Álgebras de Hopf punteadas sobre grupos finitos simples de ...ggarcia/encuentros/ggarcia-uma-rosario.pdf · Introduction Joint work with N. Andruskiewitsch and G. Carnovale. Main

Algebras de Hopf punteadas sobre grupos finitos simples de tipo Lie

Algebras de Hopf punteadas sobre grupos finitossimples de tipo Lie

Gaston Andres Garcıa

Universidad Nacional de La Plata, ArgentinaCONICET

UMA Rosario 2013

17 al 20 de septiembre de 2013


Introduction

Joint work with N. Andruskiewitsch and G. Carnovale.

Main problem:

Classify all finite-dimensional pointed Hopf algebras H over analgebriacally closed field k of characteristic zero such that G (H) isa non-abelian finite (simple) group.

We say that a finite group G collapses when everyfinite-dimensional pointed Hopf algebra H, with G (H) ' G isisomorphic to kG .


Introduction


Main problem:




Introduction


Main problem:




Introduction


Main problem:


We say that a finite group G collapses

when everyfinite-dimensional pointed Hopf algebra H, with G (H) ' G isisomorphic to kG .


Introduction


Main problem:




Introduction

Background

Background:

If G ' Z/p is simple abelian, then the classification is known:for p = 2 by [N]; for p > 7, by [AS3]; for p = 3, 5, 7, by [AS1]and [AS4].

If G ' Am, m ≥ 5 is alternating, then G collapses [AFGV1].

If G is a sporadic simple group, then G collapses, except forthe groups G = Fi22, B, M [AFGV2], [FV].

G = PSL2(q) collapses for q > 2 even.

For G not simple, non-trivial examples also exists, amongothers: S3 [AHS], S4 [GG], D4t for t ≥ 3 [FG], G associatedto an affine rack [GIV].


Introduction

Background

Background:







Introduction

Background

Background:







Introduction

Background

Background:







Introduction

Background

Background:







Introduction

Background

Background:







Introduction

On the classification of finite-dimensional pointed Hopf algebras over finite groups

Let G be a finite group and H a pointed Hopf algebra withG (H) ' G .

Let 0 = H−1 ⊂ H0 = kG (H) ⊂ H1 ⊂ . . . be the coradical filtrationof H and gr H = ⊕n∈N0Hn/Hn−1 ' R#kG (H).

R = ⊕n∈N0Rn is a graded Hopf algebra in kGkGYD. Also, the

subalgebra of R generated by V := R1 is isomorphic to the Nicholsalgebra B(V ) of V . Hencedim H <∞ ⇐⇒ dim R <∞ =⇒ dimB(V ) <∞.

Question

Determine all V ∈ kGkGYD with dimB(V ) <∞.

The following are equivalent [AFGV1]:

G collapses.

For every V ∈ kGkGYD, dimB(V ) =∞.

For every irreducible V ∈ kGkGYD, dimB(V ) =∞.


Introduction






Question



G collapses.




Introduction




R = ⊕n∈N0Rn is a graded Hopf algebra in kGkGYD.

Also, thesubalgebra of R generated by V := R1 is isomorphic to the Nicholsalgebra B(V ) of V . Hencedim H <∞ ⇐⇒ dim R <∞ =⇒ dimB(V ) <∞.

Question



G collapses.




Introduction





subalgebra of R generated by V := R1 is isomorphic to the Nicholsalgebra B(V ) of V .

Hencedim H <∞ ⇐⇒ dim R <∞ =⇒ dimB(V ) <∞.

Question



G collapses.




Introduction






Question



G collapses.




Introduction






Question



G collapses.




Introduction






Question



G collapses.




Introduction






Question



G collapses.




Introduction






Question



G collapses.




Introduction






Question



G collapses.




Introduction


Fact:

All irreducible Yetter-Drinfeld modules over kG are of theform M(O, ρ) = IndG

CG (g) ρ, O is a conjugacy class of G andρ ∈ Irr CG (g) for g ∈ O fixed.

Set B(O, ρ) := B(M(O, ρ)).

Question

Determine all pairs (O, ρ) with dimB(O, ρ) <∞.

Crucial: B(O, ρ) depends only on the underlying braided vectorspace (kO, cρ). i. e., B(O, ρ) depends only on the rack O and thenon-principal 2-cocycle arising from ρ.

Question [AFGV1]

Determine all pairs (X , q), where X is a finite rack and q is anon-principal 2-cocycle, such that dimB(X , cq) <∞.


Introduction


Fact: All irreducible Yetter-Drinfeld modules over kG are of theform M(O, ρ) = IndG

CG (g) ρ,

O is a conjugacy class of G andρ ∈ Irr CG (g) for g ∈ O fixed.

Set B(O, ρ) := B(M(O, ρ)).

Question



Question [AFGV1]



Introduction




Set B(O, ρ) := B(M(O, ρ)).

Question



Question [AFGV1]



Introduction




Set B(O, ρ) := B(M(O, ρ)).

Question



Question [AFGV1]



Introduction




Set B(O, ρ) := B(M(O, ρ)).

Question



Question [AFGV1]



Introduction




Set B(O, ρ) := B(M(O, ρ)).

Question


Crucial: B(O, ρ) depends only on the underlying braided vectorspace (kO, cρ).

i. e., B(O, ρ) depends only on the rack O and thenon-principal 2-cocycle arising from ρ.

Question [AFGV1]



Introduction




Set B(O, ρ) := B(M(O, ρ)).

Question



Question [AFGV1]



Introduction




Set B(O, ρ) := B(M(O, ρ)).

Question



Question [AFGV1]



Racks

Definition

Definition

A rack is a non-empty set X

endowed with a map . : X × X → Xsatisfying

(a) x . is a bijection for any x ∈ X ,

(b) x . (y . z) = (x . y) . (x . z) for all x , y , z ∈ X .

The archetypical example of a rack is a conjugacy class in a groupG , with x . y = xyx−1 for all x , y ∈ G .

We say that a rack is:

abelian if x . y = y for all x , y ∈ X .

decomposable if it contains two subracks R,S such thatX = R

∐S and R . S ⊆ S , S . R ⊆ R.

simple if |X | > 1 and any rack epimorphism X Y isbijective or |Y | = 1.


Racks

Definition

Definition

A rack is a non-empty set X endowed with a map . : X × X → Xsatisfying







∐S and R . S ⊆ S , S . R ⊆ R.



Racks

Definition

Definition








∐S and R . S ⊆ S , S . R ⊆ R.



Racks

Definition

Definition








∐S and R . S ⊆ S , S . R ⊆ R.



Racks

Definition

Definition




The archetypical example of a rack is a conjugacy class in a groupG ,

with x . y = xyx−1 for all x , y ∈ G .



decomposable if it contains two subracks R, S such thatX = R

∐S and R . S ⊆ S , S . R ⊆ R.



Racks

Definition

Definition








∐S and R . S ⊆ S , S . R ⊆ R.



Racks

Definition

Definition








∐S and R . S ⊆ S , S . R ⊆ R.



Racks

Definition

Definition








∐S and R . S ⊆ S , S . R ⊆ R.



Racks

Definition

Definition








∐S and R . S ⊆ S , S . R ⊆ R.



Racks

Definition

Definition








∐S and R . S ⊆ S , S . R ⊆ R.



Racks

Properties

Definition

A rack X collapses when dimB(X , q) =∞ for every finite faithful2-cocycle q.

Therefore, to solve the initial question we first need to determineall conjugacy classes in G that collapse.We have criteria that help to solve the problem without looking atthe 2-cocycle. We say that a rack X is of

Type D if it contains a decomposable subrack Y = R∐

S andelements r ∈ R, s ∈ S such that r . (s . (r . s)) 6= s.

Type F if it has a family of mutually disjoint subracks (Ra)a∈Asuch that Ra . Rb = Rb for all a, b ∈ A; for all a 6= b ∈ A,there are ra ∈ Ra, rb ∈ Rb such that ra . rb 6= rb and A hasfour elements.

Type C if it has a decomposable subrack Y = R∐

S , where|R| > 6 or |S | > 6, with elements r ∈ R, s ∈ S such thatr . s 6= s.


Racks

Properties

Definition


Therefore, to solve the initial question we first need to determineall conjugacy classes in G that collapse.

We have criteria that help to solve the problem without looking atthe 2-cocycle. We say that a rack X is of







Racks

Properties

Definition


Therefore, to solve the initial question we first need to determineall conjugacy classes in G that collapse.We have criteria that help to solve the problem without looking atthe 2-cocycle.

