# From finite groups to finite-dimensional Hopf algebras.

Introduction

Finite-dimensional Hopf algebras H have an elusive nature. On one hand they resemble finite groups in many aspects, while on the other, elementary facts about finite groups are either hard to translate or even false. In fact, Kaplansky's famous conjectures from 1975[?], are all attempts to generalize results from groups to Hopf algebras. Some of his conjectures are still open.

Even the naive translation of normal subgroups to normal Hopf subalgebras is problematic. There exist semisimple Hopf algebras of dimension 36 that have no nontrivial normal Hopf subalgebras [?], yet Burnside's theorem on solvable groups assures the existence of normal subgroups for groups of order 36. A more suitable translation of subgroups are left coideal subalgebras. When N is a subgroup of a group G then kN can be considered as a left coideal subalgebra of the Hopf algebra kG. Normal left coideal subalgebras correspond to normal subgroups in this way.

Explicitly, a Hopf algebra H is an algebra over a field k endowed with a counital coassociative coalgebra structure map [DELTA] : H [right arrow]* H [cross product] H, compatible with the algebra structure of H, an augmentation algebra map [epsilon] : H [right arrow] k and an antipode S : H [right arrow]* H. A left coideal subalgebra N of H is a subalgebra of H such that [DELTA](N) [subset] H [cross product] N. There exists also an appropriate notion of normality.

Just as normal subgroups for groups, normal left coideal subalgebras give rise to Hopf quotients. Explicitly, if N is a normal left coideal subalgebra of H then the Hopf quotient related to N is H/H[N.sup.+] where [N.sup.+] = N [intersection] ker [epsilon] and H[N.sup.+] = [N.sup.+]H is then a Hopf ideal. On the level of groups, Hopf quotients and group algebras of quotient groups coincide. For if N is a normal subgroup of G then the group algebra k(G/N) is precisely the Hopf quotient kG/(kG) [(kN).sup.+].

However, while group quotients are always of the form G/N, N a normal subgroup, Hopf quotients arise from normal left coideal subalgebras which are not necessarily Hopf subalgebras. This is the cause for a major difference between general Hopf algebras and group algebras. We avoid this problem by using integrals.

The existence of integrals for finite dimensional Hopf algebras is one of the first major achievements in their structure theory, due to Larson and Sweedler from the late 60's [12]. Integrals play the role of the averaging elements [[summation].sub.g[member of]G] g for finite groups G. Since the 90's the existence of integrals for left coideal subalgebras was established as well (e.g [11, ?, 13, 14]). Using integrals for the Hopf algebra H, for its dual Hopf algebra H* and for left coideal subalgebras of both, enable us to generalize results from groups to Hopf algebras.

In this survey we describe conjugacy classes, character tables, commutators, nilpotency and solvability for semisimple Hopf algebras. At each stage we show how our general approach coincides with the classical notions for groups.

1 Preliminaries and notations

Throughout H is a d-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0 and H* is its dual which is a semisimple Hopf algebra as well. In several of the following we assume that k = [??]. Let [epsilon] : H [right arrow] k denote the augmentation map. We use the Sweedler notation for the coalgebra structure

[DELTA](h) =[summation][h.sub.1][cross product][h.sub.2] [member of] H [cross product] H

An element h [member of] H is cocommutative if [summation][h.sub.1] [cross product] [h.sub.2] = [summation] [h.sub.2] [cross product] [h.sub.1]. for all h [member of] H.

We denote by S and s the antipodes of H and H* respectively. It is known that when H is semisimple then [S.sup.2] = Id.

We denote by [and] and [lambda] the integrals of H and H* respectively. Recall, h[and] = [and]h = ([epsilon],h) [and] for all h [member of] H and similarly p[lambda] = [lambda]p = (p, 1) [lambda] for all p [member of] H*. We choose [and] to be idempotent and [lambda] to satisfy ([lambda], [and]) = 1.

The Hopf algebra H* becomes a right and left H-module by the hit actions [??] [??] defined for all a [member of] H, p [member of] H*,

<p [??] a,a'> = <p,aa'> <a [??] p,a'> = <p,a' a>

H becomes a left and right H* module analogously.

It is known that [and] is a free generator for H as a left and a right H*-module. That is, H* [??] [and] = [and] [??] H* = H.

Denote by [.sub.ad] the left adjoint action of H on itself, that is, for all a,h [member of] H,

[h.sub.ad]a = [summation][h.sub.1]aS([h.sub.2]).

A left coideal subalgebra of H is a subalgebra N so that [DELTA](N) [subset] H [cross product] N. It is called normal if it is stable under the left adjoint action of H.

Every left coideal subalgebra of H is semisimple as an algebra and equipped with a 1-dimensional ideal of integrals, we denote by [[and].sub.N] the unique idempotent integral of N.

The most basic example of a semisimple Hopf algebra is H = kG, G a finite group, we keep this assumption throughout.

Example 1.1. Recall, kG is a Hopf algebra with [DELTA](g) = g [cross product] g, S(g) = [g.sub.-1] and <[epsilon], g> = 1 for all g [member of] G. Clearly kG is a cocommutative Hopf algebra. We have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is an idempotent integral for kG. The dual Hopf algebra kG* has a linear basis {pg} where <pg,g'> = [[delta].sub.g,g'], that is, a dual basis for the basis [{g}.sub.g[member of]G] of kG. We have [lambda] = [|G|p.sub.1].

It can be checked that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The semisimple Hopf algebra kG* is commutative and the elements {[p.sub.g]} form a set of primitive idempotents for it. We have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2 Conjugacy classes, Class sums and Character tables

The representation theories of semisimple Hopf algebras and of finite groups are closely related in the sense that they both give rise to fusion categories (that is, k -linear semisimple rigid tensor categories with finitely many simple objects and finite dimensional spaces of morphisms). These categories are tensor categories due to the fact that the tensor product of two representations is a representation as well via the coproduct structure. As a result their characters form an algebra.

Explicitly, for a finite-dimensional left H-module V denote its structure map H [right arrow]* En[d.sub.k](V) by [[rho].sub.V]. Then the character [x.sub.V] of V is defined by

<[x.sub.V],h> = Trace ([[rho].sub.V](h))

for all h [member of] H. Let d = dim H and {k = [V.sub.0], ... [V.sub.n-1]} be a complete set of nonisomorphic irreducible H-modules. We set Irr(H) = {[x.sub.i]}, the corresponding characters where [x.sub.i] = x[V.sub.i] and ([x.sub.1], 1) = [d.sub.i], the dimension of the irreducible module [V.sub.i]. Then we have

[x.sub.[V.sub.[x.sub.[W.sub. = [x.sub.[V.sub.[cross product]W,

where the product on the left coincide with the product in the Hopf algebra H*. We have also that [lambda] is the character of the left (right) regular representation of H, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Denote by R(H) [subset] H* the k-span of Irr(H). It is in fact an algebra which coincides with the algebra of cocommutative elements of H*. By ([?, 15]) it is a semisimple algebra. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a complete set of central primitive idempotents of R(H), and let {[f.sub.0], ... ,[f.sub.m-1]} be primitive orthogonal idempotents in R(H) so that [f.sub.i][F.sub.j] = [[delta].sub.ij] [f.sub.i].

In what follows we give the initial definitions that allow us to generalize results from group theory to the structure theory of Hopf algebras. We define a conjugacy class as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

We generalize also the notions of a Class sum and of a representative of a conjugacy class:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

We refer to [[eta].sub.i] as a normalized class sum. By results in [?], the elements [C.sub.i] are central in H.

When H = kG, G a finite group, the definition of conjugacy classes, class sums and representatives reduce to the usual definitions. To see this we continue with Example 1.1

Example 2.1. Let H = kG, G a finite group. Since kG* is a commutative algebra, so is R(H). It can be checked that if ([[member of].sub.i] is the ith conjugacy class of G, then the element [F.sub.i] [member of] kG*,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is cocommutative, hence belongs to R(H). In fact {[F.sub.i]} is the set of (central) primitive idempotents of R(H). In particular [F.sub.i] = [f.sub.i] in our notations.

The classical notions of conjugacy classes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and corresponding class sums [C.sub.i] are given respectively by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [g.sub.i] is an arbitrary representative of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By using the notations and the formulas in Example 1.1, they can be realized exactly as in (1) and (2). That is, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note also that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For general Hopf algebras we can not choose a representative [g.sub.i] from the conjugacy class [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We replace it by an "average" - the normalized class sum [[eta].sub.i] defined in (2). For groups it boils down to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The following result describes, in a certain sense, the "essence" of conjugacy classes. Given a Hopf algebra H, its Drinfel'd double D(H) defined in [9] is a Hopf algebra structure defined on H* [cross product] H, the tensor product of H and its dual H*. From a categorical point of view, the category of representations of D(H) is the center of the category of H-representations. We have shown in [6] that:

Proposition 2.2. Each conjugacy class [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an irreducible left D(H)-module and moreover,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as D(H)-modules, where [m.sub.i] = dim [f.sub.i]R(H).

We are ready now to define the character table for Hopf algebras. Recall, the (i,j)-th entry in the character table for G is ([x.sub.i],[g.sub.j]) where [g.sub.j] is an element of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since this value does not depend on the representative, we can replace [g.sub.j] by the 'average' representative [[eta].sub.j]. We thus define a generalized character table for H as follows:

Definition 2.3. The generalized character table ([[xi].sub.ij]) of a semisimple Hopf algebra H over [??] is given by:

[[xi].sub.ij] = <[x.sub.i], [[eta].sub.j]>,

for all irreducible characters [x.sub.i] and generalized class sums [[eta].sub.j] of H.

When R(H) is commutative then {[F.sub.i]} forms a basis of R(H) and [f.sub.i] = [F.sub.i] for all i. In this case, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the character table is a square matrix. Moreover, many other properties of character tables for groups can be generalized in this case.

The common family of Hopf algebras for which R(H) is a commutative algebra consists of quasitriangular Hopf algebras introduced by Drinfel'd [9] in the context of quantum groups. They can be described as quotients of Drinfel'd doubles. Alternatively, they can be described as Hopf algebras whose category of representations is also braided. Group algebras are always quasitriangular. Factorizable Hopf algebras are special kind of quasitriangular Hopf algebras. Drinfel'd doubles are the basic examples of factorizable Hopf algebras.

In [4] we listed properties of groups related to their character tables and showed how they can be generalized to Hopf algebras for which R(H) is a commutative algebra. We summarize:

Theorem 2.4. Let H be a semisimple Hopf algebra over [??] and assume R(H) is commutative. Then the following hold.

1. The entries of the character table [[xi].sub.ij] are algebraic integers.

2. In each row i, [[xi].sub.i0] = <[x.sub.i], 1> is maximal amongst all absolute values of the entries in this row. That is, |[[xi].sub.ij]| < <[x.sub.i], 1> for all 0 [less than or equal to] i [less than or equal to] n - 1.

3. For groups, the product of two class sums is an integral sum of class sums. The structure constants of this product are obtained from the character table.

For Hopf algebras, if [C.sub.i], [C.sub.j] are any conjugacy sums, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where for any t,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If H is quasitriangular then we have integral coefficients {[c.sub.ijt]} up to a factor of [d.sup.-3]. If H is factorizable then these coefficients are integral up to a factor of [d.sup.-1].

4. For groups, different columns are orthogonal, while the square of the norm of each column j equals [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Different rows satisfy generalized orthogonality relations.

The same result for Hopf algebras takes the following form.

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As a result, the size of each conjugacy class can be determined from the norm of the corresponding column.

5. For groups, the kernel of the irreducible representation [V.sub.i] is the union of all conjugacy classes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for which <[x.sub.i],[[eta].sub.j]> = <[x.sub.i], 1> = dim [V.sub.i].

For Hopf algebras we use the term Left kernel defined in [1] that generalizes kernels of group representations to left kernels of H-representations. While kernels of group representations are normal subgroups, left kernels of H-representations are normal left coideal subalgebras. We have:

Let [V.sub.i] be an irreducible representation of H, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

6. For groups, a collection ofconjugacy classes forms a normal subgroup if and only if it is an intersection of kernels of some irreducible representations. As a result, the order of all the normal subgroups and the inclusion relations among them can be determined from their character tables.

Similar result holds for normal left coideal subalgebras. We have,

Let L = [[direct sum].sub.j[member of]J] [C.sub.j], for some set J [subset] {0, ... ,n - 1}. Then L is a subalgebra (necessarily normal left coideal) of H if and only if there exists I [subset] {0, ... , n - 1} so that J = [J.sub.[I.sub., where

JI = {j | [x.sub.i][F.sub.j] = <[x.sub.i], 1> [F.sub.j] for all i [member of] I}.

As a result, the dimensions of all the normal left coideal subalgebras of H, and the inclusion relations among them can be determined from its character table.

When R(H) is not commutative, generalizing results from groups is more complicated. The character tables are no longer square matrices and the central idempotents [F.sub.i] in R(H) are sums of primitive idempotents [fi.sub.j]. As a result, class sums [C.sub.i] do not belong to a unique conjugacy class, but to a sum of isomorphic conjugacy classes. Explicitly, set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we choose arbitrarily [f.sub.i] = [fi.sub.j], for some j, then we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [m.sub.i] = dim [f.sub.i]R(H).

In [5] we extended as possible the results in Theorem 2.4 to the general case where R(H) is not necessarily commutative. We summarize:

Theorem 2.5. Let H be a d-dimensional semisimple Hopf algebra over [??], then

* Integr ability: All entries of the character table are algebraic integers.

* Absolute maximality: (i) Each entry of the i-th row of the character table satisfies:

|<[x.sub.i],[[eta].sub.j]>| - [m.sub.j]<[x.sub.i],1>.

(ii) Equality holds if and only if right multiplication by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] acts on [f.sub.j]R(H) as [[alpha].sub.i] Id[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where |[[alpha].sub.i]| = <[x.sub.i],1>.

(iii) <[x.sub.i], [[eta].sub.j]> = [m.sub.j]<[x.sub.i], 1> if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] acts on [f.sub.j]R(H) as <[x.sub.i], 1> [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

* Orthogonality of columns:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

* Analogues of characterizations of kernels. Let {[F.sub.j]} be the set of central primitive idempotents of R(H) and let {[V.sub.i]} be the set of irreducible representations of H. Then the following are equivalent:

(i) [F.sub.j][x.sub.i] = <[x.sub.i],1> [F.sub.j]

(ii) {[x.sub.i],[[eta].sub.j]> = [m.sub.j] <[x.sub.i], 1>.

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3 Commutators, nilpotency and solvability

The commutator subalgebra H' of a semisimple Hopf algebra H was first defined in [1]. It is a normal left coideal subalgebra of H for which H/[HH'.sup.+] is commutative and is minimal with respect to this property.

It is not hard to see that

H' = {h [member of] H|[sigma] [??] h = h[for all][sigma] [member of] G(H*)}. (3)

Here G(H*) denotes the group of group-like elements of H*, that is, the 1-dimensional (and thus multiplicative) characters on H.

Generalizing from groups we described H' in terms of Hopf algebraic commutators [6]. Let H be any Hopf algebra over k. For a,b [member of] H, define their commutator {a,b} as:

{a,b} = [summation][a.sub.1][b.sub.1]S[a.sub.2]S[b.sub.2]. (4)

Define

Com = spa[n.sub.k]{{a,b} | a,b [member of] H}. (5)

Then Com is a left coideal of H. In [6] We showed:

Proposition 3.1. Let H be a semisimple Hopf algebra then the commutator subalgebra H' of H is the algebra generated by Com.

Example 3.2. Let G be a finite group and G' its commutator subgroup. Then G/G' is an abelian group and G' is minimal with respect to this property. Moreover, all 1-dimensional characters of G, are trivial (that is, act as 1) when restricted to G'. This is the analogue of the commutator subalgebra H' of H and of (3).

Since [DELTA](g) = g [cross product] g, we find that the definition of the commutator given in (4), boils down to the group commutator [a,b] = ab[a.sup.-1][b.sup.-1] for all a, b [member of] G. The commutator group G' is the group generated by all commutators and is normal in G. Proposition 3.1 is a generalization of this fact.

Another concept which appears in the literature for group algebras is the commutator space of kG

K = S[p.sub.k]{xy - yx, x,y [member of] kG}.

The ideal generated by K is in fact kG[(kG').sup.+], which is the kernel of the algebra map kG [right arrow] kG'. Moreover, it is contained in the kernel of any algebra map kG [right arrow] k[bar.G] when [bar.G] is commutative. For general Hopf algebras, the corresponding Hopf ideal is [HH'.sup.+].

In [6] we introduced a family of elements in H' denoted by [z.sub.n], n > 1, which arise from the idempotent integral of H. This family consists of powers of the S-fixed central invertible element [z.sub.2],

[z.sub.2] = {[and], [and]'} = [summation][[and].sub.1][[and]'.sub.1]S[[and].sub.2]S[[and]'.sub.2]

where [and], [and]' are two copies of the idempotent integral of H. We refer to [z.sub.2] as an extensive commutator.

The extensive commutator [z.sub.2] has a very nice form which is the key for determining commutativity We showed

Theorem 3.3. Let H be a semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is commutative if and only if [z.sub.2] [member of] k.

Let {[E.sub.i]} be the set of central orthogonal idempotents of H. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The element [z.sub.2] generates H' in the following sense. H' is the algebra generated by the left coideal [z.sub.2] [??] H*.

What is the role played by the extensive commutator [z.sub.2] inside kG?

Example 3.4. Recall for groups [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where f is the Frobenius counting function. That is, f(g) counts the number of times g can be obtained as a commutator ab[a.sup.-1][b.sup.-1]. Clearly, if [[summation].sub.a,,b[member of]G] ab[a.sup.-1][b.sup.-1] [member of] k, then all commutators are trivial which means that G is abelian. This fact is expressed in the first statement of Theorem 3.3.

The group elements {g} that appear in {[and], [and]} with a nonzero coefficient, (that is f(g) [not equal to] 0), are precisely those that generate G'. This fact is expressed in the last part of Theorem 3.3 since [z.sub.2] [??] pg = f(g)g for all g [member of] G.

The fact that [d.sub.i]||G| is a famous property of groups. Frobenius has shown that the counting function f has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

meaning that f is a character on G. It can be seen that f and the extensive commutator [z.sub.2] are related by:

f = [lambda] [??] d[z.sub.2].

For general Hopf algebras, Kaplansky's 6th conjecture states that the dimension of any irreducible H-module divides dimH. This conjecture is still open. However, in [7] we refer to the function [f.sub.com] = [lambda] [??] d[z.sub.2] and to some other counting functions as distribution functions.

Nilpotent groups can be defined in terms of upper central series of normal subgroups,

1 = [Z.sub.0](G) [subset] [Z.sub.1](G) [subset] [Z.sub.2](G) [subset] ... [subset] [Z.sub.t](G) = G,

where [Z.sub.1] (G) = Z(G), the center of the group G, and

[Z.sub.i+1](G)/[Z.sub.i](G) = Z(G/[Z.sub.i](G)).

Equivalently, it is defined in terms of lower central series of commutators,

G [contains] [G.sub.1] [contains] [G.sub.2] [contains] ... [contains] 1,

where [G.sub.1] = [G, G], [G.sub.i] = [G, [G.sub.i-1]], the ith commutator subgroup.

Hopf commutators on one hand and integrals of certain left coideal subalgebras enable us to define nilpotent Hopf algebras as a natural generalization of nilpotent groups. The generalization of upper central series is not trivial though. While the set {[Z.sub.i](G)} are all normal subgroups of G, it makes sense to consider quotients of successive subgroups, this is no longer the case for Hopf algebras. Though we can replace normal subgroups with normal left coideal subalgebras, we do not have a meaningful replacement for quotients.

We overcome this problem by using integrals for left coideal subalgebras. Observe that [Z.sub.i+1](G)/[Z.sub.i](G) is the image of [Z.sub.i+1](G) under the group projection [[pi].sub.i] : G [right arrow] G/[Z.sub.i](G). In [8], we showed that if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a Hopf algebra projection, and [[and].sub.Zi] is the integral of Zi, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an algebra (and H-module) isomorphism. In particular, if [Z.sub.i+1] [contains] [Z.sub.i], then we have an algebra isomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This observation enables us to replace quotients with subalgebras and thus suggest a definition for an upper central series for Hopf algebras as follows:

Definition 3.5. An ascending (upper) series for H is a series of normal left coideal subalgebras

k = [Z.sub.0] [subset] [Z.sub.1] [subset] ... [subset] [Z.sub.t] [subset]

so that

[Z.sub.i+1][and][Z.sub.i] = Z(H[[and].sub.Zi]),

for all 1 [less than or equal to] i [less than or equal to] t, where [[and].sub.Zi] is the integral of [Z.sub.i] and Z(H[[and].sub.Zi]) is the center of H[[and].sub.Zi].

The lower series for H are described via Hopf commutators as follows.

Definition 3.6. A descending lower series for H is a series of iterated commutators,

H = [N.sub.0] [contains] [N.sub.1] [contains] ... [contains] [N.sub.t] [contains],

so that [N.sub.0] = H, [N.sub.i] = the normal left coideal subalgebra generated by {[N.sub.i-1], [and]}.

As for groups, when H is nilpotent the first series ends with H while the second one ends with k. They are also interrelated. Based on [7, Th. 3.7] and [8, Prop. 3.8] we have,

Theorem 3.7. Let H be a semisimple Hopf algebra over an algebraically closed field of characteristic 0, let {[Z.sub.i]} be an ascending central series and {[N.sub.i]} be a descending central series. Then

[N.sub.t] = [k.sub.1] [??] [Z.sub.t] = H.

In this case H is nilpotent and we have: [N.sub.t-i] [bar.[subset]] [Z.sub.i] for all 0 [less than or equal to] i [less than or equal to] t.

Iterated commutators of the extensive commutator z2 yield another criterion for nilpotency of H. Define an operator T : Z (H) [??] Z (H) by

T (z) = {z, [and]}.

We have:

Proposition 3.8. (i) The matrix of T with respect to the basis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is A, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

(ii) A has non-negative rational entries and the first column of A has all entries equal 1.

(iii) The first row of [A.sup.m] is (1,0, ... , 0) for all m [greater than or equal to] 0.

(iv) In the first column of [A.sup.m] we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

For all m > 0, 0 [less than or equal to] i [less than or equal to] n - 1. In particular, The first column of [A.sup.m] consists of positive rational numbers.

We refer to A as the Commutator matrix of H. Observe that this matrix depends only on the structure constants of the ring of characters on H. Eigenvalues of A are related to nilpotancy of H.

Theorem 3.9. Let H be a semisimple Hopf algebra over an algebraically closed field of characteristic 0 and assume [x.sub.i]s([x.sub.i]) [member of] Z(R(H)) for each irreducible character [x.sub.i]. Then H is nilpotent if and only if its commutator matrix A has eigenvalues {1,0} where the algebraic multiplicity of 1 is 1.

The condition [x.sub.i]s([x.sub.i]) [member of] Z(R(H)) is satisfied by an abundance of semisimple H. In particular by all quasitriangular Hopf algebras, but also by H = kG*, even though R(H) = kG is not necessarily commutative.

We next define an important family of central iterated commutators .

[[gamma].sub.0] = [and], [[gamma].sub.1] = T([and]) = {[and], [and]}, ... , [[gamma].sub.m] = [T.sub.m]([and]) = {[[gamma].sub.m-1], [and]}. (7)

We can describe the central iterated commutators [[gamma].sub.n] in terms of the coefficients of A. In particular, they are all positive rational numbers.

Proposition 3.10. Let [[gamma].sub.m] be defined as in (7) and the matrix A be defined as in Proposition 3.8. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all m [greater than or equal to] 1. Moreover, the coefficient of each [E.sub.i] in [[gamma].sub.m] is a non-zero rational number, in particular [[gamma].sub.m] is invertible.

Proposition 3.10 yields another criterion for nilpotency of H.

Theorem 3.11. Let H be a semisimple Hopfalgebra over an algebraically closed field of characteristic 0, and let [[gamma].sub.t] be defined as in (7). Assume [x.sub.i]s([x.sub.i]) [member of] Z(R(H)) for each irreducible character [x.sub.i]. Then H is nilpotent if and only if [[gamma].sub.m] = 1for some m [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Its index of nilpotency is the least integer m so that [[gamma].sub.m] = 1.

The last concept we wish to discuss is solvability. On the level of category theory there exists a notion of solvability [10] and it is customary to define solvable Hopf algebras H as those for which Rep(H) is a solvable category. However, this non-intrinsic definition is unsatisfactory as it contradicts our intuition from group theory. Commutative or nilpotent Hopf algebras are not always solvable in this sense [10, Prop. 4.5(ii),Remark 4.6(i)]. We suggested in [8] an intrinsic definition of solvability which is consistent with solvability for finite groups and as desired, commutative or nilpotent Hopf algebras are indeed solvable.

A solvable group has a subnormal series of subgroups. That is,

1 = [G.sub.0] [subset] [G.sub.1] [subset] ... [subset] G,

so that [G.sub.i] is normal in [G.sub.i+1]. G is solvable if it has a subnormal series so that [G.sub.i+1]/[G.sub.i] is an abelian group. Generalizing this definition to Hopf algebras requires an appropriate translation for subnormal series in addition to abelian quotients. This is done using integrals as follows:

Definition 3.12. Let H be a semisimple Hopf algebra. A subnormal series of left coideal subalgebras of H

k = [N.sub.0] [subset] [N.sub.1] [subset] ... [subset] [N.sub.t] = H

is a series satisfying:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all 0 [less than or equal to] i [less than or equal to] t - 1, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the integral of [N.sub.i]. H is solvable if for all a,b [member of] [N.sub.i+1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 3.13. For finite groups, the definition of solvable Hopf algebras boils down to the usual definition of solvability for groups.

To see this, let [G.sub.1] [subset] G and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] x. If [and][G.sub.1] is central in G, then for g [member of] G we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It follows that for x[member of] [G.sub.1], xg = gy for some y [member of] [G.sub.1]. Thus [G.sub.1] is normal in G.

We claim now that G/[G.sub.1] is abelian if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all a,b [member of] G. Indeed, assume [bar.a][bar.b] = [bar.b][bar.a] for all [bar.a],[bar.b] [member of] G/[G.sub.1]. Then ab[a.sup.-1] = by for some y [member of] [G.sub.1]. This implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Conversely, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then ab[a.sup.-1] = by for some y [member of] [G.sub.1]. This shows our claim.

We showed in [8] that many properties of solvable groups can be generalized to solvable Hopf algebras. In particular, any nilpotent Hopf algebra is solvable. We proved also an analogue of Burnside's [p.sup.a][q.sup.b] theorem for semisimple quasi triangular Hopf algebras.

Theorem 3.14. Let H be a quasitriangular semisimple Hopf algebra of dimension [p.sup.a][q.sup.b] over afield k of characteristic 0, p, q primes. Then H is solvable.

References

[1] S. Burciu, Kernel of representations and coideal subalgebras for Hopf algebras, Glasg. Math. J. 54 (2012), no. 1,107-119.

[2] C.W Curtis and I.W Reiner, Methods of representation theory, Vol I, John Wiley and Sons, Inc. New York, 1990.

[3] M. Cohen and S. Westreich, Conjugacy Classes, Class Sums and Character Tables for Hopf Algebras, Comm. Algebra, 39, (2011), no. 12,4618-4633.

[4] M.Cohen and S. Westreich, Recovering information from the character tables of Hopf algebras - normality, dimensions and quotients, Contemporary Math (585), Amer. Math. Soc , (2013), 213-227.

[5] M. Cohen and S. Westreich, Character tables and normal left coideal subalgebras, J. Pure Appl. Algebra 218 (2014), 1845-1866.

[6] M. Cohen and S. Westreich, Are we counting or measuring something? J. Algebra 398 (2014),111-130.

[7] M. Cohen and S. Westreich, Probabalistically nilpotent Hopf algebras, T. Am. Math. Soc, 368, No. 6, (2016), 4295-4314.

[8] M. Cohen and S. Westreich, Solvability for semisimple Hopf algebras via integrals, J. Algebra 472 (2017), 67-94.

[9] V. G. Drinfel'd, On almost cocommutative Hopf algebras, Leningrad Math. J. 1 (1990), 321-342.

[10] P. Etingof, D. Nikshych and V. Ostrik, Weakly group-theoretical and solvable fusion categories, Adv. Math. 226, (2011), 176-205.

[11] M. Kopinnen, Coideal subalgebras in Hopf algebras: freeness, integrals, smash products, Comm. Algebra 21(2), (1993), 427-444.

[12] R.G. Larson, M.E. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969) 75-93.

[13] S. Skriyabin, Projectivity and freeness over comodule algebras, Trans. of AMS, 359, no 6, (2007), 2597-2623

[14] Takeuchi. Quotient Spaces for Hopf Algebras. Comm. Alg., 22(7), (1995), 2503-2523.

[15] Y. Zhu, Hopf algebras of prime dimension, Int. Math. Res. