of the Janko,the Suzuki group and the Held group); and for a few finite groups of Lie type. References: [AF,AFZ, AFGV1, AFGV2, F, FGV]. This is done by a variety of techniques concerning specialsubracks of the different conjugacy classes of G.
References
[AF] N. Andruskiewitsch and F. Fantino. New techniques for pointed Hopf algebras. arXiv:0803.3486v1<http://arxiv.org/abs/0803.3486>. 29 pages. To appear in Proceedings of the Sixth Workshop in LieTheory and Geometry, Contemp. Math.
[AFGV1] N. Andruskiewitsch, F. Fantino, M. Grana and L. Vendramin, Finite-dimensional pointed Hopf algebraswith alternating groups are trivial, submitted.
[AFGV2] N. Andruskiewitsch, F. Fantino, M. Grana and L. Vendramin, On pointed Hopf algebras associated tosporadic groups, in preparation.
[AFZ] N. Andruskiewitsch, F. Fantino and S. Zhang. On pointed Hopf algebras associated with the symmetricgroups. arXiv:0807.2406v1 <http://arxiv.org/abs/0807.2406>. 14 pages. Manuscripta Math., accepted.
[AHS] N. Andruskiewitsch, I. Heckenberger and H.-J. Schneider, The Nichols algebra of a semisimple Yetter-Drinfeld module, arXiv:0803.2430v1. Amer. J. Math., to appear.
[F] F. Fantino, On pointed Hopf algebras associated with Mathieu groups, arXiv: 0711.3142v2 [math.QA].[FGV] S. Freyre, M. Grana and L. Vendramin, On Nichols algebras over GL(2, Fq) and SL(2, Fq),
J. Math. Phys. 48 (2007), 123513-1 – 123513-11.
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Codigos de controle da paridade e codigos de GoppaJaime Edmundo Apaza RodriguezUNESP, Brazil
Os codigos controle da paridade surgiram no seculo passado como exemplo de uma famılia decodigos detectores de um erro simples. Seu nome provem de um codigo binario que acrescentavaum sımbolo extra para que o numero de uns fosse par. Inicialmente, os codigos detectores ecorretores de erros foram criados usando unicamente conceitos de algebra e teoria dos numeros.Posteriormente, em 1977, V. D. Goppa introduziu uma nova forma de construir codigos linearesusando curvas algebricas definidas sobre corpos finitos. Esses codigos sao conhecidos hoje comoos codigos geometricos de Goppa. Neste trabalho usamos algumas das ideias de Goppa paraconstruir a classe de codigos de controle da paridade e exibimos alguns exemplos. Para issousaremos a tecnica de restricao de um codigo linear, concretamente, a restricao de um codigode Goppa racional.
Spinor class fields for lattices in function fieldsLuis Arenas-CarmonaUniversidad de Chile, Chile
The theory of spinor class fields allows the study of the set of maximal orders in a centralsimple algebra over a number field, or the set of quadratic lattices that are isometric at everycompletion of the number field. We extend this theory to the function field case through theuse os schemes to study how the spinor class field depends on the choice of an affine subset ofa projective curve.
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On Cox rings of K3 surfacesMichela ArtebaniUniversidad de Concepcion, Chile
Let X be a smooth projective surface X over C with finitely generated Picard group Pic(X).The Cox ring of X is the graded ring
R(X) =⊕
D∈Pic(X)
H0(X,OX(D)).
It is known that R(X) is a polynomial ring if and only if X is a toric surface [2] and an explicitdescription of the Cox ring is available for Del Pezzo surfaces [3]. In general, it is even difficultto decide if the Cox ring of a surface is finitely generated.
In [1] we investigate Cox rings of K3 surfaces i.e. simply connected compact complex surfaceswith trivial canonical bundle. We prove that the Cox ring of a K3 surface is finitely generatedif and only if its effective cone is polyhedral. Moreover, we compute the Cox ring of some K3surfaces which either have Picard number two or are double covers of rational surfaces. Anexample of both types is the generic K3 surface with Picard lattice isometric to
U(2) =(
0 22 0
).
In this case the Z2-graded Cox ring is:
R(X) ∼=C[a1, a2, b1, b2, c](c2 − f4,4(a, b))
,
where deg(ai) = (1, 0), deg(bi) = (0, 1) and deg(c) = (2, 2).
References
[1] M. Artebani, J. Hausen, A. Laface, On Cox rings of K3-surfaces, arXiv:0901.0369v2.[2] D. Cox, The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4(1), 17–50, (1995).[3] A. Laface, M. Velasco, Picard-graded Betti numbers and the defining ideals of Cox rings, (2007).
Cluster-tilted algebras without clustersIbrahim AssemUniversite de Sherbrooke, Canada
This is a report on a work with progress with Thomas Brustle and Ralf Schiffler. We give analgorithm allowing to start from a tilted algebra and construct the transjective component of thecorresponding cluster-tilted. In the Dynkin case, this yields the whole Auslander-Reiten quiver.We also introduce a notion of reflection allowing to obtain all tilted algebras corresponding toa given cluster-tilted.
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Partial actions of ordered groupoids on ringsDirceu BaggioUFSM, Brazil
In this joint work with A. Paques and D. Flores we introduce the notion of a partial action ofan ordered groupoid on a ring and we construct the corresponding partial skew groupoid ring.We present sufficient conditions under which the partial skew groupoid ring is either associativeor unital. Also, we show that there is a one-to-one correspondence between partial actions ofan ordered groupoid G on a ring R, in which the domain of each partial bijection is an ideal,and meet-preserving global actions of the Birget-Rhodes expansion GBR of G on R. Using thiscorrespondence we prove that the partial skew groupoid ring is a homomorphic image of theskew groupoid ring constructed through the Birget-Rhodes expansion.
Questions on towers of function fields motivated by applications in multi-party computationand secret sharingAlp BassaCWI & Leiden University, The Netherlands
In a series of papers, Cascudo, Chen, Cramer and Xing have shown how towers of algebraicfunction fields find applications in the construction of “multi-party computation friendly” secretsharing schemes. In this talk, I will try to outline this connection and discuss some of thequestions about towers, which emerge naturally from it.
Weight of codewords from evaluation codesPeter BeelenTechnical University of Denmark, Denmark
Let F be a finite field and R a finite dimensional F -algebra. Further let Ev : R → Fn
be an isomorphism of F -algebras. Error-correcting codes can be obtained from this setup byevaluating all elements in a linear subspace L of R. Many interesting codes can be describedin this way, such as algebraic geometry codes, generalized Reed-Muller codes and toric codes.In this talk we will investigate the weight of codewords from codes obtained in this way bystudying certain linear operators on R. Also we will show the connection with several otherknown techniques to estimate the weight of a codeword.
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Train algebras which satisfy the additional identity (x2 − ω(x)x)2 = 0Antonio BehnUniversidad de Chile, Chile
(Joint work with Irvin Roy Hentzel, Department of Mathematics, Iowa State University.)In the following we assume that A is a commutative algebra over a field F of characteristic
not 2. The principal powers of x ∈ A are defined as follows x1 = x and xn+1 = xnx ∀ n ≥ 1.The plenary powers of an element x ∈ A are defined as x[1] = x and x[n+1] = x[n]x[n] ∀ n ≥ 1.We observe that x[2] = x2.
If A has a non-zero algebra homomorphism ω : A→ F , then the pair (A,ω) is called a baricalgebra. A baric algebra (A,ω) is called a plenary train algebra, if there are scalars γ1, . . . , γr−1
such that every x ∈ A satisfies the following plenary identity of degree 2r−1:
(1) x[r] + γ1ω(x)n1x[r−1] + γ2ω(x)n2x[r−2] · · ·+ γr−1ω(x)nr−1x = 0
where r ≥ 2 and ni = 2r−i−1(2i − 1) for all i ∈ 1, . . . , r − 1.A baric algebra (A,w) is called a principal train algebra if there exist γ1, · · · , γt−1 ∈ K such
that every element x ∈ A satisfies the equation
(2) xn + γ1ω(x)xn−1 + · · ·+ γn−1ω(x)n−1x = 0
In recent work with Prof. I.R. Hentzel we consider plenary train algebras of arbitrary rank.Given the identity 1
we consider the roots λ1, λ2, . . . , λr of the associative polynomial xr + γ1xr−1 + γ2x
r−2 · · ·+γr−1x. A sufficient condition for the algebra to satisfy the additional identity (x2−w(x)x)2 = 0is that the pairwise products of these roots are all distinct, i.e. λiλj 6= λlλk ∀(i, j) 6= (l, k).
We also show that a train algebra with train identity (1) which satisfies the additional identity(x2 − w(x)x)2 = 0 has an idempotent as long as
r−1∑i=1
(r − i)γi 6= 0.
Another interesting fact is that if A is a commutative algebra which satisfies (x2−w(x)x)2 = 0,then A is principal train if and only if it is plenary train.
If x ∈ A is such that ω(x) = 1, then the subalgebra of A generated by x has a uniqueidempotent, and we can fully characterize its structure.
References
[1] I. M. H. Etherington. Commutative train algebras of ranks 2 and 3. J. London Math. Soc., 15:136–149, 1940.[2] J. Carlos Gutierrez Fernandez. Principal and plenary train algebras. Comm. Algebra, 28(2):653–667, 2000.[3] Alicia Labra and Avelino Suazo. On plenary train algebras of rank 4. Comm. Algebra, 35(9):2744–2752, 2007.
Derived tame local and two-point algebrasViktor BekkertUFMG, Brazil
We determine representation type of the bounded derived category of finitely generated mod-ules over finitely generated complete local and two-point algebras.
This is a joint work with Yuriy Drozd (Institute of Mathematics, Ukraine) and VyacheslavFutorny (Universidade de Sao Paulo, Brazil).
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Almost split sequences for Hopf AlgebrasFernando Araujo BorgesUFPR, Brazil
In this poster we will present results regarding almost-split sequences in the context of groupalgebras kG (following Auslander and Carlson), such as the behaviour of almost-split sequencesunder tensoring by indecomposable modules. We will also present extensions of some of theseresults for Hopf algebras (finite-dimensional, with involutive antipode); for example, if M is aindecomposable H-module, then the trivial module k is a summand of Endk(M) iff tensoring byM the almost-split sequence that ends at the trivial module k gives a non-split sequence. Fol-lowing work of Green, Marcos and Solberg, we present sufficient conditions for these equivalentproperties to hold.
The multi-Frobenius non-classical curvesHerivelto BorgesThe University of Texas at Austin, USA
An irreducible curve F defined over Fq is called q-Frobenius non-classical if the image Fr(P )of each simple point P of F under the Frobenius map lies on the tangent line at P .
Based on [2], Hefez and Voloch extended the study of the q-Frobenius non-classical curves in[1], where some interesting arithmetic and geometric properties of such curves were first pointedout.
In this talk, I will present and characterize all irreducible plane curves defined over Fq whichare simultaneously Frobenius non-classical for different powers of q. Such characterization givesrise to many previously unknown curves which turn out to have some interesting properties.For instance, for n ≥ 3 a plane curve which is both q- and qn-Frobenius non-classical will haveits number of Fqn-rational points attaining the Stohr-Voloch bound.
References
[1] A. Hefez and J.F. Voloch, Frobenius non classical curves, Arch. Math. 54, (1990) 263–273.[2] Stohr, K-O. and Voloch, J.F., Weierstrass Points and Curves over Finite Fields, Proc. London Math. Soc.(3)
52 (1986)1–19.
On the classification of irreducible modules over finite simple Lie conformal superalgebrasCarina BoyallianUniversidad Nacional de Cordoba, Argentina
We construct all finite irreducilble modules over Lie conformal superalgebras of type W , Sand K. We also give a complete description of them in terms of differential forms.
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On numerical semigroups and algebraic codesMaria Bras-AmorosUniversitat Rovira i Virgili, Spain
Given a rational point P of a curve with Weierstrass semigroup Λ = λ0 < λ1 < λ2 < . . . one can find an infinite basis z0, z1, z2, . . . of the ring of functions having poles only at P suchthat vP (zi) = −λi. Consider a set of rational points P1, . . . , Pn different from P . To each finitesubset W ⊆ N0 we associate the one-point code CW =< (zi(P1), . . . , zi(Pn)) : i ∈ W >⊥. Byextension, we say that W is the set of parity checks of CW .
The ν sequence of Λ is defined by νi = #j ∈ N0 : λi − λj ∈ Λ. The minimum set of paritychecks that are needed to correct t errors is given by R(t) = i ∈ N0 : νi < 2t+ 1 [10, 12]. Thecodes determined by R(t) are called Feng–Rao improved codes. The ν sequence is also used todefine the order bound on the minimum distance for one-point codes [9, 13, 12].
We say that the points Pi1 , . . . , Pit (Pij 6= P ) are generically distributed if no function gen-erated by z0, . . . , zt−1 vanishes in all of them. Generic errors are those errors whose non-zeropositions correspond to generically distributed points. Generic errors of weight t can be a verylarge portion of all possible errors of weight t [11]. By restricting the errors to be corrected togeneric errors the decoding requirements become weaker and we are still able to correct almostall errors. The minimum set of parity checks that are needed to correct generic errors of weightt is R∗(t) = i ∈ N0 : λi 6= λj + λk for any j, k ≥ t [7].
We can define a new sequence τ [5] that allows us to describe R∗(t) in terms of τ in asimilar way as how R(t) is described in terms of ν. Then, by studying the increasingness ofτ we can compare the codes determined by R∗(t) with standard codes and by studying therelation between the sequences ν and τ we can compare the codes determined by R∗(t) withthe Feng–Rao improved codes.
We will present some results related to classification [2, 15, 16] and characterization [2, 5]of numerical semigroups from the perspective of algebraic codes and their decoding and someresults and open questions related to counting of numerical semigroups [14, 3, 4, 6, 8, 1].
References
[1] Victor Blanco and Justo Puerto. Computing the number of numerical semigroups using generating functions.arXiv:0901.1228v1.
[2] Maria Bras-Amoros. Acute semigroups, the order bound on the minimum distance, and the Feng-Raoimprovements. IEEE Trans. Inform. Theory, 50(6):1282–1289, 2004.
[3] Maria Bras-Amoros. Fibonacci-like behavior of the number of numerical semigroups of a given genus. Semi-group Forum, 76(2):379–384, 2008.
[4] Maria Bras-Amoros. Bounds on the number of numerical semigroups of a given genus. J. Pure Appl. Algebra,213(6):997–1001, 2009.
[5] Maria Bras-Amoros. On numerical semigroups and the redundancy of improved codes correcting genericerrors. Designs, Codes and Cryptography, Accepted. 2009.
[6] Maria Bras-Amoros and Stanislav Bulygin. Towards a better understanding of the semigroup tree. SemigroupForum, Accepted. 2009.
[7] Maria Bras-Amoros and Michael E. O’Sullivan. The correction capability of the Berlekamp-Massey-Sakataalgorithm with majority voting. Appl. Algebra Engrg. Comm. Comput., 17(5):315–335, 2006.
[8] Sergi Elizalde. Improved bounds on the number of numerical semigroups of a given genus. arXiv:0905.0489v1.[9] Gui Liang Feng and T. R. N. Rao. A simple approach for construction of algebraic-geometric codes from
affine plane curves. IEEE Trans. Inform. Theory, 40(4):1003–1012, 1994.[10] Gui-Liang Feng and T. R. N. Rao. Improved geometric Goppa codes. I. Basic theory. IEEE Trans. Inform.
Theory, 41(6, part 1):1678–1693, 1995. Special issue on algebraic geometry codes.[11] Johan P. Hansen. Dependent rational points on curves over finite fields—Lefschetz theorems and exponential
sums. In International Workshop on Coding and Cryptography (Paris, 2001), volume 6 of Electron. NotesDiscrete Math., page 13 pp. (electronic). Elsevier, Amsterdam, 2001.
[12] Tom Høholdt, Jacobus H. van Lint, and Ruud Pellikaan. Algebraic geometry of codes. In Handbook of codingtheory, Vol. I, II, pages 871–961. North-Holland, Amsterdam, 1998.
[13] Christoph Kirfel and Ruud Pellikaan. The minimum distance of codes in an array coming from telescopicsemigroups. IEEE Trans. Inform. Theory, 41(6, part 1):1720–1732, 1995. Special issue on algebraic geometrycodes.
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[14] Jiryo Komeda. Non-Weierstrass numerical semigroups. Semigroup Forum, 57(2):157–185, 1998.[15] Carlos Munuera and Fernando Torres. A note on the order bound on the minimum distance of AG codes
and acute semigroups. Adv. Math. Commun., 2(2):175–181, 2008.[16] Anna Oneto and Grazia Tamone. On numerical semigroups and the order bound. J. Pure Appl. Algebra,
212(10):2271–2283, 2008.
Incidence algebrasRosali BrusamarelloUEM, Brazil
Let X = x1, x2, . . . , xn be a finite partially ordered set (poset) and K a field. The incidencealgebra I(X,K) is the K-algebra which has basis elements labelled Eij for each i, j for whichxi ≤ xj and has multiplication defined via
EijEkl =
Eil if j = k
0 if j 6= k
(i.e., the K-vector space with basis Eij with xi ≤ xj and the multiplication defined as above).Since X is finite, we can always label the elements of X in such way that xi ≤ xj implies i ≤ j,then I(X,K) can be identify with a subalgebra of the upper triangular matrices UTn(K).
These algebras were introduced in the 1960s by G.-C. Rota and R.P. Stanley, two famouscombinatorialists. From then on there have been many articles on these kinds of algebras,sometimes called structural matrix algebras, and the results are somewhat scattered in theliterature.
In this talk we will collect some of this results in order to answer the following naturalquestions.
(I) To what extent does I(X,K) determines the set X?(II) When does Z(I(X,K)) = K? That is, when is I(X,K) a central K-algebra?(III) Can we describe the group Aut(I(X,K))? Are there cases when every automorphism
of I(X,K) is inner?(IV) When does I(X,K) admit anti-automorphisms and involutions?
We will also present some new material on involutions. This talk is based on a joint workwith Prof. David W. Lewis.
Free field realizations of the elliptic affine Lie algebrasAndre Bueno
Let E be an affine complex elliptic curve with two points removed. Here, the elliptic affineLie algebra associated with E is the (three-dimensional) universal central extension of sl(2, R),where R is the ring of global sections of E. In this talk we show how to construct two free fieldrealizations of the elliptic affine Lie algebra of E. The first realization provides an analogue ofWakimoto’s construction for Affine Kac-Moody algebras, but in the setting of the elliptic affineLie algebra. The second one gives a new type of representations analogous to Imaginary Vermamodules in the affine setting. This is a joint work with B. Cox and V. Futorny.
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Classification of irreducible non-dense modules for A(2)2
Thomas BunkeIME-USP, Brazil
Let g be an affine Kac-Moody algebra with Cartan subalgebra h, root system ∆ and center Cc.A g-module V is called a weight if V =
⊕λ∈h∗ Vλ, Vλ = v ∈ V | hv = λ (h) v for all h ∈ h∗.
If V is an irreducible weight g-module then c acts on V as a scalar, called level of V . For aweight g-module V , the support is the set supp (V ) = λ ∈ h∗ | Vλ 6= 0. The root lattice Q isthe free abelian group over ∆. If V is irreducible then supp (V ) ⊂ λ + Q for some λ ∈ h∗. Anirreducible weight g-module V is called non-dense, if supp (V ) ( λ+Q,
This work contains the classification of irreducible non-dense modules for the Kac-Moodyalgebra A
(2)2 with at least one finite-dimensional weight subspace. The classification for non-
dense irreducible A(1)1 -modules with a finite-dimensional weight subspace has been done by V.
Futorny [1]. The classification problem is also solved for all affine Kac-Moody algebras fornon-zero level modules with all finite-dimensional weight subspaces (V. Futorny and A. Tsylke[2]). In these cases an irreducible module is either a quotient of a classical Verma module, orof a generalized Verma module, or of a loop module (induced from a Heisenberg subalgebra).That this will hold for all irreducible non-dense modules of affine Kac-Moody algebras hasbeen conjectured by V. Futorny [6]. With this work we confirm the conjecture for non-denseirreducible A(2)
2 -modules with a finite-dimensional weight subspace.We also obtain a classification of all possible supports for irreducible A(2)
2 -modules. The proofis elementary and involves only the combinatorics of the root system.
References
[1] Futorny, V., Irreducible non-dense A(1)1 -modules, Pacific J. of Math. Vol. 172, No. 1, 83-97, 1996
[2] Futorny, V., Tsylke, .A., Irreducible non-zero level modules with finite- dimensional weight spaces for AffineLie algebras, J. of Algebra 238, 426-441, 2001
[6] Futorny, V., Representations of Affine Lie algebras, Queen’s Papers in Pure and Applied Mathematics 106,v. 1, Kingston, 1997
[7] Chari, V., Pressley, A., Integrable Representations of Twisted Affine Lie Algebras, J. of Algebra 113, 438-46,1986
Interaction of Koszul and Ringel dualitiesPaula Andrea Cadavid SalazarIME-USP, Brazil
Joint work with Eduardo do Nascimento Marcos.In the theory of quasi-hereditary algebras there are two classical dualities: the Ringel duality,
associated with the characteristic tilting module, and the Koszul duality, associated with thecategory of linear complexes of projective modules. In [1, 2] it is shown that a certain classof Koszul quasi-hereditary algebras is stable with respect to taking of both Koszul and Ringelduals and that on this class of algebras the Koszul and Ringel dualities commute. In this posterwe present and discuss these results.
References
[1] I. Agoston, V. Dlab, E. Lukas, Quasi-hereditary extension algebras. Algebras and Representation Theory 6(2003), no. 1, 97-117.
[2] V. Mazorchuck, S. Ovsienko, A pairing in homology and the category of linear complexes of tilting modulesfor a quasi-hereditary algebra. With an appendinx by Catharina Stroppel. J. Math Kyoto Univ. 45 (2005),no. 4, 711-741.
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Codigos geometricos de GoppaJoyce dos Santos CaetanoUFABC, Brazil
Este trabalho tem o apoio financeiro da Fundacao Universidade Federal do ABC.A teoria dos codigos e hoje uma area com uma enorme atividade de pesquisa, juntando es-
forcos de matematicos e engenheiros. Seus aspectos teoricos tem motivado ainda mais as jaintensas investigacoes sobre corpos de funcoes em uma variavel (ou equivalentemente, inves-tigacoes sobre curvas algebricas) definidos sobre corpos finitos. Historicamente, os codigos deGoppa foram introduzidos por meio de relacoes polinomiais, onde V. D. Goppa abriu cam-inho para uma construcao mais geral dos codigos algebricos-geometricos, nosso objetivo seraapresentar a construcao de tais codigos sobre curvas algebricas definidas sobre um corpo finito.
On the cohomology ring of truncated quiver algebrasLeandro CaglieroFaMAF - CONICET, Argentina
By results of Cibils and Locateli it is well known that the even cohomology ring of truncatedquiver algebras over a field of characteristic zero (TQA’s) is described by certain special classof parallel paths called medals.
It is also known that, for TQA’s, the cup product of odd-degree cohomology classes is zero.Therefore the medals play a central role in the study of the structure of the cohomology ringof a TQA.
In this talk we present a joint work with Paulo Tirao in which we give a complete descriptionof the set of medals of a given quiver. This description allows us to determine the structureof the cohomology ring for a large family of TQA’s whose cup product is not zero. We thinkthat we will be able to obtain a condition on the quiver Q and on the integer N in order todetermine whether the TQA associated to Q and N has a non-trivial cohomology ring.
Una introduccion a la teorıa de representaciones de algebrasYohny Ferney Calderon HenaoUniversidad de Antioquia, Colombia
En este poster se pretende describir dos de las tecnicas que se volvieron esenciales a lo largode los ultimos anos en el estudio de la teorıa de representaciones de algebras, las cuales son:
Por un lado, los metodos diagramaticos usados por Gabriel [ ver 1] y por otro lado, lassecuencias que casi se dividen y morfismos irreducibles dadas por Auslander-Reiten en conjuntocon la aljaba de Auslander-Raiten introducida por Ringel [ver 2].
Bibliografıa.[1] Gabriel, P. Unzerlegbare Dartellungen I e II, Manuscripta Math. 6 (1972) 71-103; Symposia
Math. Inst. Naz. ALta Mat. 11 (1973) 81- 104.[2] Reiten, I. An introduction to the representation of artin algebras, Bull London Math. Soc. 17
(1985) 209-223.
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Jacobians with complex multiplicationAngel CaroccaPUC-Chile, Chile
Joint work with Herbert Lange and Rubı E. Rodrıguez.We construct and study two series of curves whose Jacobians admit complex multiplication.
The curves arise as quotients of Galois coverings of the projective line with Galois group meta-cyclic groups Gq,3 of order 3q with q ≡ 1 mod 3 an odd prime, and Gm of order 2m+1. Thecomplex multiplications arise as quotients of double coset algebras of the Galois groups of thesecoverings. We work out the CM-types and show that the Jacobians are simple abelian varieties.
Weierstrass semigroups at several points and generalized Hamming weightsCicero CarvalhoUFU, Brazil
The Weierstrass semigroup at several points is a natural generalization of the well-knownWeierstrass semigroup at a point. Its systematic study started with papers by Homma andKim in mid 90’s, and since then it has been studied by many authors. In this talk we intendto present several results on this semigroup and also some results from an ongoing work on anapplication of this semigroup to the construction of Goppa codes which have lower bounds fortheir generalized Hamming weights better than the usual bound.
Freely connected algebraDiane CastonguayUFG, Brazil
In this talk, we consider A to be a triangular algebra. To each bound quiver (Q, I) of A,one can define an homotopy relation which lead to the definition of the fundamental group ofthe bound quiver. This group also can be related to a Galois covering of the bound quiver andthus of the algebra. In this talk, we present the quiver of homotopy of an algebra as defined byP. Le Meur in [M06] together with some of the results of this work. Using this technique, weshow that a monomial algebra (that is an algebra who admits a bound quiver where the ideal isgenerated by paths) without double arrows is freely connected. An algebra is freely connected ifall its associated fundamental groups are free groups. Some known examples of freely connectedalgebras are the simply connected algebras and the finite representation type algebra.
[M06] Le Meur, P., Revetements galoisiens et groupe fondamental d’algebres de dimensionfinie, thesis de doutorado, Universite Montpellier II - Sciences et Techniques du Languedoc -(2006-02-10), Claude Cibils (Dir.)
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Divisibility of the Number of Solutions of Polynomials with Prescribed Leaders MonomialsFrancis CastroUniversity of Puerto Rico, Puerto Rico
In this presentation we consider the following problem over finite fields: To compute theexact divisibility of exponential sums of the type
S =∑
x1,...,xn∈Fp
ψ(x1 · · ·xd + xd1 + · · ·+ xdd +G(x1, . . . , xd)),
where ψ is an additive character and the degree of G is less than d. Also we compute the exactdivisibility of the number of solutions of the polynomial equation
X1 · · ·Xd +Xd1 + · · ·+Xd
d +G(X1, . . . , Xd) = 0,
over Fp, where the degree ofG is less than d. This is a jointly work with I. Rubio and R. Figueroa.
Conjugacy separability and commensurabilitySheila Campos ChagasUFAM, Brazil
A group G is called conjugacy separable if whenever x and y are non-conjugate elements of G,there exists some finite quotient of G in which the images of x and y are non-conjugate. We shalldiscuss criterions when conjugacy separability is preserved by commensurability. Several results(obtained jointly with Pavel Zalesskii) when it holds and examples of groups not satisfying thisproperty will be presented.
Grados de morfismos irreducibles y tipo de representacion de un algebraClaudia ChaioUniversidad Nacional de Mar del Plata, Argentina
El objetivo de esta charla es mostrar como el grado de ciertos morfismos irreducibles puededarnos informacion sobre el tipo de representacion de un algebra. La nocion de grado de unmorfismo irreducible fue introducido por S. Liu en 1992. En este trabajo consideraremos algebrasde dimension finita sobre un cuerpo algebraicamente cerrado. Daremos condiciones necesariasy suficientes para que el grado de un morfismo irreducible sea finito. Como aplicacion de estosresultados, obtendremos una caracterizacion de cuando un algebra es de tipo de representacionfinito.
Trabajo conjunto con P. Le Meur y S. Trepode.
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Idempotents for some cyclic codesGladys ChalomIME-USP, Brazil
Explicit expressions for idempotents of minimal codes of the ring FlCpnq ( where l, p, and qare odd primes with some additional hypothesis) were obtained by Bakshi and Raka in (*) interms of generating polynomials of the corresponding ideals. In this work (that is a joint workwith Ferraz, Guerreiro and Polcino) we present this idempotents (and some code parameters)in terms of the generators of the cyclic groups, instead.
(*) Minimal Cyclic Codes of length pnq Gurmeet K. Bashi and Madhu Raka Finite Fieldsand theirs Aplications 9 (2003) 432-448.
The Weyr form for matrices and various varieties of commuting matricesJohn ClarkUniversity of Otago, New Zealand
The Weyr form is little known but a useful alternative to the Jordan canonical form for matri-ces. In this talk we describe the Weyr form and indicate how it is useful in problems concerningapproximately simultaneously diagonalisable matrices, using algebraic geometry techniques.
Abel maps for curves of compact typeJuliana CoelhoUFF, Brazil
Fix a stable curve C of compact type. For each integer d ≥ 1 we contruct two degree-dAbel maps for C having different target spaces and we compare the fibers of both maps. Asan application we get an characterization of stable hypereliptic curves of compact type havingonly two components.
This is a joint work with Marco Pacini (UFF).
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Further examples of maximal curvesLuciane Quoos ConteUFRJ, Brazil
(Joint work with Miriam Abdon and Juscelino Bezerra.)We show that the non-singular model of the curve X (q;n) defined over Fq2n , with n ≥ 3 an
odd integer, by the affine equation:
yq2 − y = x
qn+1q+1 ,
is maximal. These curves have an interesting feature: the curve X (3; 3) is not Galois-covered bythe Hermitian curve, in contrast to the curve X (2;n) which is Galois-covered by the Hermitiancurve ([1], [2], [3]). We also explore the family of maximal curves Xb given over Fq2n by theaffine equation:
yN = −xb(x+ 1), 1 ≤ b ≤ N − 1,where N is an odd divisor of qn + 1, and gcd(N ; b) = gcd(N ; b+ 1) = 1. Explicit isomorphismsbetween them are found.
References:[1] Abdon, M., Bezerra, J., Quoos, L.: Further examples of maximal curves. Journal of Pure
and Applied Algebra, v. 213, pp. 1192-1196, 2009.[2] Garcia, A., Torres, F.: On unramified coverings of maximal curves. In: Arithmetic, Ge-
ometry and Coding Theory (AGCT-10), 2009, Luminy-Marseille. SminairesCongrs. Marseille :Societe Mathematique de France, vol. 21, pp. 35-42, 2009.
[3] Garcia, A., Stichtenoth, H.: A maximal curve which is not a Galois subcover of theHermitian curve, Bulletin of the Brazilian Mathematical Society, vol.37, pp. 139-152, 2006.
Classifying indecomposable RA LoopsMariana CornelissenUFSJ, Brazil
In this work, we will provide a complete classification of finitely-generated indecomposablegroups which quotient by its center is isomorphic to the direct product of two cyclic groups ofprime order p. Applying this result for the case p = 2, we will classify all the finitely-generatedindecomposable RA loops, further generalizing the classification done in 1995, by Eric Jespers,Guilherme Leal and Cesar Polcino Milies for the finite case.
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On partial skew Armendariz ringsWagner CortesUFRGS, Brazil
In this work we consider rings R with a partial action α of an infinite cyclic group G on R. Wewill introduce the concept of partial skew Armendariz rings and partial α-rigid rings. We willobtain that partial α-rigid rings implies partial skew Armendariz rings, the bijection betweenthe set of right annihilator in R and the set of right annihilators in R[x;α] is equivalent to Rbe partial skew Armendariz ring. We study the relationship between Baerness, ascending chaincondition on right annihilators property, right p.p.-property and right zip property between Rand R[x;α] using the concept of a partial skew Armendariz ring. It will be provided exampleswhere partial skew Armendariz, Baerness, quasi-Baerness, p.q.-Baerness and p.p-property donot imply the associativity of partial skew polynomial rings. Moreover, when (R,α) has en-veloping action (T, σ), where σ is an automorphism of T we show if the partial action is of finitetype, then R is either Baer or quasi-Baer or p.q-Baer or p.p. ring if and only if T is either Baeror quasi-Baer or p.q.-Baer or p.p ring.
TbaSeverino Collier CoutinhoUFRJ, Brazil
Fast finite field arithmetic for cryptographic applicationsRicardo DahabUNICAMP, Brazil
Finite field arithmetic is pivotal in the efficiency of cryptographic methods based on ellipticcurves. In this talk I will review some of the relevant finite field algorithms and discuss recentadvances.
Modular invariants for group-theoretical modular dataAlexei DavydovUniversity of Sydney, Australia
We classify indecomposable commutative separable (special Frobenius) algebras and theirlocal modules in group-theoretical modular categories. This gives a description of modularinvariants for group-theoretical modular data. As a bi-product we provide an answer to thequestion when (and in how many ways) two group-theoretical modular categories are equivalentas ribbon categories.
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TbaJose Antonio de la PenaUNAM, Mexico
On the loop space of a 2-categoryMatias L. del HoyoUBA, Argentina
The construction of nerves and classifying spaces associate geometric objects to categoricalstructures. The relation between categories and the homotopy types of their classifying spaceshas been widely studied, with important applications. Examples of this are Quillen’s devel-opment of higher algebraic K-theory, Thomason’s theorems on homotopy colimits, and Segal’swork on delooping topological spaces that arise from monoidal categories.
In this talk we shall discuss the notions of nerves and classifying spaces for 2-categories. Wealso state sufficient conditions under which the loop space ΩBC of the classifying space of a2-category C can be recover (up to homotopy) as the space B(Hom(x, x)), where x is an objectof C and Hom(x, x) is the category of maps x→ x in C.
This theorem says that taking loops can be done algebraically, and extends in one directionthe results of Segal about monoidal categories.
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Invariantes de algebras de grupo parciaisM. DokuchaevIME-USP, Brazil
As representacoes parciais de grupos foram introduzidas em teoria de algebras de operadoresindependentemente por R. Exel [4], e J. C. Quigg e I. Raeburn [5] como uma ferramenta rele-vante no estudo dessas algebras. O estudo algebrico das representacoes parciais foi iniciado em[1]. De modo analogo ao caso das representacoes usuais existe uma algebra, denominada algebrade grupo parcial, que controla as representacoes parciais de grupos. Um resultado estruturalsobre essa algebra foi obtido em [1] no caso de grupos finitos. Isso permite derivar consequenciassobre a estrutura das representacoes parciais de grupos finitos por um lado e, por outro, estudaruma questao natural no contexto de algebras de grupo parciais, a saber, a do Problema do Iso-morfismo: o que podemos dizer sobre os grupos G e H se KparcG ∼= KparcH como K-algebras?Com respeito a esse problema foi provado em [1] que para um domınio K de caracterısticazero, dois grupos abelianos finitos G e H sao isomorfos se e somente se KparcG ∼= KparcH.Um resultado analogo com char K > 0 foi obtido em [2]. Esses resultados mostram eviden-temente a diferenca surpreendente entre a algebra de grupo parcial e a usual, ja que para umgrupo abeliano finito G a algebra de grupo CG sobre os complexos C “lembra” so a ordem de G.
Em [3] obtemos uma lista completa de invariantes, em termos de subgrupos de G e suasK-representacoes, para algebras de grupo parciais de um p-grupo finito G sobre um corpo Kalgebricamente fechado com char K 6= p.
References
[1] M. Dokuchaev, R. Exel, P. Piccione, Partial representations and partial group algebras, J. Algebra, 226 (1),(2000), 505-532.
[2] M. Dokuchaev, C. Polcino Milies, Isomorphisms of partial group rings, Glasgow Math. J., 46, (1), (2004),161-168.
[3] M. Dokuchaev, J. J. Simon, Invariants of partial group rings of p-groups, Preprint.[4] Exel, R., Parcial Actions of Groups and Actions of Semigroups, Proc. Amer. Math. Soc. 126, no. 12 (1998),
143-151.[5] Quigg, J. C., Raeburn, I., Characterizations of Crossed Products by Parcial Actions, J. Operator Theory 37
(1997), 311-340.
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Parametrization of representations of braid groupsClaudia EgeaUniversidad Nacional de Cordoba, Argentina
In these works we give a method to produce representations of the braid group Bn of n− 1generators τ1, . . . , τn−1, (n ≤ ∞). These representations satisfy certain conditions and everyrepresentations which verify these conditions can be obtained by this method.
More specifically, the principal results are the following.Given a 5-tuple (π,X, µ, ν, U), where(1) X is a measurable space,(2) π : Bn → Aut(X) is an action,(3) µ is a π-quasi-invariant measure,(4) ν : X → N ∪ ∞ is a π-invariant measurable function,(5) U : Bn × X → B(H) is a cocycle, where H =
∫X Hxdµ(x) is a direct integral Hilbert
space associated to (X,µ, ν), (that is ν(x) = dim(Hx) for almost all x ∈ X)then we can construct a non unitary representation φ = φ(π,X,µ,ν,U) of Bn.
Moreover, if the cocycle U verifies that the operator U(τk, x) of Hx is diagonal, for almostall x ∈ X and for all k, 1 ≤ k ≤ n− 1, then φ(π,X,µ,ν,U) satisfies the following properties, for allk, j, 1 ≤ j, k ≤ n− 1,
(a) ψ(τk)ψ(τk)∗ has discrete spectral decomposition;(b) [ψ(τk)ψ(τk)∗, ψ(τj)ψ(τj)∗] = 0;(c) ψ(τk)Pψ(τk)−1 ∈ N , for all projection P ∈ N , where N is the von Neumann algebra
generated byF = ψ(τk)ψ(τk)∗ : k = 1, . . . , n− 1
Conversely, given a representations ρ of Bn which satisfies (a), (b) and (c), then there existsa 5-tuple (π,X, µ, ν, U) such that ρ is equivalent to φ(π,X,µ,ν,U).
Furthermore, we obtain conditions for these representations to be irreducible. If φ(π,X,µ,ν,U)
is a self-adjoint, µ is ergodic, ν(x) = 1 for all x ∈ X, and U is not constant, then φ(π,X,µ,ν,U) isan irreducible representation.
References
[1] C.M. Egea, E. Galina, Some Irreducible Representations of the Braid Group Bn of dimension greater thatn, to appear in Journal of Knot Theory and Its Ramifications, ArXiv math.RT/0809.4173v2 (2008).
[2] C.M. Egea, E. Galina, Parametrization of representations of Braid Groups, ArXiv math.RT/0904.0491(2009).
39
Solvable Lie bialgebrasMarco FarinatiUBA, Argentina
We propose a definition of solvability for Lie bialgebras. For these classe of Lie bialgebras,the extension problem can be “solved”, and we discuss applications to the classification problemin low dimensions.
Units in U(ZCp)Raul FerrazIME-USP, Brazil
Let Z be the ring of rational integers, p be a prime, Cp be the (cyclic) group of order p andU(ZCp) be the group of units of the integral group ring of Cp. It is known that U(ZCp) canbe writen as the direct product of ±Cp and a free abelian subgroup of finite rank of U(ZCp).In this work we give a multiplicatively independent subset of U(ZCp), which generates a directfactor to ±Cp in U(ZCp), for p less than or equal to 67.
Relative invariant theoryWalter FerrerUniversidad de La Republica, Uruguay
We continue with the development of a relative viewpoint in geometric invariant theory. Inparticular we prove that if an algebra group acts in a linearly reductive way on an affine variety,then the variety can be split as a product of the unipotent radical of the group with an affinevariety where the action is given by the action by a reductive group. The study of the actionof the original group on the unipotent radical, is given by a twisting of the conjugation action.The study of this twisting seems to be crucial in order to understand the new features providedby a relative reductive action.
Partial actions on semiprime rings: GlobalizationMiguel FerreroUFRGS, Brazil
In this talk we find conditions under which a partial action of a group on a semiprime ring,which does not necessarily have an identity element, has an enveloping action. In the case thisenveloping action does exist it is also semiprime and is unique unless equivalence. In particular,if the semiprime ring has an identity element we recover the well-known result on the existenceof an enveloping action formerly obtained by M. Dokuchaev and R. Exel.
40
Polynomial identities for graded tensor products of algebrasJose Antonio O. FreitasUNICAMP, Brazil
Joint work with Plamen Koshlukov.According to Kemer’s theory the T-prime algebras are fundamental in describing the T-
ideals in characteristic 0. Recall that A is T-prime if its T-ideal is prime inside the class ofthe T-ideals in K(X). Such T-ideals are called T-prime as well. Kemer proved that the onlyT-prime T-ideals in characteristic 0 are 0, K(X), T (Mn(K)), T (Mn(E)), and T (Mk,l). HereMn(K) and Mn(E) are the full matrix algebra over K and over the Grassmann algebra E,
respectively. Further Mk,l is the subalgebra of Mk+l(E) that consists of all matrices(u vw t
)where u ∈ Mk(E0), t ∈ Ml(E0), v ∈ Mk×l(E1), w ∈ Ml×k(E1). Kemer proved that the tensorproduct of any two T-prime algebras is PI equivalent to a T-prime algebra.Kemer’s Theorem Let charK = 0. ThenMa,b ⊗ E ∼Ma+b(E); Ma,b ⊗Mc,d ∼Mac+bd,ad+bc; M1,1 ∼ E ⊗ E.The remaining tensor products of T-prime algebras from the list above yield isomorphisms.One of the first and most important results in the combinatorial PI theory was Regev’s
A ⊗ B theorem: the tensor product of two PI algebras is still a PI algebra. Regev’s theoremhas motivated a lot of research on PI algebras.
Another kind of a tensor product, is the Z2-graded one. Assume A and B are Z2-graded,A = A0 ⊕A1 and B = B0 ⊕B1. The Z2-graded tensor product A⊗B of A and B is the vectorspace A ⊗ B with the multiplication (a1⊗b1)(a2⊗b2) = (−1)|b1||a2|a1a2⊗b1b2 for ai ∈ A0 ∪ A1,bi ∈ B0 ∪ B1. Here if a ∈ A0 ∪ A1 is a homogeneous element we denote by |a| its Z2-degree.Regev and Seeman proved the analog of the A⊗B theorem when one replaces the tensor productby the graded one. They studied the PI equivalence of graded tensor products, and conjecturedthat the graded tensor product of two T-prime algebras is PI equivalent to another T-primealgebra in characteristic 0. They proved several cases of such equivalences.
We worked with the remaining cases and thus confirm Regev and Seeman’s conjecture. Weshow that in positive characteristic one cannot expect such a behaviour from the graded tensorproduct as in characteristic 0. More precisely we prove that whenever charK = 0 one hasT (Mk,l⊗E) = T (Mk+l(E)) and T (Mk,l⊗Mr,s) = T (Mp,q) where p = kr + ls, q = ks + lr,exactly the same behaviour as in the case of ungraded tensor products. If charK = p > 2 weprove that T (M2(E)) ⊂ T (M1,1⊗E), a proper inclusion.
Finite dimensional pointed Hopf algebras over S4
Agustın Garcıa IglesiasFaMAF - UNC, Argentina
Let k be an algebraically closed field of characteristic 0. We conclude the classification offinite-dimensional pointed Hopf algebras whose group of group-likes is S4. We also describeall pointed Hopf algebras over S5 whose infinitesimal braiding is associated to the rack oftranspositions.
41
Quantum groups and Hopf algebrasGaston Andres GarcıaUniversidad Nacional de Cordoba, Argentina
We will introduce the notion of quantum group and we will show its relation with Hopfalgebras, and in particular with the classification problem of finite-dimensional Hopf algebrasover an algebraically closed field of characteristic zero.
The quantum groups, introduced in 1986 by Drinfeld [Dr], form a certain class of Hopfalgebras. They can be presented as deformations in one or more parameters of associativealgebras related to linear algebraic groups or semisimple Lie algebras.
One of the main open problems in the theory of Hopf algebras is the classification of finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero. The firstobstruction in solving the classification is the lack of enough examples. Hence, it is necessaryto find new families of Hopf algebras. From the beginning, this role was played by the quantumgroups. They consists of a large family with different structural properties and were used withprofit to solve the classification problem for fixed dimensions.
After defining quantum groups and Hopf algebras, we will give some basic examples and wewill study properties that characterize the known quantum groups. Finally, we will show howthey get into the scene of the classification problem of Hopf algebras of dimension p3 and ingeneral, of pointed Hopf algebras with abelian coradical.
References
[Dr] V. Drinfeld, ‘Quantum groups’, Proc. Int. Congr. Math., Berkeley 1986, Vol. 1 (1987), 798-820.
Representation type and Hochschild cohomology of toupie algebrasMaria Andrea GaticaUniversidad Nacional de La Pampa, Argentina
Joint work with Artenstein, D. and Lanzilotta, M.The goal of this talk is to describe a family of finite dimensional algebras, called toupie al-
gebras, that are a generalisation of the canonical algebras introduced by Ringel in 1984. Moreprecisely, we study the simple connectedness, rigidity and representation type using combina-torial parameters associated to their quivers.
In the first part of this talk, we compute the Hochschild cohomology groups of toupie algebras.Then we will see how these groups are useful to determine the simple connectedness and rigidityof them. In the second part, we determine the representation type of toupie algebras.
[AGL] Artenstein, D.; Gatica, M. Lanzilotta, Representation type of toupie algebras, prepint.[CDHL] D. Castonguay, J. Dionne, F. Huard, M. Lanzilotta, Algebras toupie, prepint.[GL] Gatica, M. A., Lanzilotta, M., Hochschild cohomology of a generalisation of canonical
algebras, prepint.[R] Ringel, C.M., Tame algebras and integral quadratic forms, Lecture Notes in Mathematics,
1099, Springer-Verlag, Berlin, 1984.
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Grassmann codes and their relativesSudhir R. GhorpadeIndian Institute of Technology Bombay, India
Grassmann codes are linear error correcting codes associated to Grassmann varieties overfinite fields with their natural Plucker embedding. The study of these codes goes back toRyan (1987) and has been of much recent interest. A number of extensions and variants ofGrassmann codes, such as Schubert codes, codes associated to flag varieties, and most recently,affine Grassmann codes, have been considered. It is seen that questions motivated by codingtheory concerning these codes also give rise to interesting problems and results concerning thegeometry of Grassmann varieties and their linear sections, combinatorics of hypergraphs, andmultilinear algebra of exterior powers of finite dimensional vector spaces.
In this talk, we shall give an overview of some of the known results and open questionsconcerning Grassmann codes and their relatives. This will include parts of our joint work with(i) G. Lachaud, (ii) M. A. Tsfasman, (iii) A. R. Patil and H. K. Pillai, (iv) T. Johnsen, and (v)P. Beelen and T. Høholdt.
Graded polynomial identites and exponential growthAntonio GiambrunoUniversita di Palermo, Italy
Let A be an algebra over a field of characteristicf zero graded by a finite abelian group G. Westudy a growth function related to the graded polynomial identities satisfed by A by computingthe exponential rate of growth of the sequence of graded codimensions of A.
Right-division in Moufang loopsMaria de Lourdes Merlini GiulianiUFABC, Brazil
Joint work with Kenneth W. Johnson (Penn State University).Starting from a group (G, .) consider the quasigroup (G, ∗), obtained from a G using right
division x ∗ y = x.y−1. It is well known that (G, ∗) satisfies the identity:
(1) (x ∗ z) ∗ (y ∗ z) = x ∗ yOur purpose is construct a quasigroup (L, ∗) starting from a Moufang loop (L, .) instead ofthe group G. First, we have to find an identity similar to (1), which we call the FundamentalIdentity (2). Then we consider the inverse problem: given a quasigroup (H, ∗) satisfying (2),can we define another operation so that (H, ) is a Moufang loop? If not, what propertiesdoes (H, ) satisfy? In this work I used program Prover9.
43
p-adic diagonal equationsHemar GodinhoUnB, Brazil
Joint work with P.H.A. Rodrigues.In this communication we are going to present some recent results concerning the solubility
of systems of additive forms of degree k in n variables of the type
a11Xk1 + · · · + a1NX
kN = 0
......
...aR1X
k1 + · · · + aRNX
kN = 0.
where the coefficients are over the p-adic integers OK. The guideline for this study is thelongstanding conjecture of E. Artin that states: if the number of variables n > Rk2, then thissystem has p-adic zeros for all prime p. We would like to present results for general p-adic fieldsK, and also in the special case of Qp, where the research is much more developed.
Bass cyclic units as factors in a free product in integral group ring unitsJ. Z. Goncalves and Angel del RioIME-USP, Brazil and Universidad de Murcia, Spain
Let U(ZG) be the group of units of the integral group ring ZG, of the finite group G overthe ring of integers Z. Let G be a group of odd order, and let a be a noncencentral element ofprime order of G. We prove that if u is a Bass cyclic unit based on the element a , then thereexists either a Bass cyclic unit or a bicyclic unit v, such that < u, vn > is a nonabelian freegroup for n sufficiently large.
On smooth cubic projective hypersurfacesVictor Gonzalez AguileraUTFSM, Chile
The smooth cubic hypersurfaces Xn of the complex projective space Pn+1 are a classicalsubject of Algebraic Geometry, to Xn we can associate several algebraic geometric objets asits group of regular automorphisms, its Fano scheme of lines, its middle cohomology and itscorresponding moduli spaces. In this conference we will review some results for n = 3, 4, 5,mainly their relations with the moduli spaces of 5-dimensional and 21-dimensional principallypolarized abelian varieties, the deformation of symmetric products of K3 surfaces and somequotients of complex hyperbolic manifolds. Also we present some new computational methodobtained in a joint work with A. Liendo that allow to classify smooth cubic hypersurfaces Xn
admiting an automorphism of prime order.
44
Idempotents in abelian group algebrasMarines GuerreiroUFV, Brazil
In this work we present techniques for the calculation of idempotents in abelian group al-gebras, over a field of characteristic two and under some restrictions, for abelian groups ofexponent pq, for p and q distinct prime numbers. These idempotents are generators of minimalabelian codes. The main idea extends previous work done by C. Polcino Milies and R. A. Ferraz.This is a joint work with R. A. Ferraz, C. Polcino Milies and A. G. Chalom from IME-USP.
On left Jordan centralizersClaus HaetingerUNIVATES, Brazil
In this work we concentrate our study on Jordan left centralizers. The first result willcharacterize rings with a Jordan centralizer T . Such rings have a T invariant ideal I, and Iis the union of an ascending chain of nilpotent ideals. Also, we include an application of thisresult.
This short communication is based on chapter 5 of the PhD thesis of M.S. Tammam El-Sayiad(Beni-Suef University, Egypt), under supervision of M.N. Daif (Al-Azhar University, Egypt),I.R. Hentzel (Iowa State University, USA), H.A. El-Saify (Beni-Suef University, Egypt) andC. Haetinger (UNIVATES, Brazil).
Representations of Yang-Mills algebrasEstanislao HerscovichUniversidad de Buenos Aires, Argentina
The aim of this article is to describe families of representations of the Yang-Mills algebrasYM(n) (n ≥ 2) defined in by A. Connes and M. Dubois-Violette.
We first describe some irreducible finite dimensional representations. Next, we provide fam-ilies of infinite dimensional representations of YM(n), big enough to separate points of thealgebra. In order to prove this result, we prove and use that all Weyl algebras Ar(k) areepimorphic images of YM(n).
This is a joint work with Andrea Solotar, to appear in Annals of Mathematics. For thecomplete article, seehttp://pjm.math.berkeley.edu/scripts/coming.php?jpath=annals
On the coalgebra description of OCHAEduardo HoefelUFPR, Brazil
OCHA is the homotopy algebra of open-closed strings. It can be defined as a sequence ofmultilinear operations on a pair of DG spaces satisfying certain relations which include the L∞relations in one space and the A∞ relations in the other. In this paper we show that the OCHAstructure is intrinsic to the tensor product of the symmetric and tensor coalgebras. We alsoshow how an OCHA can be obtained from A∞-extesions and define the universal envelopingA∞-algebra of an OCHA as an A∞-extension of the universal enveloping of its L∞ part by itsA∞ part.
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On the number of points of a plane curve over a finite fieldMasaaki HommaKanagawa University, Japan
Joint work with Seon Jeong Kim.In the paper [7], Sziklai posed a conjecture on the number of points of a plane curve over a
finite field. Let C be a plane curve of degree d over Fq without an Fq-linear component. Thenhe conjectured that the number of Fq-points Nq(C) of C would be at most (d− 1)q+ 1. But hehad overlooked the known example of a curve of degree 4 over F4 with 14 points ([6], [1]). Sowe must modify this conjecture.
Modified Sziklai’s Conjecture. Unless C is a curve over F4 which is projectively equivalentto(3) X4 + Y 4 + Z4 +X2Y 2 + Y 2Z2 + Z2X2 +X2Y Z +XY 2Z +XY Z2 = 0
over F4, we might have
(4) Nq(C) ≤ (d− 1)q + 1.
This conjecture makes sense only if 2 ≤ d ≤ q+ 1 because the conjectural bound exceeds theobvious bound Nq(C) ≤ #P2(Fq) = q2 + q + 1 if d ≥ q + 2.
In [3], we proved the inequality
Nq(C) ≤ (d− 1)q + (q + 2− d),
which guarantees the truth of the conjecture for d = q+ 1, and presented an example of a curveof degree q + 1 having q2 + 1 Fq-points. Moreover, we observed that if a curve of degree 4 overF4 has more than 13 F4-points, then this curve is projectively equivalent to the curve (1) overF4.
Recently we have found two facts concerning this conjecture. The first one is that theinequality (2) holds if d = q > 4, and for each q, there exists a nonsingulsr curve of degree qover Fq with (q− 1)q+ 1 rational points. Note that the truth of the inequality (2) for d = q = 3is classical [4], and it is well known for d = q = 2. The second one is that (2) holds if the curveC is nonsingular of degree d ≤ q − 1, which is an easy cosequence of results of Stohr - Voloch[5] and Hefez - Voloch [2]. Therefore, together with our previous results, the following theoremhas been established.
Theorem. The modified Sziklai’s conjecture is true for nonsingular curves. Moreover there isan example of a nonsingular curve for which equality holds in (2) for d = q + 2, q + 1, q, q −1,√q + 1(when q is square), and 2.
References
[1] G. van der Geer and M. van der Vlugt, Tables of curves with many points,http://www.science.uva.nl/~geer/
[2] A. Hefez and J. F. Voloch, Frobenius nonclassical curves, Arch. Math. (Basel) 54 (1990), 263–273; Correction,Arch. Math. (Basel) 57 (1991), 416.
[3] M. Homma and S. J. Kim, Around Sziklai’s conjecture on the number of points of a plane curve over a finitefield, to appear in Finite Fields and Their Applications.
[4] B. Segre, Le geometrie di Galois, Ann. Mat. Pura Appl. (4) 48 (1959) 1–96.[5] K.-O. Stohr and J. F. Voloch, Weierstrass points and curves over finite fields, Proc. London Math. Soc. (3)
52 (1986), 1–19.[6] J. P. Serre, Nombres de points des courbes algebriques sur Fq , Sem. de Theorie des Nombres de Bordeaux
1982–1983, exp. 22; Oeuvres III, No. 129, 664–668.[7] P. Sziklai, A bound on the number of points of a plane curve, Finite Fields Appl. 14 (2008) 41–43.
46
Exceptional sequences, braid groups and clustersKiyoshi IgusaBrandeis University, USA
Auslander Reiten quivers of type An and Bn can be viewed as braid diagrams and clustermutations can be seen as homotopies of these braids. These are examples of more generaltheorems about the “dual braid monoid” and exceptional sequences proved by Bessis, Brady-Watt, Ingalls-Thomas and myself with Ralf Schiffler. I will use the Orlik-Solomon algebra tointerpret some of the well-known results about cluster combinatorics in types An and Bn interms of the cohomology of the corresponding braid groups.
Group algebras satisfying certain homological properties - A surveySurender K. JainOhio University, USA
The purpose of this talk is to give various homological properties of group algebras beginningwith classical result of Cconnell, Renault and others on self–injectivity. The group algebra ofan infinite dihedral group over a field of characteristic different from 2 is not self-injective buthas an important property possessed by self-injectve algebras, namely that each complementright or left ideal is a direct summand of the algebra. Such algebras are known in the literatureas CS algebras. We will also discuss other properties of self-injectve group algebras like vonNeumann continuity and quasi-continuity. We will talk about an interesting result on primeCS group algebras of polycyclic-by-finite groups that such groups are either infinite dihedralgroups or torsion-free abelian groups. We will talk about a semiprime CS group algebra KGof a polycyclic-by-finite group which are hereditary and if the field is algebraically closed fieldK, then this group algebra is a direct sum of matrix rings over the group algebra KN , whereN is either infinite dihedral group or Z. We will also talk about the question as to when asemilocal group algebra KG is continuous? Complete answer is known in some cases whichinclude when G is nilpotent. It is open, for example, when G is solvable. Another question thatis of interest is: whether a CS group algebra of a locally finite group over rationals is artinian(this algebra is von-Neumann regular)? We will close with the question as to when a generalizedgroup algebra L1(G,A) of a locally compact group G over a Banach algebra A with identity isright self-injective.The corresponding question regarding self-injectvity of L2(G) which is alsoof interest to those in Harmonic Analysis remains open.
Decomposability criterion for coherent sheavesMarcos JardimUNICAMP, Brazil
Using the theory of representations of quivers, we describe a topological criterion for thedecomposability of coherent sheaves on projective varieties. As an application, we provide anew proof of Grothendieck’s Theorem and show when a Steiner bundle on a projective space isdecomposable.
47
On graded central polynomials of the graded algebra M2(E)Sandra Mara Alves JorgeCEFET-MG, Brazil
By considering E the infinite dimensional Grassmann algebra over a field of characteristiczero and the T-prime algebra M = M2(E) with the Z2-grading M =
(E 0
0 E
)⊕(
0 E
E 0
),
in this work we describe the space of the graded central polynomials modulo the ideal of thegraded identities of M.
Representation type of the universal enveloping algebras of alternative algebrasIryna KashubaIME-USP, Brazil
This a joint work with I. Shestakov. The talk is devoted to the problem of the classificationof indecomposable alternative bimodules over finite dimensional alternative algebras.
We assume the base field k to be algebraically closed and of characteristic 6= 2, 3. Recall,that for an alternative algebra A the category A-bimod of k-finite dimensional A-bimodulesis equivalent to the category U-mod of (left) finitely dimensional modules over an associativealgebra U = U(A), which is called the universal enveloping algebra of A. If A has finitedimension the algebra U is finite dimensional as well. It allows us to apply to the category A-bimod all the machinery developed in the representation theory of finite dimensional algebras. Inparticular, in accordance with the representation type of the algebra U one can define alternativealgebras of the finite, tame and wild representation types. As in the case of associative algebrathe distinction of the objects of finite and tame representation type is an interesting problem,especially because in these cases we can obtain a complete classification of finite dimensionalbimodules over A.
We introduce two new notions for alternative algebras: a diagram and a tensor algebra ofa module, which prove to be very useful. In particular, we give a classification in the case ofalgebras with radical square zero. The results obtained are similar to the corresponding classicalresults for associative algebras.
Homology (of) Hopf algebrasChristian KasselUniversite de Strasbourg and CNRS, France
In joint work with Julien Bichon (arXiv:0807.1651) we answered the question of finding asuitable homology theory for arbitrary Hopf algebras; namely we associate to each Hopf algebraH two commutative Hopf algebras H1(H) and H2(H) in such a way that, if H is the algebra ofa group G, then Hi(H) is the algebra of the homology group Hi(G). When H is a cosemisimpleHopf algebra over an algebraically closed field of characteristic zero, then H1(H) is the algebra ofthe universal abelian grading group of the category of corepresentations of H. The computationof H1(H) and H2(H) for the Sweedler algebra shows that Hi(H) is not always a group algebra.Therefore, quite surprisingly, the right structure for a homology theory for general Hopf algebrasis not that of a group, but of a commutative Hopf algebra.
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Bielliptic Weierstrass pointsTakao KatoYamaguchi University, Japan
Let X be a non-hyperelliptic curve of genus g ≥ 5 over C. X is called bielliptic if X is a two-sheeted covering of an elliptic curve E. Then, there exists an automorphism σ of X, called it bya bielliptic involution, such that X/〈σ〉 is equivalent to E. By the Riemann-Hurwitz relation,it is easy to see that there exist 2g − 2 fixed points of σ. For a fixed point P of σ, by thepull-back of meromorphic functions on E, it is shown that 2k (k = 2, 3, 4, . . . ) are Weierstrassnon-gaps at P . More precisely there are two types Weierstrass gap sequences for fixed points ofelliptic involutions. For convenience, we list up them by their order sequences (of holomorphicdifferentials), instead of the gap sequences :Type Ig: 0, 1, 2, 4, . . . , 2j, . . . , 2g − 6, 2g − 4,Type IIg: 0, 1, 2, 4, . . . , 2j, . . . , 2g − 6, 2g − 2.
The main result of this talk is that the converse is true, namely, the existence of a Weierstrasspoint P ∈ X of Type Ig or Type IIg implies the existence of a bielliptic involution of X and P isa fixed point of the bielliptic involution. After this result, we can call P a bielliptic Weierstrasspoint . For g ≥ 8, this fact was already proved in 1979. Thus, in this talk, we shall give anoutline of the proof of the above result for 5 ≤ g ≤ 7.
As an easy consequence of our result, for instance, we obtain that there does not exist a curveof genus 5 having exactly k Weierstrass points of Type II5 for 21 ≤ k ≤ 23, etc.
As algebras de Maltsev livresAlexandr KornevIME-USP, Brazil
Trabalho conjunto com Ivan Shestakov.Um dos problemas importantes da teoria de algebras nao-associativas e a estrutura de algebras
livres. As questoes tıpicas consideradas sao a construcao de bases efetivas, descricao de radicais,centros, e ideais de identidades destas algebras.
Para as algebras alternativas livres An em n geradores livres, em trabalhos de Kleinfeld eHumm, Shestakov, Filippov e Iltiakov foram provados os fatos seguintes:i) a algebra An e semiprima se e somente se n ≤ 3;ii) o nil-radical de An coincide com o conjunto de elementos nilpotentes dela;iii) para todo n ≥ 1, o ideal de identidades T (An+1) da algebraAn+1 esta contido estritamente
no ideal T (An);iv) uma base efetiva de algebra A3 foi construida.As algebras de Malcev tem relacoes fortes com as algebras alternativas. Mesmo assim, para
as algebras de Malcev livres Mn em n geradores so um analogo do fato iii) foi provado emtrabalhos de Shestakov e Filippov para todo n 6= 3. Para n = 3, esta questao continua emaberto.
Nos construimos uma base efetiva para a algebra M3 e provamos que ela e semiprima eespecial, isso e, pode ser imersa na algebra comutador A(−)
3 da algebra A3. Alem disso, provamoso seguinte analogo do fato iv) acima: o radical primo da algebra Mn coincide com o conjuntode todos os elementos engelianos desta algebra.
49
Graded identities in Lie algebrasPlamen KoshlukovIMECC-UNICAMP, Brazil
Let K be an infinite field, G an abelian group, and L a G-graded Lie algebra over K. Wediscuss several results concerning the G-graded identities of L.
Assume the characteristic of K different from 2. If L = sl2(K) then there are three naturalnontrivial gradings on L. The first of these is by C2, the cyclic group of order 2, the second isby the integers and is concentrated in −1, 0, 1, and the third is by C2×C2. Recall that if K isalgebraically closed then these are all nontrivial finite abelian gradings on sl2(K). We describethe graded identities in each of the three cases. It turns out the corresponding ideals of gradedidentities are finitely based; we exhibit such bases.
If the characteristic of K is 2 one may consider gl2(K) with its natural grading by C2. Wedescribe the graded identities in this case as well (a joint result with A. Krasilnikov). In contrastwith the above here the ideal of graded identities is not finitely based. In fact we give an explicitconstruction of a just nonfinitely based based variety of C2-graded Lie algebras (in characteristic2). Here we recall that the associative algebra M2(K) admits a finite basis of its C2-gradedidentities even in characteristic 2.
Finally we consider the Lie algebra W1 of the derivations of the polynomial ring in onevariable. This algebra is graded by the integers in a canonical way. We describe its gradedidentities. We exhibit a list of graded identities of degrees 2 and 3 that generate the ideal ofgraded identities of W1. This list is infinite and consists of independent identities. Thereforethe ideal of graded identities of W1 is not finitely based. (These are joint results with J. Freitasand A. Krasilnikov.)
50
A class of locally nilpotent commutative algebrasAlicia LabraUniversidad de Chile, Chile
Let A be the free commutative (nonassociative) algebra on a finite set t1, . . . , tk of gener-ators.
Let dim[n, k] be the dimension of the subspace of A which is spanned by terms of degree lessthan n. Thus dim[n, k] is the number of distinct monomials of A with degree less than n.
This talk deals with the variety of commutative non associative algebras satisfying the identityL3x + γLx3 = 0, γ ∈ K. In [4] it is proved that if γ = 0, 1 then any finitely generated algebra is
nilpotent. We generalize this result by proving that if γ 6= −1, then any such algebra is locallynilpotent. Our results require characteristic 6= 2, 3. The main result is:
Theorem 1. Any commutative algebra over a field of characteristic 6= 2, 3, satisfying the iden-tity L3
x + γLx3 = 0 with γ 6= 0,±1,±12 is locally nilpotent.
More precisely, any commutative algebra generated by k elements over a field of character-istic 6= 2, 3, satisfying the identity L3
x + γLx3 = 0 with γ 6= 0,±1,±12 is nilpotent of index at
most 24ndim[n,k]+2(n−2), where n is the index of nilpotency of the free commutative algebra on kgenerators satisfying the identity x3 = 0.
We also see the five cases which arose as exceptions in the above Theorem and in the onlycase where the Theorem cannot be extended is γ = −1.
This is a joint work with Antonio Behn and Alberto Elduque.REFERENCES
(1) A. A. Albert, On power-associative rings, T. Am. Math. Soc. 64 (1948), 552-593.(2) W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user
language. J. Symbolic Comput., 24(3-4):235-265, 1997(3) L. Carini, I. R. Hentzel, G. M. Piacentini-Cattaneo, Degree four identities not imply by
commutativity, Commun. in Algebra 16 (2) (1988), 339-356.(4) I. Correa, I. R. Hentzel, A. Labra,, Nilpotency of Commutative Finitely Generated Al-
gebras Satisfying L3x + γLx3 = 0, γ = 1, 0, to appear in J. Algebra.
(5) I. Correa, I. R. Hentzel, Commutative finitely generated algebras satisfying ((yx)x)x=0are solvable, to appear in Rocky Mountain J. of Math.
(6) L. Elgueta, A. Suazo, J. C. Gutierrez Fernandez, Nilpotence of a class of commutativepower-associative nilalgebras, J. Algebra 291 (2005), 492-504.
(7) M. Gerstenhaber and H. C. Myung, On commutative power-associative nilalgebras oflow dimension, Proc. Amer. Math. Soc. 48 (1975), 29-32.
(8) J. C. Gutierrez Fernandez, A. Suazo, Commutative power-associative nilalgebras ofnilindex 5, Results Math. 47 (2005), 296-304.
(9) J. C. Gutierrez Fernandez, Commutative finite-dimensional algebras satisfying x(x(xy)) =0 are solvable, to appear in Comm. in Algebra.
(10) I. R. Hentzel, A. Labra, On left nilalgebras of left nilindex four satisfying an identity ofdegree four, Inter. J. of Algebra and Comp. 17 (1) (2007), 27-35.
(11) J. M. Osborn, Commutative non associative algebras and identitities of degree four,Can. J. Math. 20 (1968), 769-794.
(12) W. Stein, Sage Mathematics Software (Version 3.1.1), The SAGE Group, 2008,http://www.sagemath.org.
(13) D. A. Suttles, Counterexample to a conjecture of Albert, Notices Amer. Math. Soc. 19(1972), A-566.
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Cox rings of rational surfacesAntonio LafaceUniversidad de Concepcion, Chile
Let X be a smooth rational surface defined over the complex numbers. In this talk I willdiscuss about the finite generation of the Cox ring of X:
R(X) :=⊕
D∈Cl(X)
H0(X,OX(D)).
Let κ := κ(X,−KX) be the anticanonical Kodaira dimension of X; then R(X) is known to befinitely generated if κ = 2 by [2]. I will show that if κ = 1, then R(X) is finitely generated ifand only if the effective cone Eff(X) is polyhedral and will give an idea of how to classify suchsurfaces. This result employes methods of [1]. Moreover, in case κ = 0, I will provide examplesof rational surfaces with polyhedral effective cone and not finitely generated Cox ring.
References
[1] M. Artebani, J. Hausen, A. Laface: On Cox rings of K3-surfaces, arXiv:0901.0369v2[2] D. Testa, A. Varilly, M. Velasco: Big rational surfaces, arXiv:0901.1094v2
Coverings of laura algebrasPatrick Le MeurENS Cachan, France
Joint work with I. Assem and J. C. Bustamante.In the eighties, Gabriel and some of his students introduced (Galois) coverings of algebras in
order to classify representation finite algebras. The achievement of this classification is based onthe so-called universal cover of a representation finite algebra. This is a covering of the algebrawhich happens to be Galois with group isomorphic to the fundamental group of the Auslander-Reiten quiver if the algebra is standard. When the representation finite algebra is standard andwhen its universal cover is trivial, the algebra is called simply connected. Buchweitz and Liuproved in 2004 that a standard representation finite algebra is simply connected if and only ifits first Hochschild cohomology group vanishes.
On the other hand, in 2003, Assem and Coelho, and Reiten and Skowronski independentlydefined the class of laura algebras which comprises both representation finite and quasi-tiltedalgebras.
This work is an attempt to generalise to laura algebras the above results of Gabriel’s school onrepresentation finite algebras. We shall see that a laura algebra (with a connecting component)has a covering which happens to be Galois with group isomorphic to the fundamental groupof the connecting component if it is standard. Moreover, a standard laura algebra is simplyconnected if and only if its first Hochschild cohomology group vanishes.
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A generalization of the Jacobson radicalThierry Petit LobaoUFBA, Brazil
In this work we discuss a generalization of the concept of quasi-regularity and prove thatit is a radical property according to Divinsky; thus it determines a kind of Kurosh-Amitsurradical, in fact, it generates a whole enumerable family of Jacobson-like radicals: the Jk radicals,with k integer. We study some properties of these radicals such as extensibility (closenessunder extensions), matrix and polynomial extensibility, Amitsur property among others; wealso prove that they contain all nilpotent ideals and are hereditary, so they are supernilpotent.We investigate the representation of these radicals as upper and lower radical classes and tryto localize their position in the lattice of some important radicals as well.
A generalisation of the dual Kummer surfaceVictor Gonzalo Lopez-NeumannUFU, Brazil
For a curve C of genus 2 the notion of divisor in general position is defined. For such acurve, the dual Kummer surface K∗ is the variety parametrizing divisors of degree 3 in generalposition, up to linear equivalence and antipodal involution. It is known that K∗ is birationallyequivalent to K, the Kummer surface belonging to the curve C. We generalize the notion ofdivisor in general position for a hyperelliptic curve of genus g. Then we show that, for such acurve, there is a variety parametrizing the classes of divisors of degree g+ 1 in general position,up to antipodal involution. This is a generalization of the Kummer surface. The main result isthat it is birationally equivalent to the Kummer variety belonging to the curve C.
Chevalley groups and hyperalgebrasTiago MacedoUNICAMP, Brazil
We present some results which relate the algebra of distributions of a Chevalley group andthe so called hyperalgebras. The latter are Hopf algebras obtained by reduction modulo p ofthe Kostant integral form of a simple Lie algebra. Then we try to rebuild algebraic groups fromHopf algebras which are their algebras of distribution.
2− d-Koszul algebrasEduardo N. MarcosIME-USP, Brazil
This is a report on some results on a joint paper with Ed Green.Koszul algebras and “kozulity” properties in general are very ubiquitous. Some generalization
appear in print. We stay in the setting of quotient of quiver algebras. In this setting the firstgeneralization where the ideal of relations is not generated in one degree was the 2− d Koszulalgebras. We give the proposed definitions and show using some Grobner basis theory, that thesome of these algebras have finitely generated Yoneda-Ext Algebras.
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Max Noether theorem for singular curvesRenato Vidal MartinsUFMG, Brazil
The Max Noether theorem we mention above is the following: “Let ω be the dualizing sheafof a nonhyperelliptic nonsingular curve C. Then Symn(H0(ω)) surjects H0(ωn).” The assertionremains true, with the same proof, if ω is invertible, ie, C is Gorenstein. The talk is about howto generalize the result to curves which are non-Gorenstein.
TbaGuillermo MateraUniversidad Nacional de General Sarmiento, Argentina
On modular unipotent representation of GL(n,K)Olivier MathieuLyon, France
Let K be a finite field. Unipotent representations of GL(n,K) in characteristic zero havebeen fully described by classical authors (Green, James). Here we are interested in unipotentrepresentation in finite characteristic. We show that a certain class of irreducible character canbe computed. The proof requires tilting theory.
Compactified Picard stacks over the moduli space of curves with marked pointsMargarida MeloUniversidade de Coimbra, Portugal
Let Picd,g,n be the stack parametrizing degree d line bundles over smooth curves of genus gwith n marked points. In this talk I will give a construction of smooth and irreducible algebraicstacks yeldying a modular compactification of Picd,g,n over the moduli stack of n-pointed stablecurves,Mg,n. By this we mean an algebraic stack with a proper (or at least universally closed)map onto øMg,n, containing Picd,g,n as a dense open substack. These stacks parametrize whatwe will call balanced line bundles over n-pointed quasistable curves, generalizing L. Caporaso’scompactification of the universal degree d Picard variety over Mg. In fact, for n = 0, we justgive a stack theoretical description of Caporaso’s compactification and then, following the linesof Knudsen’s construction of Mg,n, we go on by induction on the number of points.
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On the center of the relatively free algebra of Ma,b and the ring of generic matricesThiago Castilho de MelloUNICAMP, Brazil
The study of the ring of generic matrices is an important subject in ring theory. One descrip-tion of a generic matrix algebra is that it is a relatively free algebra for Mn(F ), the algebra ofn× n matrices over the field F [2]. In [1] Berele describes the algebra Uk(Ma,b), the relativelyfree algebra in k generators for Ma,b, and shows that its center is a direct sum of the field anda nilpotent algebra. He also asks whether or not such center contains non-scalars elements. Inthis poster we present such results and we give an answer for Berele’s question when a = b = 1.References[1] Berele, A., Generic verbally prime PI-algebras and their GK-dimensions, Communicationsin algebra, 21(5) (1993) 1487-1504.[2] Formanek, E., The Polynomial Identities and Invariants of n× n Matrices, CBMS RegionalConference Series in Mathematics, 78, AMS (1991).
The poset of p-subgroups of a finite groupGabriel MinianUniversidad de Buenos Aires, Argentina
Given a finite group G and a prime p, we denote by Sp(G) the poset of nontrivial p-subgroupsof G ordered by inclusion. In 1975, K. Brown investigated the relationship between the topo-logical properties of the poset Sp(G) and the algebraic properties of G and proved a variationof Sylow’s Theorems for the Euler characteristic of Sp(G) [4]. In 1978, D. Quillen showed thatif G has a nontrivial normal p-subgroup, Sp(G) is contractible. He proved that the converse ofthis statement is true for solvable groups and conjectured that it is true for all finite groups [5].This conjecture is still open, although it has be shown to be true in many cases [1].
In the first part of my talk I will recall Brown and Quillen’s constructions and results. In thesecond part of the talk I will show how this conjecture can be reformulated and analyzed froma very different point of view, using the topology and combinatorics of finite topological spaces.
References
1 M. Aschbacher and S. D. Smith. On Quillen’s conjecture for the p-groups complex.Ann. of Math. 137 (1993), 473-529.
2 J.A. Barmak. Equivariant simple homotopy types and Quillen’s conjecture on the posetof p-subgroups of a group. (Work in progress)
3 J.A. Barmak and E.G. Minian. Simple homotopy types and finite spaces. Adv. Math. 218(2008), Issue 1, 87-104.
4 K. Brown. Euler characteristics of groups: The p-fractional part. Invent. Math. 29(1975), 1–5.
5 D. Quillen. Homotopy properties of the poset of nontrivial p-subgroups of a group.Adv. Math. 28 (1978) 101–128.
6 R.E. Stong. Group actions on finite spaces. Discrete Mathematics 49 (1984) 95–100.
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The homology of reduced lattices and some combinatorial duality theoremsGabriel MinianUniversidad de Buenos Aires, Argentina
A reduced lattice is a finite poset L with the property that every upper bounded set of L hasa supremum. The homology of a finite poset can be defined (and computed) in two different,but equivalent, ways:
(1) A finite poset is a finite topological space and one can compute, in this way, its (singular)homology.
(2) Associated to any finite poset P there is a simplicial complex K(P ), whose simplicesare the nonempty chains of P . One can define the homology of P as the homology ofthe polyhedron K(P ).
By an old result of McCord, one can easily prove that both definitions coincide. In this talk Iwill show how to compute, in an alternative (and simpler) way, the homology of reduced latticesusing new combinatorial and topological techniques. I will also exhibit generalizations of someclassical duality results in homology, such as Poincare duality and Alexander duality. Thesegeneralizations will allow to study geometrical problems of polyhedra from a new point of view.
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Methods of poset representation theory in steganographyAgustın Moreno CanadasUniversidad Nacional de Colombia, Colombia
Joint with : Oscar Espinel Montano, and Jaleydi Cardenas Poblador.The representation theory of posets was introduced in the 70’s by Nazarova and Roiter.
The main aim of their study was to determine indecomposable matrix representations of agiven poset P over a fixed field k or indecomposable filtered k-linear representations of Pintroduced by Gabriel in connection with the investigation of oriented graphs having finitelymany isomorphism classes of indecomposable linear representations. In this case a k-linearrepresentation is a system U = (U0 ; Ux | x ∈ P ), where U0 is a finite dimensional k-vectorspace and for each x ∈ P , Ux is a subspace Ux ⊂ U0 such that Ux ⊂ Uy if x ≤ y in P [4].
On the other hand, image algebra is a mathematical theory concerned with the transfor-mation and analysis of images [3]. Although much emphasis is focused on the analysis andtransformation of digital images, the main goal is the establishment of a comprehensive andunifying theory of images transformations, image analysis, and image understanding in the dis-crete as well as continuous domain. Much of the mathematics associated with image algebraand its implication to computer vision remains largely unchartered territory which awaits dis-covery. For example very little work has been done in relating image algebra to computer visiontechniques which employ tools from representation theory.
In this talk we shall describe how the steganography allows us to give a connection betweenthe theories described above. In particular we will describe how we can use representations ofposets over some finite fields in order to construct systems of multiple watermarks. We notethat these types of systems have been used to give security to some banknotes and stamps [1,2].
References
[1] E. Becker, W. Buhse, D Gunnewig, and N. Rump, Digital Rights Management, Springer,2003.
[2] I. Moskowitz, Information Hiding, Springer, Proc. 4th International Workshop, April,2001, Pittsburgh, USA, 2001.
[3] G. Ritter and J. Wilson, Handbook of Computer Vision Algorithms in Image Algebra,CRC Press, New York, 2001.
[4] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories,Gordon and Breach, London, 1992.
TbaC. MorenoCity University of New York, USA
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Plane arcs from plane curvesBeatriz MottaUNICAMP, Brazil
An (n, d)-arc is a set of n points in the projective plane P2(Fq), Fq being the finite field oforder q, such that no line intersects the set in more than d points; the arc is complete if cannotbe embedded in an (n+ 1, d)-arc. In this talk we are concerned with arcs arising from the set ofrational points of plane curves over finite fields; as a matter of fact we obtain complete arcs fromcurves of type yd = f(x). There exist several examples of such curves which, in a natural way,appear in the context of Arithmetic Theory as well as Coding Theory. We have for examplethe Hermitian curve y`+1 = x`+1 + 1 (q = `2) or the Norm-Trace curve y`
2+`+1 = x`2
+ x` + x(q = `3). Since both curves have a large number of rational points compared to their genus, thecorresponding arcs have a large number of points compared to their degrees.
References
[1] H. Borges, On multi-Frobenius non-classical plane curves, preprint.[2] H. Borges, On complete (N, d)-arcs derived from plane curves, Finite Fields and Their Applications 15
(2009), no. 1 82–96.[3] A. Garcia, The curves yn = f(x) over finite fields, Arch. Math. 74 (1990), no.1 36–44.[4] O. Geil, On codes from norm-trace curves, Finite Fields and Their Applications 9 (2003), 351–371.[5] M. Giulietti, F. Pambianco, F. Torres and E. Ughi, On complete arcs arising from plane curves, Designs,
codes and Cryptography 25 (2002), 237–246.[6] B. Motta and F. Torres, Plane arcs from plane curves, work in progress.
Tensor products, characters, and blocks of finite-dimensional representations of quantumaffine algebras at roots of unityAdriano MouraIMECC-UNICAMP, Brazil
The blcok decomposition of the category of finite-dimensional representation of quantumaffine algebras was first studied by the speaker and P. Etingof. The original approach usedanalytic properties of the action of the R-matrix which are true only if the quantization pa-rameter q is not in the unity circle. Later, the speaker and V. Chari used a different approachin connection with the theory of q-characters which could be used for generic q. One of themain tools used in both approaches was a consequence of a result of Chari on tensor products ofirreducible representations. Namely, part of the study relied on the fact that a tensor productof fundamental representations could always be re-oredered in such away that it becomes iso-morphic to a Weyl module. We will talk about a joint work with D. Jakelic where we considerthe root of unity case. It turns out that the aforementioned result about tensor products offundamental representations is no longer valid. After reviweing the basics on finite-dimensionalrepresentations of quantum affine algebras, we will discuss the techniques we used to overcomethis issue.
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Genus 3 curves: a world to exploreEnric NartUniversitat Autonoma de Barcelona, Spain
In the last twenty years, curves of genus 1 and 2 over finite fields have been extensively studiedand a lot of computational issues have been addressed to facilitate their manipulation in regardsto their applications to coding theory and/or cryptography. This talk is a survey on the problemsthat concern the development of similar techniques for genus 3 curves. Emphasis will be laid onthe case of non-hyperelliptic genus 3 curves (plane quartics), and in the application of classicalanalytical tools (of Riemann, Klein, Weber ...) to study some modern problems concerningarithmetic properties of these curves over finite fields.
Etale homotopy types of moduli stacksFrank NeumannUniversity of Leicester, UK
Using the machinery of etale homotopy theory as developed by Artin and Mazur we willdetermine the etale homotopy type of certain moduli stacks of algebraic curves and abelianvarieties and relate their etale fundamental groups to the absolute Galois group Gal(Q/Q)extending previous results obtained by Oda. This is joint work with Paola Frediani (Pavia).
On a 4-gonal curve of genus 9Akira OhbuchiTokushima University, Japan
Joint work with T.Harui and T.Kato.Let C be an algebraic curve defined over C and let
sC(2) = mind |∃g2d : birational.
It is known that sC(2) ≤ g + 2 where g = genus of C, sC(2) = g + 2 if and only if C ishyperelliptic and sC(2) = g + 1 if and only if C is bielliptic. Harui, Kato and Ohbuchi provethat if sC(2) = g, then C is a doble cover of genus 2 curve when g ≥ 10. In case g ≤ 8, theclassification of a curve with sC(2) = g is very easy. But in case g = 9, it is not easy to classifya curve with sC(2) = g. Today’s aim is to construct a counterexample in case genus 9, i.e. toconstruct a curve of genus 9 which is a birational g2
g = g29 but is not a double cover of genus 2.
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Leavitt path algebrasJoacir Lucas de OliveiraUFPR, Brazil
Joint work with Marcelo Muniz Silva Alves (supervisor).Let Q = (Q0, Q1, s, t) be a quiver, i.e., Q0 and Q1 are two nonempty sets whose elements are
called the vertices (or points) and arrows (or edges) of Q, respectively, and s, t : Q1 → Q0 are thefunctions that assign to each arrow α ∈ Q1 its source s(α) ∈ Q0 and its target t(α) ∈ Q0. Givena field K, the path algebra K(Q) over K is the free K-algebra K[Q0∪Q1] with relations vivj =δi,jvi, vi, vj ∈ Q0 and α = s(α)α = αt(α), α ∈ Q1. In order to define the Leavitt path algebraof Q we consider the extended quiver Q = (Q0, Q1 ∪ (Q1)∗, s′, t′), where (Q1)∗ = α∗; α ∈ Q1and the functions s′, t′ are given by s′|Q1 = s, t′|Q1 = t, s′(α∗) = t(α), t′(α∗) = s(α). In theextended quiver the elements of Q1 are the real edges and the elements of Q∗1 are called the ghostedges; the extended graph has a canonical involution that takes α to α∗, which then induces aninvolution in K(Q).
The Leavitt path algebra L(Q) = LK(Q) of the quiver Q is the path algebra K(Q) with theCuntz-Krieger relations: (i) α∗β = δα,βα for all α, β ∈ Q1, and (ii) v =
∑α∈Q1: s(α)=v αα
∗,whenever v ∈ Q0 and s−1(v) 6= ∅ (v is not a sink). In order that the second Cuntz-Kriegerrelation make sense we consider only quivers where s−1(v) is a finite set for all v ∈ Q0 (whichare called row-finite quivers). We will be mainly interested in giving necessary and sufficientconditions on Q for the Leavitt path algebra L(Q) to be a simple algebra. We will see thatthe Leavitt path algebra L(Q) is simple if, and only if, Q satisfies (i) the only hereditary andsatured subsets of Q0 are ∅ and Q0, and (ii) every cycle in Q has an exit.
References
[1] Gene Abrams and Gonzalo Aranda Pino; The Leavitt path algebra of a graph, Jornal of Algebra 293(2005) 319-334
[2] Ibrahim Assem, Algebre et Modules, Masson 1997.[3] P. J. Hilton; U. Stammbach, A course in homological algebra, second edition, Springer 1997[4] Dummit, David S.; Foote, Richard M. Abstract Algebra, third edition John Wiley & Sons, Inc 2004.[5] Ibrahim Assem, Daniel Simson, Andrzej Skowronski, Elements of the Representation Theory of
Associative Algebras, 1: Techniques of Representation Theory, London Mathematical Society 2006.
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On commutativity and finiteness in groupsRicardo Nunes de OliveiraUFG, Brazil
The notion of weak commutativity between groups was introduced by Said Sidki (1980), byintroducing the group of weak commutativity χ(H) =
⟨H,Hψ| [h, hψ] = 1, ∀h ∈ H
⟩, where
ψ : H → Hψ an isomorphism. It was shown that χ, seen as an operator on the class of groups,preserves a number of properties such as finiteness, nilpotency and solubility. In a recent jointpaper with Sidki (to appear in the Bulletin of SBM) we consider a more general form of weakcommutativity, where given two isomophic groups H, H and a bijection f : H \ e → H \ e,we define
χ(H, f) =⟨H, H | [h, f(h)] = 1, ∀h ∈ H
⟩.
We produce results which support the conjecture that χ(H, f) preserves nilpotency. Finitenessfollows from a general theorem in the same paper of 1980.
The question of extending the finiteness criterion and other properties to a group involvingthree or more copies of H, leads us to consider quotients of the group
χ(H,n) =⟨H,Hψ, . . . ,Hψn−1 | [hψi
, hψj] = 1, ∀h ∈ H, 1 6 i, j 6 n− 1
⟩where ψn = 1. We will give both theoretical and computational results in this direction.
Low complexity normal bases in finite fieldsDaniel PanarioCarleton University, Canada
Let Fqn be an extension of Fq, and α ∈ Fqn . A normal basis of Fqn over Fq is a basis of theform N = α, αq, . . . , αqn−1. In this case, we say that α is a normal element of Fqn .
Let αi = αqi
for 0 ≤ i ≤ n− 1, and let T = (tij) be the n× n matrix given by
ααi =n−1∑j=0
tijαj , 0 ≤ i ≤ n− 1, tij ∈ Fq.
The complexity of the normal basis N , denoted by cN , is the number of non-zero entries in T .Mullin et al (1989) proved that cN ≥ 2n− 1. The normal basis N is optimal when cN = 2n− 1.
Optimal normal bases over finite fields were completely characterized in a fundamental paperdue to Gao and Lenstra (1992). Optimal normal elements do not exist for all finite fields andall extensions.
Normal bases are widely used in applications of finite fields in areas such as coding theoryand cryptography. In particular, optimal normal bases are desirable. When no optimal normalbasis exists, it is useful to have normal elements of low complexity, say of complexity boundedby cn for some small constant c. However, when no optimal normal basis exists, the problemof classifying all low complexity normal bases is still open.
In this talk, first, we briefly review some old constructions of low complexity normal elements.Then we comment on some new constructions of low complexity normal elements based on thetrace of an optimal normal element. Finally, we give some experimental results and conjecturesabout the distribution of the complexity of normal elements in F2n , for several values of n.
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Near orders on higher dimensional varietiesRafael PeixotoUNICAMP, Brazil
Goppa constructed codes based on algebraic curves; many authors generalize this constructionbased on higher dimensional varieties such as Hermitian varieties, Toric varieties, zero schemesof vector bundles, etc. The study of these codes, which is based on resources from algebraicgeometry, is usually difficult. Høholdt, van Lint and Pellikaan presented a construction of codesby using linear algebra and semigroup theory only. This construction is based on the concept ofweight on algebras (over finite fields). Matsumoto noticed that the aforementioned constructionallows us to work mainly with one-point Goppa codes. A similar picture for the case of two-points Goppa codes was worked out by using the concept of near weight. The very generalconcept of weight on an arbitrary variety was given by Geil, Pellikaan and Little. Here we willintroduce the notion of near weight on arbitrary algebraic varieties.
The Kirillov-Reshetikhin modules associated to E6
Fernanda de Andrade PereiraUNICAMP, Brazil
Let g be a finite-dimensional simple Lie algebra over the field of complex numbers. Considerits loop algebra g and the corresponding quantized enveloping algebras Uq(g) and Uq(g). It isknown that, unless g is of type A, there is no quantum group analogue of the evaluation mapsg → g. In particular, the concept of evaluation representations is not available in the contextof the quantum affine algebra Uq(g) in general. Chari and Pressley introduced and studied theconcept of minimal affinizations which place the role of evaluation modules in the sense thatthey are the simplest of the irreducible representations. A special class of minimal affinizationsis that of Kirillov-Reshetikhin modules. These modules are subject of intense studies in thelast decade because of their applications to mathematical physics. One problem of particularinterest is that of describing their characters. Chari and Moura have solved this problem wheng is of type A,B,C,D, or G by transferring the problem to the context of current algebras.More recently, Moura has initiated a program for extending this method to general minimalaffinizations. We will present some new results in this direction when g is of type E6.
Acknowledgements: This project is part of the author’s Master Thesis. She is grateful forthe guidance of Adriano Moura and the financial support from FAPESP.
References
[1] Humphreys, James E. Introduction to Lie Algebras and Representation Theory. New York: Springer, 1972.[2] Wan, Zhe-xian. Kac-Moody Algebra. Beijing: World Scientific, 1991.[3] Kac, Victor G. Infinite dimensional Lie algebras. Cambridge: Cambrige University Press, 1985.[4] Chari, Vyjayanthi; Moura, Adriano. The restricted Kirillov-Reshetikhin modules for the current and twisted
current algebras. Comm. Math. Phys. 266 no. 2, 2006. 431-454.[5] Chari, Vyjayanthi; Moura, Adriano. Kirillov-Reshetikhin modules associated to G2. Contemp. Math. 442
(2007), 41-59.[6] Moura, Adriano. Restricted limits of minimal affinizations. arXiv:0812.2238 [math.RT].[7] Hatayama, Goro; Kuniba, Atsuo; Okado, Masato; Takagi, Taichiro; Yamada, Yasuhiko. Remarks on the
Fermionic Formula. Contemp. Math. 248, 1999.
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The Gerstenhaber bracket in the Hamiltonian formalismLuiz Henrique PereiraUFPR, Brazil
We study applications of the Gerstenhaber bracket on the description of the evolution equa-tions in the context of the classical Hamiltonian formalism. We begin by comparing the wellknown rule played by the Lie, Poisson and Nijenhuis-Schouten brackets in that formalism. Next,we study the Gerstenhaber bracket and its relations to the above mentioned brackets as well asapplications in the description of classical Hamiltonian systems.
Polynomial identities of ternary algebrasLuiz A. PeresiIME-USP, Brazil
(Joint work with Murray R. Bremner, University of Saskatchewan.)The classical triple products are: Lie, Jordan and anti-Jordan. Each one of these products,
together with the polynomial identities it satisfies, defines the corresponding variety of triplesystems. Using the representations of the symmetric group S3, we classify up to equivalencethe ternary operations over the field of rational numbers.
For one representative of each equivalence class, we obtain the polynomial identities of degree≤ 5 satisfied by the ternary operation in every totally associative ternary algebra. Thesepolynomial identities define new varieties of triple systems.
The polynomial identities are obtained as the nullspace of a large matrix with integer entries.In the most difficult cases, our methods use the representation theory of the symmetric group,the Hermite normal form of an integer matrix and the Lenstra-Lenstra-Lovasz algorithm forlattice basis reduction.
References
[1] M. R. Bremner, L. A. Peresi, Classification of trilinear operations, Comm. Algebra 35: 2932-2959 (2007).[2] M. R. Bremner, L. A. Peresi, An application of lattice basis reduction to polynomial identities for algebraic
structures, Linear Alg. Appl. 430: 642-659 (2009).
Right coideal subalgebras in the quantum Borel algebra of type G2
Barbara PogorelskyUFRGS, Brazil
In this paper we describe the right coideal subalgebras containing all group-like elements ofthe multiparameter quantum group U+
q (g), where g is a simple Lie algebra of type G2, while themain parameter of quantization q is not a root of 1. If the multiplicative order t of q is finite,t > 4, t 6= 6, then the same classification remains valid for homogeneous right coideal subalgebrasof the positive part u+
q (g) of the multiparameter version of the small Lusztig quantum group.
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Lie nilpotence of skew symmetric elements in group ringsCesar Polcino MiliesIME-USP, Brazil
Let G be a group with an involution ∗ extended linearly to an involution of FG, the groupalgebra of G over a field F of characteristic different from 2. We shall survey some of the knownresults regarding the Lie structure of (FG)+ = x ∈ FG | x∗ = x and FG− = x ∈ FG | x∗ =−x the sets of symmetric and skew symmetric elements of FG respectively. Also, we shall givenecessary and sufficient conditions on G for FG− to be Lie nilpotent, in the case when G is atorsion group with no elements of order 2. This result is joint work with A. Giambruno andS.K. Sehgal.
p-adic representation of cyclic groups and pro-p groupsAnderson L.P. PortoUnB, Brazil
This study establishes that only (n+1)(n+2)2 of indecomposable ZpCpn-lattices can be obtained
from a semidirect product F o Cpn of a free pro-p group F and a cyclic group of order pn byfactoring out the commutator subgroup [F, F ].
Hereditary abelian categoriesMaria Julia RedondoUniversidad Nacional del Sur, Argentina
Given an abelian category A, we find conditions for it being hereditary by considering ahomology theory h : T → A, where T is a triangulated category. Joint work with T. Pirashvili.
The endomophisms monoid of a homogeneous vector bundleAlvaro RittatoreUniversidad de la Republica, Uruguay
A vector bundle over an abelian variety E → A is said to be homogeneous if for every a ∈ A,E ∼= t∗aE, where ta : A→ A is the translation by a. One can say that the study of homogeneousvector bundles has its origin in the work of Atiyah on vector bundles over elliptic curves (late´50s). In the ´70s, Miyanishi and Mukai described the basic properties of the category ofhomogeneous vector bundles, as well as the algebraic structure of the automorphisms groupof these bundles. Recently, Brion and Rittatore have shown that any algebraic monoid (notnecessarily affine) can be immersed as as closed submonoid of the endomorphisms monoid ofsuch a bundle. Thus, the endomorphisms monoid of homogeneous vector bundles are called toplay a role in the description of an arbitrary algebraic monoid similar to the one played by then× n-matrices in the affine case.
In this talk we will show the basic algebraic and geometric properties of the endomorphismsmonoid Endhb)(E) of a homogeneous vector bundle E → A. In particular, we will endowEndhb)(E) with a structure of vector bundle, and stablish the relationship between this structureand the structure of E → A.
This is a join work with Leticia Brambila-Paz.
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Serre relations for Lie algebras associated to quasi-Cartan matrix of type Bn and CnAntonio Daniel RiveraUAEM, Mexico
A square matrix with integer coefficients A is called a quasi-Cartan matrix, if it is symmetriz-able (that is, there exists a diagonal matrix D with positive diagonal entries such that DA issymmetric) and Aii = 2 for all i. A quasi-Cartan matrix is called a Cartan matrix, if it ispositive definite, that is, all principal minors are positive, and Aij ≤ 0 for all i 6= j.
Every semisimple Lie algebra defines a root system on the dual space of a Cartan subalgebraand a Cartan matrix, which expresses the dual of the Killing form on a root base. Serre’sTheorem gives then a representation of the given Lie algebra in generators and relations interms of the Cartan matrix. In this work, we generalize Serre’s Theorem to give an explicitrepresentation in generators and relations for any semisimple Lie algebra associated to positivequasi-Cartan matrix of type Bn or Cn.
On non-abelian tensor powers of a groupNoraı R. RoccoUnB, Brazil
Joint work with Ticianne P. Bueno - UFG.In this talk we are concerned with non-abelian tensor powers of a group G. The first task is
to extend the definition of the tensor square T2(G) = G ⊗ G to contruct a n-th tensor powerTn(G), n ≥ 2, in a reasonable way. We do that recursively by using the natural action ofG on Tn−1(G), whereas for the action of Tn−1(G) on G we take that of γn−1(G) acting byconjugation on G: the action of a monomial x1 ⊗ · · · ⊗ xn−1 on an element g ∈ G is defined bygx1⊗···⊗xn−1 := g[x1,...,xn−1] (here the conjugator is a left normed simple commutator of weightn− 1). These actions are compatible, so that the non-abelian tensor product Tn−1(G)⊗G maybe defined. There is an isomorphism Tn−1(G)⊗G ∼= G⊗Tn−1(G), x⊗ g 7→ (g⊗x)−1 and hencewe are able to define Tn(G) := Tn−1(G)⊗G.
Hecke algebras and quantum Clifford algebrasRoldao da RochaUFABC, Brazil
In the quantum Clifford algebraic framework, it is shown that the Atiyah-Bott-Shapiro mod 8periodicity theorem is modified and Hecke algebras are constructed as a very particular quantumClifford algebra. Some extensions and representations are provided.
References:[1] C. Brouder, B. Fauser, A. Frabetti, and R, Oeckl, Quantum field theory and Hopf algebracohomology, J. Phys. A37 (2004) 5895-5927[2] B. Fauser, Quantum Clifford Hopf Gebra for Quantum Field Theory, Adv. Appl. Clifford Al-gebras 13 (2003) 115-125; On the Hopf algebraic origin of Wick normal-ordering, J. Phys. A34(2001) 105-116.[3] R. da Rocha and J. Vaz, Extended Grassmann and Clifford algebras, Adv. Appl. CliffordAlgebras 16 (2006) 103-125.
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Covering coalgebras and dual non-singularityVirginia RodriguesUFSC, Brazil
Localization is an important technique in ring theory and yields the construction of severalrings of quotients. Colocalization in comodule categories has been investigated by (Jara et al.,Commun. Algebra, 34(8): 2843-2856, 2006 and Nastasescu and Torrecillas, J. Algebra, 185:203-220, 1994). In this work we look at possible coalgebra covers π : D → C that could play therole of a coalgebra colocalization. Codense covers will dualize rational extensions; a maximalcodense cover construction for coalgebras with projective covers is proposed. Also we look fordual non-singularity concept for modules which turns out to be comodule-theoretic propertythat turns the dual algebra of a coalgebra into a nonsingular ring and we look at coprimecoalgebras and Hopf algebras which are non-singular as coalgebras. This is a join work withC. Lomp (UP - Porto - Portugal).
Prym-Tyurin varieties via Hecke algebrasAnita RojasUniversidad de Chile, Chile
Joint work with A. Carocca, H. Lange, R. E. Rodrıguez.We present some of the results developed in [1]. Denote by G a finite group and π : Z → Y
a Galois covering of smooth projective curves with Galois group G. For every subgroup H ofG there is a canonical action of the corresponding Hecke algebra Q[H\G/H] on the Jacobianof the curve X = Z/H. To each rational irreducible representation W of G we associate anidempotent in the Hecke algebra, which induces a correspondence of the curve X and thus anabelian subvariety P of the Jacobian JX. We give sufficient conditions onW, H, and the actionof G on Z for P to be a Prym-Tyurin variety. We obtain many new families of Prym-Tyurinvarieties of arbitrary exponent in this way.
References
[1] Angel Carocca, Herbert Lange, Rubı Rodrıguez and Anita Rojas. Prym-Tyurin varieties and Hecke Algebras.Journal fur die reine und angewandte Mathematik (Crelle’s journal) (2009), Accepted.
Trace forms and ideals on commutative algebras satisfying an identity of degree fourCarlos Rojas-BrunaDuocUC, Chile
We study a variety of commutative algebras satisfying the identity :
((xy)z)t− ((xy)t)z + ((yt)x)z − ((yt)z)x+ ((yz)t)x− ((yz)x)t = 0.
These algebras appeared in the classification of the degree four identities given by L. Carini,I.R. Hentzel and Piaccentini-Cattaneo. We prove the existence of a trace form, and also thatthe existence of a non degenerate trace form give us a relation between these algebras and thevariety defined by the identity :
2β
(xy)2 − x2y2
+ γ
((xy)x)y + ((xy)y)x− (y2x)x− (x2y)y
= 0.
Finally we prove that Ass[A] and N(A) are ideals in these algebras.
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Subgroups embedded in R. Thompson’s group VOlga Patricia Salazar-DıazUniversidad Nacional de Colombia, Colombia
Joint work with Collin Bleak.We explore, using the dynamics of the elements of Richard Thompson’s group V , whether itcontains subgroups isomorphic to the direct products ZZ2 ∗ ZZ or ZZn ∗ ZZm.
On the rank of the fibres of rational elliptic surfacesCecılia SalgadoUniversite Paris VII, France
We compare the generic and special ranks of the fibres of rational elliptic surfaces over numberfields. We show, for a large class of these surfaces, that there are infinitely many fibres withrank at least the generic rank plus two.
Galois correspondence for α-partial Galois Azumaya extensionsAlveri A. Sant’AnaUFRGS, Brazil
In 1965 appeared three diferent papers, independently authored by F. R. DeMeyer, T. Kan-zaki and M. Harada, which investigated central Galois algebras, i. e., Galois algebras Λ over kwith group G such that k is the center of Λ. Later, R. Alfaro and G. Szeto (1995 and 1997)generalized the class of central algebras to the class of Galois Azumaya extensions, i.e., Galoisextensions of an Azumaya algebra, where they characterized such extensions in terms of theAzumaya property and H-separability of the induced skew group ring. In the 97’s paper, Alfaroand Szeto gave two nice one-to-one correspondence theorems for a Galois Azumaya extensionsgeneralizing the corresponding DeMeyer’s result.
In this talk we give a version of these correspondence theorems for α-partial Galois Azumayaextensions obtaining an extension of the Alfaro and Szeto’s results to the setting of partialGalois theory.
This is a joint work with A. Paques (UFRGS/Brasil) and V. Rodrigues (UFSC/Brasil).
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Somas de quadrados e formas hermitianasClotilzio Moreira dos SantosIBILCE - UNESP, Brazil
Escrever −1 como soma de quadrados em um corpo F (de caracterıstica diferente de dois) eum problema antigo. O menor numero de parcelas (se existir) tal que isto ocorre e dito nıvel(level) do corpo F , caso contrario o nıvel e infinito. Em 1965 A. Pfister [1] demonstrou quese o nıvel de um corpo e finito, entao ele e uma potencia de dois. Neste mesmo trabalho,A. Pfister resolve parcialmente, mas no caso interessante, o problema de escrever um produtode n (n > 0) quadrados como soma de n quadrados; a saber, no caso em que n e uma potenciade 2. A agitacao entusiasmada causada pela elegante demonstracao de Pfister tem contribuıdomuito para o rapido desenvolvimento da teoria algebrica de formas quadraticas, depois de 1965.
O nıvel de um corpo e um invariante importantıssimo na teoria de formas quadratica, poisum corpo possui nıvel infinito se, e somente se, ele possui pelo menos uma ordem. Vale a penacitar tambem o celebre princıpio Local-Global de Pfister que nos da, como consequencia, que seo nıvel de um corpo e finito, entao o anel de Witt das formas quadraticas sobre ele e de torcao.Em se tratando de formas quadraticas sobre aneis mais gerais, em 1980 Dai, Lam e Peng [2]usando argumentos topologicos, provou que qualquer numero natural maior que zero pode sero nıvel de algum anel. Sobre aneis com divisao existem contribuicoes recentes de varios autoresna tentativa de se calcular o nıvel. Por exemplo, Lewis [3], demonstrou que existem algebras dequaternios com divisao com nıveis 2k e 2k + 1, k > 0. No caso de nıvel hermitiano, o resultadomais recente que conhecemos e de 1988, devido a Lewis [6].
A grande maioria das tecnicas usadas para o calculo do nıvel esta relacionada a formasquadraticas de Pfister. Neste trabalho apresentaremos alguns destes resultados, Apresentare-mos tambem, alguns resultados de pesquisa sobre o nıvel hermitiano relacionado com formashermitianas de Pfister sobre alguns aneis com divisao. Existem varios problemas interessantesem aberto nesta teoria. Um deles e o de se saber quando uma forma (quadratica ou hermitiana)de Pfister e hiperbolica, caso for isotropica; ou quando o conjunto de valores representado porela e um grupo, fatos que ocorrem quando se trata de formas de Pfister sobre corpos. Seraexibida situacoes onde ja se conhece e situacoes onde nao se tem ainda uma resposta.
Bibliografia:[1] A. Pfister “Zur Darstelling von -1 als Summe von Quadraten in einem Koper” J. London
Math. Soc. 40 (1965), 159-165.[2] Z.D. Dai, T.Y. Lam and C.K. Peng “Levels in algebra and topology” Bull A.M.S.v.3 n.2
(1980) 845-848.[3] D.W. Lewis “Levels of quaternions algebra”, Rocky Mountain J. 19(1989), 787-792.[4] D.B. Leep “Levels of division algebras” Glasgow Math. J. 32(1990), 365-370.[5] D.W. Hoffann “Levels of quaternion algebras” Arch. Math. 90(2008), 401-411.[6] D.W. Lewis “Sums of hermitian squares” J. Of Algebra 115 (1988), 466-480.
Homologıa de Hochschild y homologıa cıclica de algebras de dimension finitaNatalia Abad SantosUniversidad Nacional del Sur, Argentina
Toda k-algebra de dimension finita sobre un cuerpo algebraicamente cerrado k es Moritaequivalente a un algebra basica, y esta se puede ver como el algebra de caminos kQ de un carcajQ modulo un ideal bilatero I, esto es, A = kQ/I, donde carcaj es un grafo orientado finito. Eneste trabajo se estudia la homologıa de Hochschild y la homologıa cıclica para ciertas algebrasde caminos.
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Classification of involutions on incidence algebrasEdnei A. Santulo Jr.UEM, Brazil
Joint with: Rosali Brusamarello and Erica Zancanella.We classify the involutions on an incidence algebra of a connected finite poset X over a field
K. In other words, let X be a connected finite poset, K be a field and I(X,K) be the incidencealgebra of X over K. Let ρ1, ρ2 : I(X,K)→ I(X,K) be two K-linear involutions on I(X,K).We give necessary and sufficient conditions for the existence of an automorphism Ψ of I(X,K)such that Ψρ1 = ρ2Ψ.
Cluster algebras and cluster categoriesRalf SchifflerUniversity of Connecticut, USA
In this mini-course we will introduce cluster algebras, cluster categories and cluster-tiltedalgebras, and we will explain results that are specific to each of them as well as the connectionsbetween all three of them. The (tentative) outline of the lectures is as follows:
Lecture 1: Cluster algebras: motivation, sketch of definition, examples, cluster variables andcoefficients, Laurent phenomenon, positivity conjecture, classifications.
Lecture 2: Cluster categories: definition, cluster-tilting theory, relation to cluster algebras,generalizations.
Lecture 3: Cluster-tilted algebras: various definitions, relation to tilted algebras, examples.Lecture 4: Cluster algebras of finite mutation type: triangulated surfaces, Laurent expansion
formulas, positivity theorem.
On Kac Wakimoto conjecture about dimension of simple representation of Lie superalgebraVera SerganovaUniversity of California, USA
Let g be a simple classical Lie superalgebra. Some time ago Kac and Wakimoto conjecturedthat a simple finite dimensional g module has a non zero superdimension if and only if its degreeof atypicality equals the defect of g. We will give a proof of this conjecture in some cases.
Construcao de algebras primas degeneradas atraves de algebra de GrassmannIvan ShestakovIME-USP, Brazil
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Free unitary and symmetric pairs in group rings of characteristic 2Mazi ShirvaniUniversity of Alberta, Canada
Let kG be the group ring of a finite group of odd order G over a field k of characteristic2, and let ∗ be an involution of G, extended in the natural way to an involution of kG. Wedetermine when kG contains two ∗-symmetric (respectively, ∗-unitary) units which generate afree group of rank 2.
State-closed groupsSaid SidkiUnB, Brazil
Automorphisms of one-rooted regular trees T (Y ) indexed by finite sequences from a finiteset Y of size m ≥ 2, have a natural interpretation as automata on the alphabet Y , and withstates which are again automorphisms of the tree. A subgroup of the group of automorphismsA (Y ) of the tree is said to be state-closed, in the language of automata (or self-similar in thelanguage of dynamics) of degree m provided the states of its elements are themselves elementsof the same group.
There is a very large variety of types of state-closed groups. Normally groups in this classare considered under additional conditions such as having finite number of generators, beingfinite-state, or satisfying certain identities. Some of the examples are the torsion groups ofGrigorchuk and of Gupta-Sidki, the torsion-free BSV and Basilica weakly branch groups, allfinitely generated and finite-state. There are also the faithful representations of affine groupsover the integers and of free groups as finite-state and state-closed groups.
State-closed free abelian groups of degree 2 and having finite rank were characterized byNekrashevych-Sidki in 2004 and finitely generated state-closed torsion-free nilpotent groups ofdegree m ≥ 2 were studied by Berlatto-Sidki in 2007.
We will review in this lecture known results and recent work by the author and Andrew Brun-ner on state-closed general abelian groups without restrictions on finite generation or degree.
On Z2-graded identities and central polynomials of the Grassmann algebraViviane Ribeiro Tomaz da SilvaUFMG, Brazil
Let E be the infinite-dimensional Grassmann algebra over a field F of characteristic zero andconsider L the F -vector space spanned by all generators of E. Let ϕl be any fixed automorphismof E of order 2 such that L is an homogeneous subspace. In this talk, we present the generators ofthe ideal of the Z2-graded identities (based on joint work with Prof. Onofrio Mario Di Vincenzofrom University of Basilicata, Italy) and we get a complete description of the graded centralpolynomials of the superalgebra (E,ϕl).
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The hyperbolic propertyA. C. Souza FilhoEACH-USP, Brazil
Hyperbolic groups were firstly studied by Gromov, see [1]. An important result states that ifa group is hyperbolic it has no subgroup isomorphic to Z× Z. According to this result we saythat an associative algebra has the hyperbolic property if any Z-order of the algebra is such thatthe unit group of the order has no subgroup isomorphic to Z×Z. In [3], it was studied the finitesemigroups which the semigroup algebra over the racional numbers has the hyperbolic property.A classification of these semigroups was started in [2]. In a recent article we completely classifiedthe finite semigroups with this property. Following the techniques developed in [3], we also statea structure theorem for alternative finite dimensional algebras over the racional numbers suchthat, for any Z-order, the unit loop of the order has no subgroup isomorphic to Z× Z.
References
[1] M. Gromov, Hyperbolic Groups, in Essays in Group Theory, M. S. R. I. publ. 8, Springer, 1987, 75-263.[2] E. Iwaki, S. O. Juriaans, A. C. Souza Filho, Hyperbolicity of Semigroup Algebras, J. Algebra, (319)(12)(2008),
500-515.[3] A.C. Souza Filho, Sobre uma Classificacao dos Aneis de Inteiros, dos Semigrupos Finitos e dos RA-Loops
com a Propriedade Hiperbolica (On a Classification of the Integral Rings, Finite Semigroups and RA-Loopswith the Hyperbolic Property), PhD. Thesis, IME-USP, Sao Paulo, 2006, 108 pages.
Galois geometries and coding theory: two interacting research areasLeo StormeGhent University, Belgium
In coding theory, a linear [n, k, d]-code C over the finite field of order q is a k-dimensionalsubspace of the n-dimensional vector space V (n, q) over the Galois field Fq of order q. Theminimum distance d of the code C is the minimal number of positions in which two distinctcodewords of C differ.
A finite projective space PG(N, q) of dimensionN over the Galois field Fq of order q arises fromthe (N + 1)-dimensional vector space V (N + 1, q) over Fq, when one identifies the vector linesof V (N + 1, q) as being the points of PG(N, q). These finite projective spaces PG(N, q) are alsocalled Galois geometries. In Galois geometries, many different substructures are investigated.
Many problems in coding theory have an equivalent statement as problems on specific sub-structures in Galois geometries. This means that techniques from Galois geometries can be usedto solve problems in coding theory.
Concrete examples include the link between linear MDS codes and arcs in Galois geometries,linear codes meeting the Griesmer bound and minihypers in Galois geometries, covering radiusand saturating sets, linear codes arising from the incidence matrices of geometrical structures,and functional codes.
During this talk, we will describe a number of these links and show how techniques fromGalois geometries contribute to coding theory.
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Projection lattices pointsJoao Eloir StrapassonUFPR, Brazil
Lattices which are the projection of Zn are important for the continuous source coding /channel. Present a construction in which lattices can be good for approximate projections.
On the quasi-cyclicity and linearity of the Gray image of a code over a Galois ringHoracio Tapia-RecillasUniversidad Autonoma Metropolitana-I, Mexico
(Joint work with C.A. Lopez-Andrade)In ([1]) necessary and sufficient conditions for the Gray image of a code over the ring Zp2 where
p is any prime to be quasi-cyclic are given. Also in ([2]), necessary and sufficient conditions forthe Gray image of a linear code over the ring Z4 can be found. In this talk similar results arepresented for codes over a Galois ring of the form GR(p2,m).
References
[1] Ling, S. and Blackford, J.T. “Zpk+1 -Linear Codes.” IEEE Trans. Inf. Theory, vol.48, pp. 2592-2605, (2002).[2] A.R. Jr. Hammonds, P.V. Kummar, R. Calderbank, N.J.A. Sloane, and P. Sole, “The Z4-linearity of Kerdock,
Preparata, Goethals and related codes.” IEEE Trans. Inf. Theory, vol.40, pp. 301-319, (1994).
An introduction to central simple algebras and the Brauer groupEduardo TenganICMC-USP, Brazil
Central Simple Algebras are finite dimensional algebras over a field K with no non-trivialtwo-sided ideals (they are ‘simple’) and whose centre is precisely K (they are ‘central’). Familiarexamples include the ring Mn(K) of n × n matrices and division algebras (‘non-commutativefields’) over K. For a given field K, the central simple algebras over K can be organized in agroup Br(K), called the Brauer group of K, which can be thought of as a ‘directory’ of alldivision algebras over K. Alternatively, Br(K) can also be defined in terms of Galois cohomolgyof K, and therefore it can be also be viewed as an arithmetic invariant of K. The interplaybetween these two points of view is a very rich one, with applications not only to the theoryof Central Simple Algebras, but also to Number Theory, Field Theory, Quadratic Forms andAlgebraic Geometry.
This mini-course covers some of the classical theory of central simple algebras and the Brauergroup. We will start with the basic theory and then briefly review some of the results of Galoiscohomology that we will need. We end up with two non-trival computations of the Brauergroup, for local and global fields.
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The algebra of differential operators associated to a weight matrixJuan TiraoUniversidad Nacional de Cordoba, Argentina
Given a weight matrix W (x) of size N on the real line one constructs a sequence of matrixvalued orthogonal polynomials, (Pn). We will introduce the algebra D(W ) of all differentialoperators D with matrix coefficients such that PnD = L(n)Pn, with L(n) in the algebra A ofN × N complex matrices. Then we will exhibit certain representations of this algebra, provethat it is a ∗-algebra and give a precise description of its isomorphic image inside the directproduct of a countable number of copies of A.
TbaGordana TodorovNortheastern University, USA
On the representations of the 1D N-Extended SuperalgebraFrancesco ToppanCBPF, Brazil
We discuss the recent progresses in the construction and classification of the representa-tions of the 1D N-Extended Superalgebra (the graded algebra of the Supersymmetric QuantumMechanics) in terms of a finite number of time-dependent fields. General properties of thesupersymmetric invariants are also discussed.
Castle curves and codesFernando TorresUNICAMP, Brazil
We introduce two types of curves of interest for Coding Theory purposes: the so-called Castleand weak Castle curves. We discuss the main properties of codes arising from these curves.
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Generalized octonionic structures on S7 and Clifford algebrasMarcio Andre TraeselUFABC, Brazil
In this paper we investigate the deformations in the octonionic algebra — from the vectorbundle to the whole exterior and Clifford bundles on the 7-sphere S7, using the Clifford algebra-parametrized octonionic formalism. Using the products between multivectors of C`0,7 (theClifford algebra over the metric vector space R0,7) and octonions, resulting in an octonion, andleading to the non-associative standard octonionic product in a particular case, we generalizethe formalism previously introduced in [arXiv:math-ph/0603053v1], also associated with thetransformation rules for bosonic and fermionic fields on the tangent bundle over the 7-sphereS7. This generalization is accomplished also in the formalism of u-products, where u ∈ C`0,7is fixed, but arbitrary. The octonionic product generalization on S7 is explicitly dependent onthe point where the tangent bundle, and the exterior and the Clifford bundles are regarded.The formalism presented in this paper concerns how to extend the octonionic product on thetangent bundle on the basis manifold S7 to the whole exterior bundle and the Clifford bundleon S7.
On the representation dimension of some classes of tame algebrasSonia TrepodeUniversidad Nacional de Mar del Plata, Argentina
Joint with I. Assem and F. Coelho. This is a work in progress. The notion of representationdimension was introduced by Auslander in the 70’s as a measure of how far an algebra is to be offinite representation type. He proved that representation finite algebras are characterised by thefact that their representation dimensions is at most two. So it is reasonable to ask if the tamealgebras have representation dimensions at most 3. We study the representation dimensions ofsome particular classes of tame algebras
Irreducible bimodules over alternative superalgebrasMaria TrushinaMoscow City Pedagogical University, Russia
Let F be an algebraically closed field of characteristic 6= 2. In a joint work with I. P. Shestakovwe describe the irreducible bimodules over all finite dimensional alternative superalgebras overF.
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Multiplicity formulae for admissible restriction of Discrete Series representationsJorge VargasFAMAF-CIEM, Argentina
We fix H ⊂ G reductive, connected matrix groups and π an irreducible square integrablerepresentation of G. Whenever G is compact (resp. H is a maximal compact subgroup) themultiplicity of each irreducible H-factor of π restricted to H may be computed in terms of:infinitesimal characters (resp. Harish Chandra parameters), Weyl group, and ONE partitionfunction, Kostant-Heckman (resp. Blattner) formulae. Assume π, H are so that π restrictedto H is admissible, in joint work with Michel Duflo, we have obtained a sufficient conditionto express the multiplicity of each irreducible factor in the framework of above. We also showthe formula holds for (G,H) a symmetric pair. In order to show our formula, we consider thecoadjoint orbit Ω attached to the Harish-Chandra parameter of π. Then, by means of “discrete”and “continuos” Heaviside functions we relate the multiplicity of each irreducible H-factor of πrestricted to H and the push forward to h∗ of the Liouville measure for Ω.
Some algebras with directed gluingsRosana Retsos Signorelli VargasEACH-USP, Brazil
For an algebra A denote by LA = X ∈ indA : pdY ≤ 1 for each predecessor Y of Xand RA = X ∈ indA : idY ≤ 1 for each successor Y of X. These subcategories played animportant role in the study of some algebras like quasi-tilted algebras, shod algebras and lauraalgebras. In this work we study the class of algebras satisfying the property: all indecomposableprojective A-modules Px and all indecomposable injective A-module Ix lies in LA ∪ RA. Sucha class includes for exemplo, shod algebras.
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Hopf algebra of dimension 16Cristian VayUniversidad Nacional de Cordoba, Argentina
We complete the classification of Hopf algebras of dimension 16 over an algebraically closedfield of characteristic zero. We show that a non-semisimple Hopf algebra of dimension 16, haseither the Chevalley property or its dual is pointed. This work was conducted with GastonGarcia.
About Quasi-Jordan algebras generated by dialgebras and their relation with Leibniz algebrasRaul VelasquezUniversidad de Antioquia, Colombia
Velasquez and Felipe introduced the notion of a quasi-Jordan algebra (see [3]). These algebrassatisfy the right-commutativity a(bc) = a(cb) and the right quasi-Jordan identity (ba2)a =(ba)b2.
More recently, Kolesnikov, used techniques of conformal algebras related with dialgebras(see [2]), and Bremner, used polynomial identities (see [1]), to introduce a new definition ofquasi-Jordan algebras based on the prouct
x y :=12
(x a y + y ` x),
where x, y are elements in a dialgebra (D,a,`). These product satisfies the right-commutativity,the right quasi-Jordan and the associative-derivation identity: (a, b2, c) = 2(a, b, c)b.
In this talk we show that there are inner derivations (classical and left derivations) for quasi-Jordan algebras introduced by Kolesnikov and Bremner (K-B). Additionally, we show a rela-tionship between K-B quasi-Jordan algebras and Leibniz algebras and few results about splitquasi-Jordan algebras.
References
[1] M. R. Bremner, A note on quasi-Jordan algebras, preprint (2009) (http://math.usask.ca/ brem-ner/research/publications/qjnote.pdf).
[2] P. S. Kolesnikov, Varieties of dialgebras and conformal algebras, Sibirsk. Mat. Zh. Vol. 49, No. 2 (2008),322-339.
[3] R. Velasquez, R. Felipe, Quasi-Jordan algebras, Comm. Algebra Vol. 36, No. 4 (2008), 1580-1602.[4] R. Velasquez, R. Felipe, Split dialgebras, split quasi-Jordan algebras and regular elements, J. Algebra Appl.Vol. 8 No. 2 (2009), 191-218.
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Free subgroups of U(ZG) generated by alternating unitsPaula Murgel VelosoIME-USP, Brazil
In this work, done in collaboration with Jairo Goncalves, we search for free subgroups in theunit group U(ZG) of the group ring ZG, using an alternating unit and a bicyclic unit, and apair of alternating units.
In a similar investigation [1], Goncalves and del Rio show that in the integral group ring ZG,with G a nonabelian group with order coprime with 6, there always exists a pair formed by aBass cyclic unit and a bicyclic unit, such that the subgroup they generate is not 2-related.
Let G be a group containing x ∈ G an element of odd order n, and c ∈ N, 1 ≤ c < 2n suchthat (c, 2n) = 1. Then, according to [4, Lemma 10.6], the element
gc(x) :=c−1∑i=0
(−x)i = 1− x+ x2 − . . .+ xc−1 ∈ ZG
is a unit in ZG, called an alternating unit.Just like some other types of units in group rings [2], [3], alternating units defined in a
homomorphic image of ZG may be lifted to alternating units in ZG. So the research techniquemust involve studying the behavior of pairs formed by an alternating unit unit and a bicyclicunit, and pairs of alternating units in group rings ZH, with H minimal groups that could becounter-examples to the result, and classifying such groups as well.
References
[1] J. Z. Goncalves, A. del Rio, Bicyclic units, Bass cyclic units and free groups, J. Group Theory 11 (2008),247–265.
[2] J. Z. Goncalves, D. S. Passman, Free unit groups in group algebras, J. Algebra 246 (2001), 226–252.[3] J. Z. Goncalves, P. M. Veloso, Special units, unipotent units and free groups in group algebras, AMS
Contemporary Mathematics, 2009 (submitted).[4] S. K. Sehgal, Units in Integral Group Rings, Longman Scientific & Technical Press, Harlow, 1993.
Deformations of restricted simple Lie algebras in positive characteristicFilippo VivianiUniversidade Roma Tre, Italy
Simple Lie algebras over an algebraically closed field F of positive characteristic (differentfrom 2 and 3) have recently been classified by Block-Wilson-Premet-Strade, answering positivelyto a conjecture of Kostrikin and Shafarevich. Among the simple Lie algebras, a very importantsubclass is formed by the restricted simple Lie algebras, that are the ones which occur as Liealgebras of group schemes over F . In this talk, I will report on the explicit computation of theordinary and restricted infinitesimal deformations of the restricted simple Lie algebras.
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On a group of permutation polynomialsQiang (Steven) WangCarleton University, Canada
A polynomial f over a finite field Fq is called a permutation polynomial if the mappingf : Fq → Fq permutes the elements of Fq. Permutation polynomials were first investigatedby Hermite, and since then, many studies concerning them have been devoted. In the last 20years there has been a revival in the interest for permutation polynomials, in part due to theircryptographic applications. Let r, d be a positive integers satisfying d | q − 1 and G(d, q) bethe set of permutation polynomials of form xrf(xq−1)/d such that deg f(x) < d, (r, (q-1)/d)=1, and 1 ≤ r ≤ (q − 1)/d. It was shown by Wan and Lidl that G(d, q) is isomorphic to ageneralized wreath product G oR H, where G = Z(q−1)/d is the additive group, R = (Z(q−1)/d)∗
is the multiplicative group acting on G, and H = Sd is the symmetric group on Zd. In thistalk I will give a brief introduction to the structure of group G(d, q) and then describe how theinverse of an element in G(d, q) looks like explicitly.
Some classification results on nonassociative graded algebrasLuis Alberto Wills-ToroUniversidad Nacional Colombia, Colombia
We present recent results on finite dimensional nonassociative algebras graded by a finiteabelian group. In particular, we explore nonassociative algebras over the reals and providesome classification results. We include as well their cohomological characterization.
Spectra of quantum Schubert cells and quantum flag varietiesMilen YakimovLouisiana State University, USA
De Concini, Kac, and Procesi defined a family of subalgebras Uwq of a quantized universalenveloping algebra Uq(g), associated to the elements of the corresponding Weyl group W . Theyare deformations of universal enveloping algebras of nilpotent Lie algebras and can be consideredas quantized algebras of functions on Schubert cells.
We will describe explicitly all torus invariant prime ideals of the algebras Uwq , constructefficient generating sets, and describe the poset of those ideals. We will then apply these resultsto classify the torus invariant prime ideals of quantum partial flag varieties.
Genus for groupsPavel ZalesskiiUnB, Brazil
We intruduce a notion of genus for a group with respect to some natural family of groups asa number of isomorphism classes of groups of the family having the same profinite completion.The results on finiteness of genus for several important families of groups and calculation ofgenus in several classes of groups will be presented.
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Grupos agindo sobre arvores e automatosFlavia Ferreira Ramos ZapataUnB, Brazil
O grupo de automorfismos A da arvore regular n-aria tem despertado grande interesse nosultimos anos por ser uma fonte de exemplos de novos fenomenos em Teoria Combinatoria deGrupos bem como sua conexao com outras areas, tais como teoria do automato e sistemasdinamicos. Cada automorfismo de A pode ser interpretado de uma maneira simples comoautomato. Vamos apresentar exemplos classicos de grupos de automorfismos de arvores bemcomo problemas envolvendo tais. Dentre os problemas que nos interessam esta o de saber se paracertos automorfismos de infinitos estados, existe um automato de finitos estados que reconhecea mesma liguagem que o anterior.
Pro-finite limit groupsTheo ZapataUnB, Brazil
We begin a study of a pro-finite analogue of limit groups via extensions of centralizers. Somepartial results on group theoretic structure and homological properties on this new class ofpro-finite groups are presented.
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List of participantsRicardo Alfaro (University of Michigan-Flint, USA)
Eli Aljadeff (Technion, Israel)[email protected]
Hellen Angelica Silva Almeida (PUC-RJ, Brazil)[email protected]
Edson Ribeiro Alvares (UFPR, Brazil)[email protected]
Marcelo Muniz S. Alves (UFPR, Brazil)[email protected]
Nicolas Andruskiewitsch (Universidade de Cordoba, Argentina)[email protected]
Jaime Edmundo Apaza Rodriguez (UNESP, Brazil)[email protected]
Luis Arenas-Carmona (Universidad de Chile, Chile)[email protected]
Michela Artebani (Universidad de Concepcion, Chile)[email protected]
Dalia Artenstein (Udelar, Uruguay)[email protected]
Ibrahim Assem (Universite de Sherbrooke, Canada)[email protected]
Samir Assuena (IME-USP, Brazil)[email protected]
Dirceu Baggio (UFSM, Brazil)[email protected]
Alp Bassa (CWI & Leiden University, The Netherlands)[email protected]
Peter Beelen (Technical University of Denmark, Denmark)[email protected]
Antonio Behn (Universidad de Chile, Chile)[email protected]
Viktor Bekkert (UFMG, Brazil)[email protected]
Laerte Bemm (UFRGS, Brazil)[email protected]
German Benitez Monsalve (Universidad de Antioquia, Colombia)[email protected]
Vagner Rodrigues de Bessa (UnB, Brazil)[email protected]
Fernando Araujo Borges (UFPR, Brazil)[email protected]
Herivelto Borges (The University of Texas at Austin, USA)[email protected]
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Carina Boyallian (Universidad Nacional de Cordoba, Argentina)[email protected]
Maria Bras-Amoros (Universitat Rovira i Virgili, Spain)[email protected]
Matheus Batagini Brito (UNICAMP, Brazil)[email protected]
Rosali Brusamarello (UEM, Brazil)[email protected]
Andre Bueno (USP, Brazil)[email protected]
Thomas Bunke (IME-USP, Brazil)[email protected]
Juan Manuel Burgos Mieres (UDELAR, Uruguay)jmburgos [email protected]
Juan Carlos Bustamante (Universidad San Francisco de Quito, Ecuador)[email protected]
Paula Andrea Cadavid (IME-USP, Brazil)[email protected]
Joyce Caetano (UFABC, Brazil)[email protected]
Antonio Cafure (UNGS, Argentina)[email protected]
Leandro Cagliero (FaMAF - CONICET, Argentina)[email protected]
Yohny Ferney Calderon Henao (Universidad de Antioquia, Colombia)[email protected]
Jaleydi Cardenas Poblador (Universidad de La Salle, Colombia)[email protected]
Angel Carocca (PUC-Chile, Chile)[email protected]
Cicero Carvalho (Universidade Federal de Uberlandia, Brazil)[email protected]
John Hermes Castillo (IME-USP, Brazil)[email protected]
Diane Castonguay (UFG, Brazil)[email protected]
Francis Castro (University of Puerto Rico, Puerto Rico)[email protected]
Felipe L. Castro (UFRGS, Brazil)[email protected]
Sheila Chagas (UFAM, Brazil)[email protected]
Claudia Chaio (Universidad Nacional de Mar del Plata, Argentina)[email protected]
82
Gladys Chalom (IME-USP, Brazil)[email protected]
John Clark (University of Otago, New Zealand)[email protected]
Flavio Coelho (IME-USP, Brazil)[email protected]
Juliana Coelho (UFF, Brazil)[email protected]
Luciane Quoos Conte (UFRJ, Brazil)[email protected]
Mariana Cornelissen (UFSJ, Brazil)[email protected]
Wagner Cortes (UFRGS, Brazil)[email protected]
Severino Collier Coutinho (UFRJ, Brazil)[email protected]
Ricardo Dahab (UNICAMP, Brazil)[email protected]
Alexei Davydov (University of Sydney, Australia)[email protected]
Jose Antonio de la Pena (UNAM, Mexico)[email protected]
Matias L. del Hoyo (UBA, Argentina)[email protected]
Michael Dokuchaev (IME-USP, Brazil)[email protected]
Gisele Ducati (UFABC, Brazil)[email protected]
Claudia Egea (Universidad Nacional de Cordoba, Argentina)[email protected]
Oscar Andres Espinel Montana (Universidad Nacional de Colombia, Colombia)[email protected]
Eduardo Esteves (IMPA, Brazil)[email protected]
Sebastian Falu (Universidade Nacional del Sur, Argentina)[email protected]
Marco Farinati (UBA, Argentina)[email protected]
Frederico Sercio Feitosa (UFJF, Brazil)[email protected]
Bojana Femic (Math. Inst. of Serbian Academy of Sci. and Arts, Serbia)[email protected]
Elsa Fernandez (Univ. Nacional de la Patagonia San Juan Bosco, Argentina)[email protected]
83
Edison Alberto Fernandez Culma (Universidad Nacional de Cordoba, Argentina)[email protected]
Raul Antonio Ferraz (IME-USP, Brazil)[email protected]
Vitor Ferreira (IME-USP, Brazil)[email protected]
Walter Ferrer Santos (Universidad de la Republica, Uruguay)[email protected]
Miguel Ferrero (UFRGS, Brazil)[email protected]
Saradia Della Flora (UFRGS, Brazil)[email protected]
Daiana Flores (UFRGS, Brazil)[email protected]
Erica Zancanella Fornaroli (UEM, Brazil)[email protected]
Daiane Silva de Freitas (UFRGS, Brazil)[email protected]
Jose Antonio O. Freitas (UNICAMP, Brazil)[email protected]
Vyacheslav Futorny (IME-USP, Brazil)[email protected]
Arnaldo Garcia (IMPA, Brazil)[email protected]
Gaston Andres Garcia (Universidad Nacional de Cordoba, Argentina)[email protected]
Agustin Garcia Iglesias (FaMAF - UNC, Argentina)[email protected]
Maria Andrea Gatica (Universidad Nacional de La Pampa, Argentina)[email protected]
Sudhir Ghorpade (Indian Institute of Technology, India)[email protected]
Antonio Giambruno (Universita di Palermo, Italy)[email protected]
Maria de Lourdes Giuliani (UFABC, Brazil)[email protected]
Paula Olga [email protected]
Luciane Gobbi (UFRGS, Brazil)[email protected]
Hemar Godinho (UnB, Brazil)[email protected]
Jairo Zacarias Goncalves (IME-USP, Brazil)[email protected]
84
Luciano Javier Gonzalez (Universidad Nacional de La Pampa, Argentina)[email protected]
Victor Gonzalez Aguilera (UTFSM, Chile)[email protected]
Viviana Gubitosi (Canada)[email protected]
Marines Guerreiro (UFV, Brazil)[email protected]
Claus Haetinger (UNIVATES, Brazil)[email protected]
Estanislao Herscovich (Universidad De Buenos Aires, Argentina)[email protected]
Eduardo Hoefel (UFPR, Brazil)[email protected]
Alexander Holguin-Villa (IME-USP, Brazil)[email protected]
Masaaki Homma (Kanagawa University, Japan)[email protected]
Kiyoshi Igusa (Brandeis University, USA)[email protected]
Surender K. Jain (Ohio University, USA)[email protected]
Marcos Jardim (UNICAMP, Brazil)[email protected]
Sandra Mara Alves Jorge (CEFET-MG, Brazil)[email protected]
Iryna Kashuba (IME-USP, Brazil)[email protected]
Christian Kassel (CNRS & Universite de Strasbourg, France)[email protected]
Takao Kato (Yamaguchi University, Japan)[email protected]
Patricia Massae Kitani (IME-USP, Brazil)[email protected]
Alexandr Kornev (IME-USP, Brazil)[email protected]
Plamen Koshlukov (IMECC-UNICAMP, Brazil)[email protected]
Alicia Labra (Universidad de Chile, Chile)[email protected]
Antonio Laface (Universidad de Concepcion, Chile)[email protected]
Joao Roberto Lazzarin (UFSM, Brazil)[email protected]
85
Patrick Le Meur (ENS Cachan, France)[email protected]
Fernando Levstein (UNC FaMAF, Argentina)[email protected]
Igor Lima (IMECC-UNICAMP, Brazil)[email protected]
Victor Luiz Salgado de Lima (UFPR, Brazil)[email protected]
Thierry Petit Lobao (UFBA, Brazil)[email protected]
Victor Gonzalo Lopez-Neumann (Universidade Federal de Uberlandia, Brazil)[email protected]
Tiago Macedo (UNICAMP, Brazil)[email protected]
Ulisses Alves Maciel Neto
Eduardo Marcos (IME-USP, Brazil)[email protected]
Oscar Marquez (Universidad Nacional de Cordoba, Argentina)[email protected]
Renato Alessandro Martins (IME-USP, Brazil)[email protected]
Renato Vidal Martins (UFMG, Brazil)[email protected]
Guillermo Matera (Universidad Nacional de General Sarmiento, Argentina)[email protected]
Olivier Mathieu (Lyon, France)[email protected]
Francisco Medeiros (IME-USP, Brazil)[email protected]
Thiago Castilho de Mello (IMECC-UNICAMP, Brazil)[email protected]
Margarida Melo (Universidade de Coimbra, Portugal)[email protected]
Gabriel Minian (Universidad de Buenos Aires, Argentina)[email protected]
C. Moreno (City University of New York, USA)
Agustin Moreno Canadas (Universidad Nacional de Colombia, Colombia)[email protected]
Andrea Morgado (UFRGS, Brazil)andrea [email protected]
Beatriz Motta (UNICAMP, Brazil)[email protected]
86
Adriano Moura (UNICAMP, Brazil)[email protected]
Enric Nart (UAB, Spain)[email protected]
Carlos Nascimento (UNICAMP, Brazil)[email protected]
Ruth Nascimento (UFPR, Brazil)ruth [email protected]
Frank Neumann (University of Leicester, UK)[email protected]
Akira Ohbuchi (Tokushima University, Japan)[email protected]
Joacir Lucas de Oliveira (UFPR, Brazil)joacir [email protected]
Peterson Oliveira (Instituto Federal do Sul de Minas, Brazil)[email protected]
Ricardo Nunes de Oliveira (Universidade Federal de Goias, Brazil)[email protected]
Marco Pacini (UFF, Brazil)[email protected]
Daniel Panario (Carleton University, Canada)[email protected]
Antonio Paques (UFRGS, Brazil)[email protected]
Eddy Pariguan (Pontificia Universidad Javeriana, Colombia)[email protected]
Aline Cristina Pegas (UFPR, Brazil)[email protected]
Rafael Peixoto (UNICAMP, Brazil)[email protected]
Fernanda de Andrade Pereira (UNICAMP, Brazil)[email protected]
Luiz Henrique Pereira (UFPR, Brazil)[email protected]
Luiz Antonio Peresi (IME-USP, Brazil)[email protected]
Tanise Carnieri Pierin (UFPR, Brazil)[email protected]
Hector Pinedo (IME-USP, Brazil)[email protected]
Rosemary Pires (IME-USP, Brazil)[email protected]
Maria Ines Platzeck (Universidad Nacional del Sur, Argentina)[email protected]
87
Barbara Pogorelsky (UFRGS, Brazil)[email protected]
Cesar Polcino Milies (IME-USP, Brazil)[email protected]
Anderson L.P. Porto (UnB, Brazil)[email protected]
Luis Enrique Ramirez (IME-USP, Brazil)[email protected]
Ana Rodriguez Raposo (Universidad da Coruna, Spain)[email protected]
Maria Julia Redondo (Universidade Nacional del Sur, Argentina)[email protected]
Julio Cesar dos Reis (UNICAMP, Brazil)[email protected]
Marcio Alexandre de O. Reis (IME-USP, Brazil)[email protected]
Evander Pereira de Rezende (UnB, Brazil)[email protected]
Alvaro Rittatore (Universidad de la Republica, Uruguay)[email protected]
Daniel Rivera (UAEM, Mexico)[email protected]
Noraı Romeu Rocco (UnB, Brazil)[email protected]
Roldao da Rocha (UFABC, Brazil)[email protected]
Paulo Henrique Rodrigues (UFG, Brazil)[email protected]
Rodrigo Lucas Rodrigues (IME-USP, Brazil)[email protected]
Virginia Rodrigues (UFSC, Brazil)[email protected]
Anita Rojas (Universidad de Chile, Chile)[email protected]
Carlos Rojas (DuocUC, Chile)[email protected]
Nadina Elizabeth Rojas (Universidad Nacional de Cordoba, Argentina)[email protected]
Olga Patricia Salazar-Diaz (Universidad Nacional de Colombia, Colombia)[email protected]
Diego Zurawski Saldanha (UFRGS, Brazil)[email protected]
Cecılia Salgado (Universite Paris VII, France)[email protected]
88
Alveri Sant´Ana (UFRGS, Brazil)[email protected]
Clotilzio Santos (UNESP, Brazil)[email protected]
Natalia Abad Santos (Universidad Nacional del Sur, Argentina)[email protected]
Ednei A Santulo Jr (UEM, Brazil)[email protected]
Cristian Scarola (Universidad Nacional de La Pampa, Argentina)[email protected]
Ralf Schiffler (University of Connecticut, USA)[email protected]
Vera Serganova (University of California, USA)[email protected]
Ivan Shestakov (IME-USP, Brazil)[email protected]
Mazi Shirvani (University of Alberta, Canada)[email protected]
Said Sidki (UnB, Brazil)[email protected]
Anderson Tiago da Silva (IME-USP, Brazil)[email protected]
Leonardo Amorim Silva (UNICAMP, Brazil)[email protected]
Renata Silva (IME-USP, Brazil)[email protected]
Ricardo Paleari da Silva (UFPR, Brazil)[email protected]
Tiago Martins da Silva (UFRGS, Brazil)[email protected]
Viviane Silva (UFMG, Brazil)[email protected]
Manuela da Silva Souza (UNICAMP, Brazil)[email protected]
Antonio Calixto Souza Filho (EACH-USP, Brazil)[email protected]
Tertuliano C. de Souza Neto (UnB, Brazil)[email protected]
Cristina Spohr (IME-USP, Brazil)[email protected]
Leo Storme (Ghent University, Belgium)[email protected]
Joao Eloir Strapasson (UFPR, Brazil)[email protected]
89
Thaisa Tamusiunas (UFRGS, Brazil)[email protected]
Horacio Tapia Recillas (Autonomous Metropolitan University, Mexico)[email protected]
Eduardo Tengan (ICMC-USP, Brazil)[email protected]
Juan Tirao (FaMAF, Universidad Nacional de Cordoba, Argentina)[email protected]
Gordana Todorov (Northeastern University, USA)[email protected]
Francesco Toppan (CBPF, Brazil)[email protected]
Fernando Torres (UNICAMP, Brazil)[email protected]
Marcio Andre Traesel (UFABC, Brazil)[email protected]
Sonia Trepode (Universidad Nacional de Mar del Plata, Argentina)[email protected]
Maria Trushina (Moscow City Pedagogical University, Russia)[email protected]
Israel Vaisencher (UFMG, Brazil)[email protected]
Jorge Vargas (Universidad Nacional de Cordoba, Argentina)[email protected]
Rosana Retsos Signorelli Vargas (EACH-USP, Brazil)[email protected]
Cristian Vay (Universidad Nacional de Cordoba, Argentina)[email protected]
Raul Velasquez (Universidad de Antioquia, Colombia)[email protected]
Paula Murgel Veloso (IME-USP, Brazil)[email protected]
Leandro Vendramin (UBA, Argentina)[email protected]
Luciana Lima Ventura (UnB, Brazil)[email protected]
Quimey Vivas (Universidad de Buenos Aires, Argentina)[email protected]
Filippo Viviani (Universita di Roma 3, Italy)[email protected]
Heily Wagner (IME-USP, Brazil)[email protected]
Qiang (Steven) Wang (Carleton University, Canada)[email protected]
90
Luis Alberto Wills-Toro (Universidad Nacional de Colombia, Colombia)[email protected]
Milen Yakimov (Louisiana State University, USA)[email protected]
Pavel Zalesskii (UnB, Brazil)[email protected]
Flavia Ferreira Ramos Zapata (UnB, Brazil)[email protected]
Theo Zapata (UnB, Brazil)[email protected]
Ignacio Zurrian (FaMAF, UNC, Argentina)[email protected]
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Name Index
Alfaro, R., 11, 21, 81Aljadeff, E., 5, 9, 21, 81Almeida, H., 81Alvares, E., 8, 21, 81Alves, M., 9, 22, 81Andruskiewitsch, N., 5, 6, 22, 81Apaza, J., 12, 23, 81Arenas-Carmona, L., 23, 81Artebani, M., 7, 24, 81Artenstein, D., 81Assem, I., 8, 24, 81Assuena, S., 81
Baggio, D., 8, 25, 81Bahturin, Y., 5Bassa, A., 11, 25, 81Beelen, P., 11, 25, 81Behn, A., 10, 26, 81Bekkert, V., 9, 26, 81Bemm, L., 81Benitez, G., 12, 81Bessa, V., 81Borges, F., 9, 27, 81Borges, H., 11, 27, 81Boyallian, C., 10, 27, 82Bras-Amoros, M., 11, 28, 82Brito, M., 82Brusamarello, R., 8, 29, 82Bueno, A., 10, 29, 82Bunke, T., 10, 30, 82Burgos, J., 82Bustamante, J., 82
Cadavid, P., 12, 30, 82Caetano, J., 12, 31, 82Cafure, A., 82Cagliero, L., 9, 31, 82Calderon, Y., 12, 31, 82Cardenas, J., 82Carocca, A., 7, 32, 82Carvalho, C., 11, 32, 82Castillo, J., 82Castonguay, D., 8, 32, 82Castro, F., 11, 33, 82Castro, F.L., 82Chagas, S., 7, 33, 82Chaio, C., 8, 33, 82Chalom, G., 5, 12, 34, 83Clark, J., 8, 34, 83Coelho, F., 5, 83Coelho, J., 7, 34, 83Conte, L., 11, 35, 83Cornelissen, M., 12, 35, 83Cortes, W., 8, 36, 83Coutinho, S., 6, 36, 83
Dahab, R., 11, 36, 83Davydov, A., 9, 36, 83de la Pena, J., 6, 37, 83del Hoyo, M., 9, 37, 83Dokuchaev, M., 8, 38, 83
Ducati, G., 83
Egea, C., 7, 39, 83Espinel, O., 83Esteves, E., 5, 83
Falu, S., 83Farinati, M., 9, 40, 83Feitosa, F., 83Femic, B., 83Fernandez, E., 83Fernandez-C., E., 84Ferraz, R., 5, 8, 40, 84Ferreira, V., 5, 84Ferrer, W., 5, 9, 40, 84Ferrero, M., 5, 8, 40, 84Flores, D., 84Flora, S., 84Fornaroli, E., 84Freitas, D., 84Freitas, J., 12, 41, 84Futorny, V., 5, 84
Garcia, A., 5, 84Garcia, G., 6, 42, 84Garcia-I., A., 9, 41, 84Gatica, M., 8, 42, 84Ghorpade, S., 11, 43, 84Giambruno, A., 5, 8, 43, 84Giuliani, M., 10, 43, 84Gneri, P., 84Gobbi, L., 84Godinho, H., 7, 44, 84Goncalves, J., 5, 8, 44, 84Gonzalez, L., 85Gonzalez, V., 7, 44, 85Gubitosi, V., 85Guerreiro, M., 12, 45, 85
Haetinger, C., 8, 45, 85Herscovich, E., 9, 45, 85Hoefel, E., 10, 45, 85Holguin-Villa, A., 85Homma, M., 11, 46, 85
Igusa, K., 9, 47, 85
Jain, S., 6, 47, 85Jardim, M., 7, 47, 85Jorge, S., 12, 48, 85
Kashuba, I., 10, 48, 85Kassel, C., 6, 48, 85Kato, T., 7, 49, 85Kharchenko, V., 5Kitani, P., 85Kornev, A., 10, 49, 85Koshlukov, P., 10, 50, 85
Labra, A., 5, 10, 51, 85Laface, A., 7, 52, 85Lazzarin, J., 85
93
Le Meur, P., 8, 52, 86Levstein, F., 86Lima, I., 86Lima, V., 86Lobao, T., 8, 53, 86Lopez, V., 7, 86
Macedo, T., 12, 53, 86Maciel, U., 86Marcos, E., 5, 9, 53, 86Marquez, O., 86Martins, R.A., 86Martins, R.V., 7, 54, 86Matera, G., 11, 54, 86Mathieu, O., 10, 54, 86Medeiros, F., 86Mello, T., 12, 55, 86Melo, M., 7, 54, 86Minian, G., 7, 9, 55, 56, 86Moreno, A., 12, 57, 86Moreno, C., 11, 57, 86Morgado, A., 86Motta, B., 12, 58, 86Moura, A., 10, 58, 87Murakami, L., 5
Nart, E., 11, 59, 87Nascimento, C., 87Nascimento, R., 87Natale, S., 5Neumann, F., 7, 59, 87
Ohbuchi, A., 7, 59, 87Oliveira, J., 12, 60, 87Oliveira, P., 87Oliveira, R., 7, 61, 87
Pacini, M., 87Panario, D., 5, 61, 87Paques, A., 87Pariguan, E., 87Pegas, A., 87Peixoto, R., 12, 62, 87Pereira, F., 12, 62, 87Pereira, L., 12, 63, 87Peresi, L., 10, 63, 87Pierin, T., 87Pinedo, H., 87Pires, R., 87Platzeck, M., 5, 87Pogorelsky, B., 9, 63, 88Polcino, C., 5, 64, 88Porto, A., 12, 64, 88
Ramirez, L., 88Raposo, A., 88Redondo, M., 5, 9, 64, 88Reis, J., 88Reis, M., 88Rezende, E., 88Rittatore, A., 7, 64, 88Rivera, D., 8, 65, 88Rocco, N., 7, 65, 88Rocha, R., 10, 65, 88
Rodrigues, P., 88Rodrigues, R., 88Rodrigues, V., 8, 66, 88Rojas, A., 7, 66, 88Rojas, C., 10, 66, 88Rojas, N., 88
Salazar-Diaz, O., 7, 67, 88Saldanha, D., 88Salgado, C., 7, 67, 88Sant’Ana, A., 8, 67, 89Santos, C., 8, 68, 89Santos, N., 9, 68, 89Santulo, E., 8, 69, 89Scarola, C., 89Schiffler, R., 6, 69, 89Sehgal, S., 5Serganova, V., 5, 6, 69, 89Shestakov, I., 5, 10, 69, 89Shirvani, M., 8, 70, 89Sidki, S., 5, 6, 70, 89Silva, A., 89Silva, L., 89Silva, R., 89Silva, R.P., 89Silva, T., 89Silva, V., 8, 70, 89Souza, A., 8, 71, 89Souza, M., 89Souza, T., 89Spohr, C., 89Storme, L., 11, 71, 89Strapasson, J., 12, 72, 89
Tamusiunas, T., 90Tapia, H., 11, 72, 90Tengan, E., 6, 72, 90Tirao, J., 5, 10, 73, 90Todorov, G., 8, 73, 90Toppan, F., 10, 73, 90Torres, F., 11, 90Traesel, M., 12, 74, 90Trepode, S., 8, 74, 90Trushina, M., 10, 74, 90
Vainsencher, I., 5Vaisencher, I., 90Vargas, J., 10, 75, 90Vargas, R., 8, 75, 90Vay, C., 9, 76, 90Velasquez, R., 10, 76, 90Veloso, P., 8, 77, 90Vendramin, L., 90Ventura, L., 90Vivas, Q., 90Viviani, F., 7, 77, 90
Wagner, H., 90Wang, Q., 11, 78, 90Wills-Toro, L., 10, 78, 91
Yakimov, M., 10, 78, 91
Zalesskii, P., 7, 78, 91
94
Zapata, F., 12, 79, 91Zapata, T., 12, 79, 91Zurrian, I., 91
95
