José Antonio de la Peña Guanajuato · José Antonio de la Peña Auslander-Reiten theory. Consider...

José Antonio de la Peña

Instituto de Matemáticas, UNAM and

Centro de Investigación en Matemáticas,

Guanajuato.

Mérida. December 2014.

On the occasion of the 60. Anniversary of José Seade

o Representations of algebras: general concepts.

o The Coxeter polynomial of an algebra.

o Mahler measure of polynomials and Lehmer’s conjecture

o Algebras associated to singularities.

o Algebras of cyclotomic type.

o Examples.

Lenzing-de la Peña: Extended canonical algebras and

Fuchsian singularities. Math Z. (2010).

de la Peña: Algebras whose Coxeter polynomial are

products of cyclotomic polynomials. Algebras and

Representation Th. (2014)

de la Peña: On the Mahler measure of the Coxeter

polynomial of a finite dimensional algebra. Adv. Math.

(2014)

de la Peña: Cyclotomicity of the Coxeter matrix and the

representation type of algebras. In preparation.

Mroz-de la Peña: Tubes in derived categories. J. Algebra.

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José Antonio de la Peña

José Antonio de la Peña

A finite dimensional algebra kQ accepts only finitely many

indecomposable representations (up to isomorphisms) if and

only if Q has an underlying Dynkin diagram:

After work of Tits and Gelfand et al. there is a bijection:

X dim X

between indecomposable modules and roots of the (Tits) form

qA(u) =u,uA

José Antonio de la Peña

Auslander-Reiten theory.

Consider a finite dimensional algebra A and X an

indecomposable non-projective module. There exists an exact

sequence

0 X E X 0

Such that

• X is an indecomposable non-injective module;

• the sequence almost split, that is, for an indecomposable

module Y non-isomorphic to X, the following is exact:

0 Hom (Y,X) Hom (Y, E) Hom (Y, X) 0

In particular, there is a natural isomorphism:

Ext1(Z,X) D Hom (X, Z),

where D=Homk( - , k) is the natural duality.

José Antonio de la Peña

Spectral theory of Coxeter. transformations

Let A be a finite dimensional algebra of finite global dimension

and C the associated Cartan matrix. The Coxeter matrix is

= - C C-T

In case A= kQ

o for X an indecomposable non-projective module holds

[X] = [X], for the classes in the Grothendieck group.

o let be the spectral radius of then:

• if Q is of Dynkin type =1 but 1 is not eigenvalue;

• If Q is extended Dynkin =1 is an eigenvalue;

• If Q is wild, then > 1 and defines the growth rate of

(dim n X )n

for non-preprojective indecomposable modules

Singularities of Dynkin type

Derived categories and invariant transformations.

Notation and definitions.

Let A be a triangular finite dimensional algebra.

𝜑𝐴 denotes the Coxeter transformation of 𝐾0 𝐴 = ℤ𝑛

𝜒𝐴(T) denotes the Coxeter polynomial (=characteristic polynomial of 𝜑𝐴)

Write:

𝜒𝐴(T) =𝑎0 + 𝑎1 𝑇 + 𝑎2𝑇2 +⋯+ 𝑎𝑛𝑇

𝑛 an integral polynomial

𝜒𝐴 T = (𝑇 − 𝜆𝑖)𝑛𝑖=1 for the eigenvalues 𝜆𝑖

Then

• 𝑎𝑖 = 𝑎𝑛−𝑖 we say that 𝜒𝐴(T) is self-reciprocal;

• 𝑎0 = 𝑎𝑛 = (−1)𝑛det𝜑𝐴 = 1

• 𝑎1 = − 𝜆𝑖 = −𝑡𝑟𝑛𝑖=1 𝜑𝐴

We say that A is of cyclotomic type if all eigenvalues satisfy || 𝜆𝑖 ||=1

Derived categories and invariant transformations.

Serre duality: Ext1 𝑋, 𝑌 ≅ 𝐷Hom(𝑌, 𝜏𝑋)

Euler bilinear form:

< [𝑋], [𝑌] >= (−1)𝑗dimExt𝑗(𝑋, 𝑌)∞𝑗=0

Therefore

< [𝑋], [𝑌] >= −< 𝑌 , 𝜏𝑋 >

1 0 0 0 0 0 0 -1

C= 1 1 0 0 = 0 0 1 1 (t)= t4+t3+t+1= (t+1)2(t2-t+1)

1 0 1 0 0 1 0 1

1 1 1 1 -1 -1 -1 -1

José Antonio de la Peña

Hereditary algebras

Algebras A=kQ are:

representation finite if Q is Dynkin;

Tame if Q is extended Dynkin and there are one

parametric families of modules of dimension u if qA(u) = 0

(Dlab-Ringel);

• Wild, else.

V. Kac: There are infinitely many indecomposable modules

X with dim X=u if u is a connected vector with qA(u) ≤ 0.