We say that a rack X is of







Racks

Properties

Definition









Racks

Properties

Definition









Racks

Properties

Definition





Type F if it has a family of mutually disjoint subracks (Ra)a∈Asuch that

Ra . Rb = Rb for all a, b ∈ A; for all a 6= b ∈ A,there are ra ∈ Ra, rb ∈ Rb such that ra . rb 6= rb and A hasfour elements.




Racks

Properties

Definition





Type F if it has a family of mutually disjoint subracks (Ra)a∈Asuch that Ra . Rb = Rb for all a, b ∈ A;

for all a 6= b ∈ A,there are ra ∈ Ra, rb ∈ Rb such that ra . rb 6= rb and A hasfour elements.




Racks

Properties

Definition





Type F if it has a family of mutually disjoint subracks (Ra)a∈Asuch that Ra . Rb = Rb for all a, b ∈ A; for all a 6= b ∈ A,there are ra ∈ Ra, rb ∈ Rb such that ra . rb 6= rb

and A hasfour elements.




Racks

Properties

Definition









Racks

Properties

Definition









Racks

Properties

Remarks:

I If O is a conjugacy class in a finite group G , then O is of type Dif and only if there exist x , y ∈ O such that x , y are not conjugatedin 〈x , y〉 and (xy)2 6= (yx)2.

I If Z is a finite rack that admits a rack epimorphism Z X ,where X is of type D (F, C), then Z is of type D (F, C).

I If Z is indecomposable, then it admits a rack epimorphismZ X with X simple.

Theorem [AFGV1], [H], [ACG]

A rack X of type C, D or F collapses.

I A rack X is cthulhu when it is neither of type C, D, F.

I A rack X is sober if every subrack is either abelian orindecomposable. A sober rack is cthulhu.


Racks

Properties

Remarks:I If O is a conjugacy class in a finite group G ,

then O is of type Dif and only if there exist x , y ∈ O such that x , y are not conjugatedin 〈x , y〉 and (xy)2 6= (yx)2.








Racks

Properties

Remarks:I If O is a conjugacy class in a finite group G , then O is of type Dif and only if there exist x , y ∈ O such that x , y are not conjugatedin 〈x , y〉 and (xy)2 6= (yx)2.








Racks

Properties









Racks

Properties









Racks

Properties









Racks

Properties









Racks

Properties







I A rack X is sober if every subrack is either abelian orindecomposable.

A sober rack is cthulhu.


Racks

Properties









Finite groups of Lie type

Definition

We consider finite-dimensional pointed Hopf algebras with finitesimple group of Lie type, [MaT].

Let p be a prime number, m ∈ N, q = pm and Fq the field with qelements.

Let G be a semisimple algebraic group defined over Fq. ASteinberg endomorphism F : G→ G is an abstract groupautomorphism having a power equal to a Frobenius map. Thesubgroup GF is called a finite group of Lie type.

Assume G is a simple simply connected algebraic group. ThenG/Z (G) is a simple abstract group but GF is not simple ingeneral. In fact G := GF/Z (GF ) is a simple finite group except for8 examples that appear in low rank and with q = 2 or 3. These Gare called finite simple groups of Lie type.



Definition


Let p be a prime number,

m ∈ N, q = pm and Fq the field with qelements.





Definition


Let p be a prime number, m ∈ N,

q = pm and Fq the field with qelements.





Definition


Let p be a prime number, m ∈ N, q = pm and

Fq the field with qelements.





Definition







Definition



Let G be a semisimple algebraic group defined over Fq.

ASteinberg endomorphism F : G→ G is an abstract groupautomorphism having a power equal to a Frobenius map. Thesubgroup GF is called a finite group of Lie type.




Definition



Let G be a semisimple algebraic group defined over Fq. ASteinberg endomorphism F : G→ G is an abstract groupautomorphism having a power equal to a Frobenius map.

Thesubgroup GF is called a finite group of Lie type.




Definition







Definition




Assume G is a simple simply connected algebraic group.

ThenG/Z (G) is a simple abstract group but GF is not simple ingeneral. In fact G := GF/Z (GF ) is a simple finite group except for8 examples that appear in low rank and with q = 2 or 3. These Gare called finite simple groups of Lie type.



Definition




Assume G is a simple simply connected algebraic group. ThenG/Z (G) is a simple abstract group but GF is not simple ingeneral.

In fact G := GF/Z (GF ) is a simple finite group except for8 examples that appear in low rank and with q = 2 or 3. These Gare called finite simple groups of Lie type.



Definition




Assume G is a simple simply connected algebraic group. ThenG/Z (G) is a simple abstract group but GF is not simple ingeneral. In fact G := GF/Z (GF ) is a simple finite group except for8 examples that appear in low rank and with q = 2 or 3.

These Gare called finite simple groups of Lie type.



Definition







Definition

There are three families of finite simple groups of Lie type,according to the classes of Steinberg endomorphisms:

Chevalley groups. Correspond to Fq-split Steinberg maps: thereexists an F -stable torus T such that F (t) = tq for all t ∈ T . ThenF is called a Frobenius map and GF = G(Fq) is the finite group ofFq-points: PSLn(q), n ≥ 2 (except PSL2(2) ' S3 andPSL2(3) ' A4); PSp2n(q), n ≥ 2; PΩ2n+1(q), n ≥ 3, q odd;PΩ+

2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).

Steinberg groups. Correspond to twisted Steinberg maps, i. e. F isthe product of a Frobenius map with an automorphism of Ginduced by a non-trivial Dynkin diagram automorphism: PSUn(q),n ≥ 3 (except PSU3(2)); PΩ−2n(q), n ≥ 4; 3D4(q), 2E6(q).

Suzuki-Rees groups. Related to very twisted Steinberg maps:2B2(22n+1), n ≥ 1; 2G2(32n+1), n ≥ 1; 2F4(22n+1), n ≥ 1.



Definition


Chevalley groups.

Correspond to Fq-split Steinberg maps: thereexists an F -stable torus T such that F (t) = tq for all t ∈ T . ThenF is called a Frobenius map and GF = G(Fq) is the finite group ofFq-points: PSLn(q), n ≥ 2 (except PSL2(2) ' S3 andPSL2(3) ' A4); PSp2n(q), n ≥ 2; PΩ2n+1(q), n ≥ 3, q odd;PΩ+

2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).





Definition


Chevalley groups. Correspond to Fq-split Steinberg maps:

thereexists an F -stable torus T such that F (t) = tq for all t ∈ T . ThenF is called a Frobenius map and GF = G(Fq) is the finite group ofFq-points: PSLn(q), n ≥ 2 (except PSL2(2) ' S3 andPSL2(3) ' A4); PSp2n(q), n ≥ 2; PΩ2n+1(q), n ≥ 3, q odd;PΩ+

2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).





Definition


Chevalley groups. Correspond to Fq-split Steinberg maps: thereexists an F -stable torus T such that F (t) = tq for all t ∈ T .

ThenF is called a Frobenius map and GF = G(Fq) is the finite group ofFq-points: PSLn(q), n ≥ 2 (except PSL2(2) ' S3 andPSL2(3) ' A4); PSp2n(q), n ≥ 2; PΩ2n+1(q), n ≥ 3, q odd;PΩ+

2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).





Definition


Chevalley groups. Correspond to Fq-split Steinberg maps: thereexists an F -stable torus T such that F (t) = tq for all t ∈ T . ThenF is called a Frobenius map and GF = G(Fq) is the finite group ofFq-points:

PSLn(q), n ≥ 2 (except PSL2(2) ' S3 andPSL2(3) ' A4); PSp2n(q), n ≥ 2; PΩ2n+1(q), n ≥ 3, q odd;PΩ+

2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).





Definition



2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).





Definition



2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).

Steinberg groups. Correspond to twisted Steinberg maps, i. e. F isthe product of a Frobenius map with an automorphism of Ginduced by a non-trivial Dynkin diagram automorphism:

PSUn(q),n ≥ 3 (except PSU3(2)); PΩ−2n(q), n ≥ 4; 3D4(q), 2E6(q).




Definition



2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).





Definition



2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).


Suzuki-Rees groups. Related to very twisted Steinberg maps:

2B2(22n+1), n ≥ 1; 2G2(32n+1), n ≥ 1; 2F4(22n+1), n ≥ 1.



Definition



2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).





Reduction to semisimple and unipotent classes

I Take x ∈ G ;

we want to investigate the orbit OGx .

I If x = xsxu is the Chevalley-Jordan decomposition in G, thenxs , xu ∈ G .

I Let K = CG(xs), a reductive subgroup of G. ThenK = K ∩ G = CG (xs).

Since xu ∈ K , OKxu is a subrack of OG

x and we can reduce our studyto the case when x is either unipotent or semisimple.




I Take x ∈ G ; we want to investigate the orbit OGx .









I If x = xsxu is the Chevalley-Jordan decomposition in G,

thenxs , xu ∈ G .

















I Let K = CG(xs), a reductive subgroup of G.

ThenK = K ∩ G = CG (xs).





















Unipotent classes in PSLn(q)

For a ∈ (F∗q)n−1, define

ra =

1 a1 0 . . . 00 1 a2 . . . 0...

. . .. . . 0

0 . . . . . . 1 an−1

0 . . . . . . 0 1

.

A unipotent element u ∈ GLn(q) is of type λ = (λ1, . . . , λk) if it isconjugate to the element

u =

u1 0 . . . 00 u2 . . . 0...

. . ....

0 . . . . . . uk

where ui = r1 ∈ Fλi×λiq .




For a ∈ (F∗q)n−1, define

ra =

1 a1 0 . . . 00 1 a2 . . . 0...

. . .. . . 0

0 . . . . . . 1 an−1

0 . . . . . . 0 1

.

A unipotent element u ∈ GLn(q) is of type λ = (λ1, . . . , λk) if it isconjugate to the element

u =

u1 0 . . . 00 u2 . . . 0...

. . ....

0 . . . . . . uk

where ui = r1 ∈ Fλi×λiq .




We describe some results for G = SLn(q).

We begin with unipotent classes. By isogeny arguments, thefollowing theorem will imply the result for G = PSLn(q) and it iscrucial for the proof for all Chevalley groups.

Theorem

Let O be a unipotent conjugacy class in G . If O is not listedbelow, then it collapses.

n type q Remark

2 (2) even or not a square sober

3 (3) 2 sober(2, 1) 2 cthulhu

4 (2, 1, 1) 2 cthulhu





We begin with unipotent classes.

By isogeny arguments, thefollowing theorem will imply the result for G = PSLn(q) and it iscrucial for the proof for all Chevalley groups.

Theorem


n type q Remark



4 (2, 1, 1) 2 cthulhu






Theorem


n type q Remark



4 (2, 1, 1) 2 cthulhu






Theorem


n type q Remark



4 (2, 1, 1) 2 cthulhu




For non-semisimple and non-unipotent classes in SLn(q) we havethe following

Proposition

Let x ∈ G = SLn(q) with Chevalley-Jordan decompositionx = xsxu. Assume that xs is not central and xu 6= e. Then OG

x

collapses.

Nevertheless, for G = PSLn(q) we do not have the complete resultyet:

Proposition

Let x ∈ SLn(q) with Chevalley-Jordan decomposition x = xsxu.Assume that xs is not central and xu 6= e. If xu is not listed below,then OKxu collapses. In consequence, if x = π(x) ∈ G, then OG

x

collapses.





Proposition


x

collapses.


Proposition


x

collapses.





Proposition


x

collapses.


Proposition


x

collapses.





Proposition


x

collapses.


Proposition


x

collapses.




n = h1Λ1 + · · ·+ h`Λ` xu = (u1, . . . , u`) q = (qµ1 , . . . , qµ`)

n = 2Λ1 > 2, ` = 1 xu = u1 all

h1 = 2 (u1, id, . . . , id) odd andhi ≥ 2 for 2 ≤ i ≤ ` ui = id for i 6= 1 9 or not a square

hj = 2 (u1, . . . , u1, id, . . . , id) q = 3#j : uj 6= id ≥ 2 ui = id for j < i ≤ `hi ≥ 2 for j < i ≤ `

h1 = 2 (u1, id, . . . , id) q = 3

h1 = 3 (u1, id, . . . , id) q = 2

h1 = 4 (u1, id, . . . , id) q = 2u1 of type (2, 1, 1)

hj = 2 (u1, . . . , u1, id, . . . , id) q = 2#j : uj 6= id ≥ 2 ui = id for j < i ≤ `

h1 = 2, (u1, id, . . . , id) q even



Collapsing unipotent classes in a Chevalley group

Now we summarize the results (still on progress) on collapsingunipotent classes in a Chevalley group.

Let G be a Chevalleygroup, G 6= PSLn(q).

Theorem





Now we summarize the results (still on progress) on collapsingunipotent classes in a Chevalley group. Let G be a Chevalleygroup, G 6= PSLn(q).

Theorem





Now we summarize the results (still on progress) on collapsingunipotent classes in a Chevalley group. Let G be a Chevalleygroup, G 6= PSLn(q).

Theorem





G q type or representative

PSp2n(q), even alln ≥ 2 odd & 6= (1r1 , 2)

9 (1r1 , 2)3 (1r1 , 2r2 , 3r3), r2r3 > 0

PΩ2n+1(q), n ≥ 3 3 (1r1 , 2r2 , 3r3), r2r3 > 0

PΩ+2n(q), even all

n ≥ 4 3 (1r1 , 2r2 , 3r3), r2r3 > 0

E6(q) 2,4 all except xα1(1)

E7(q) 2 all except y119

4 all excepty113, y115, y117, y118, y119

E8(q) 2 all except z195

4 all except z189, z193, z194, z195

p=2,3,5 ⊂ D8(a7)

F4(q) 2,3,4 all except x4


GRACIAS


References

N. Andruskiewitsch, F. Fantino, M. Grana and L. Vendramin,Finite-dimensional pointed Hopf algebras with alternatinggroups are trivial, Ann. Mat. Pura Appl. (4), 190 (2011),225–245.

N. Andruskiewitsch, F. Fantino, M. Grana and L. Vendramin,Pointed Hopf algebras over the sporadic simple groups. J.Algebra 325 (2011), pp. 305–320.

N. Andruskiewitsch, I. Heckenberger and H.-J. Schneider, TheNichols algebra of a semisimple Yetter-Drinfeld module, Amer.J. Math. 132, no. 6, 1493–1547.

N. Andruskiewitsch and H.-J. Schneider, Finite quantumgroups and Cartan matrices, Adv. Math. 154 (2000), 1–45.

N. Andruskiewitsch and H.-J. Schneider, Finite quantumgroups over abelian groups of prime exponent, Ann. Sci. Ec.Norm. Super. 35 (2002), 1–26.


References

N. Andruskiewitsch and H.-J. Schneider, On the classificationof finite-dimensional pointed Hopf algebras. Ann. Math. Vol.171 (2010), 375–417.

S. Caenepeel and S. Dascalescu, On pointed Hopf algebras ofdimension 2n, Bull. London Math. Soc. 31 (1999), pp. 17–24.

F. Fantino and G. A. Garcıa. On pointed Hopf algebras overdihedral groups. Pacific J. Math, Vol. 252 (2011), no. 1,69–91.

F. Fantino and L. Vendramin, On twisted conjugacy classes oftype D in sporadic simple groups. Contemp. Math., 585(2013) 247–259.

S. Freyre, M. Grana and L. Vendramin, On Nichols algebrasover SL(2, q) and GL(2, q). J. Math. Phys. 48, (2007) 123513.


References

S. Freyre, M. Grana and L. Vendramin, On Nichols algebrasover PGL(2, q) and PSL(2, q). J. Algebra Appl. 9 (2010),195–208.

G. A. Garcıa and A. Garcıa Iglesias, Finite dimensional pointedHopf algebras over S4, Israel J. Math., 183 (2011), 417–444.

A. Garcıa Iglesias and C. Vay, Finite-dimensional Pointed orCopointed Hopf algebras over affine racks, Journal of Algebra,to appear.

G. Malle and D. Testerman, Linear Algebraic Groups andFinite Groups of Lie Type, Cambridge Studies in AdvancedMathematics 133 (2011).

W.D. Nichols, Bialgebras of type one, Commun. Alg. 6 (1978),1521–1552.

Documents

Álgebras de Hopf punteadas sobre grupos finitos simples de ...ggarcia/encuentros/ggarcia-uma-rosario.pdf · Introduction Joint work with N. Andruskiewitsch and G. Carnovale. Main