0 10 20 30 40 50 60 70 800

1

2

3

4

5

6

trace(𝜑𝑘)

N.B. given two cyclotomic polynomials Φ𝑛 and Φ𝑚

o Φ𝑛⨂Φ𝑚 is product of cyclotomic polynomials;

o Φ𝑛⨂Φ𝑚 = Φ𝑘𝑒𝑘

𝑘 its cyclotomic decomposition: open problem

o Mahler measure is multiplicative, ie 𝑀 𝑓𝑔 = 𝑀 𝑓 𝑀 𝑔 .

L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, Crelle, Oeuvres I (1857), 105-108

𝐴𝑚+1 = 𝐴𝑚[𝑀𝑛] one-point extension s.t. 𝐸𝑥𝑡𝑘 𝑀𝑛, 𝑀𝑛 = 0, for 𝑘 ≥ 1

where 𝐵 𝑀 =𝐵 𝑀0 𝑘

with the usual matrix operations.

0 10 20 30 40 50 60 70 80 90 10010

0

101

102

103

10,22,30,42,50,62,70,82,90,102 are not cyclotomic

José-Antonio de la Peña

José-Antonio de la Peña

José-Antonio de la Peña

Finite dimensional algebras and singularities José Antonio de la Peña

José Antonio de la Peña

m=2

Spectral radius = 1.106471… < 1.1762… Mahler measure= 1.224278…

joint work with Helmut Lenzing

• Dolgachev: Math. Annalen 265 (1983)

• Wagreich: Proc. Symp. Pure Math. AMS 40 (1983),

José Antonio de la Peña

Exhaustive Search: It is possible to determine all polynomials of a given degree D

having bounded Mahler measure. Searches performed using measure bound 1.3:

1980 D ≤ 16 (Boyd) 1989 D = 18 and 20 (Boyd)

Jan 1996 D = 22 May 1996 D = 24 2003 D ≤ 40 (Flammang, Rhin, Sac-Epee, Wu)

2008 D ≤ 44 (Rhin, Mossinghoff, Wu)

2008 D ≤ 54 (Rhin, Mossinghoff, Wu) (From Mossinghoff’s Web page).

Lehmer added the following remark in his 1933 paper (using Ω to denote

the measure):

“We have not made an examination of all 10th degree symmetric

polynomials, but a rather intensive search has failed to reveal a better

polynomial than x10 + x9 − x7 − x6 − x5 − x4 − x3 + x + 1, Ω = 1.176280818.

“All efforts to find a better equation of degree 12 and 14 have been

unsuccessful.” Despite extensive searches, Lehmer's polynomial remains

the world champion

8 9 10

A the corresponding hereditary algebra has 𝑀 𝜒𝐴 = 1.1762… minimal

known.

=

Happel (1997):

Clearly, if 𝐴 is a cyclotomic algebra with 𝑛 vertices, then 𝑡𝑟 𝜑𝑘 ≤ 𝑛, for any 𝑘 ≥ 0

Thank you!

José Antonio de la Peña

jap@matem.unam.mx

jap@cimat.mx

Let B be an accessible critical of non cyclotomic type, then either

𝜒𝐵 has a unique root outside the unit circle (Pissot polynomial), M(𝜒𝐵 )=𝜌𝐵

or

𝜒𝐵 has two roots outside the unit circle, M(𝜒𝐵 )=𝜌𝐵2

Theorem 1: A an accessible algebra, then

either A is of cyclotomic type

or

there is a convex subcategory B of A such that Mahler measure

M(𝜒𝐵) ≥ 𝜇0 (Lehmer’s number).

0 10 20 30 40 50 60 70 800

1

2

3

4

5

6

0 10 20 30 40 50 60 70 80 90 10010

0

101

102

103

10,22,30,42,50,62,70,82,90,102 are not cyclotomic

m the m-th cyclotomic polynomial has as roots the primitive roots of unity, hence it has degree (m).

Finite dimensional algebras and singularities José Antonio de la Peña

There is a functor modZ k[x0, x1] coh P1,

that takes each graded k[x0, x1]-module M to the triple ((Mx0)0, (Mx1)0, σM),

where y acts on the degree zero part of Mx0 via the identification y = x1/x0,

the variable y−1 acts on the degree zero part of Mx1 via the identification y−1

=x0/x1, and the isomorphism σM equals the obvious identification

[(Mx0)0]x1/x0 =[(Mx1)0]x0/x1 .

Proposition (Serre). The above functor induces an equivalence

modZ k[x0, x1]/mod0Z k[x0, x1] coh P1

The category coh P1 is a k-linear hereditary category satisfying Serre

duality. More precisely, there is a functorial k-linear isomorphism

DExt1(F, G ) ≈ Hom(G,F(−2)) for all F, G coh P1.

Finite dimensional algebras and singularities José Antonio de la Peña

Theorem: (1) [Geigle-Lenzing]: coh(X) is a hereditary category with Serre

duality. (2) [Happel]: if H is a hereditary category then the bounded derived

category Der(H) is triangulated equivalent to Der(mod H) for some

hereditary algebra H or Der(coh(X)) for some weighted projective line X.

R is the translation algebra

Finite dimensional algebras and singularities José Antonio de la Peña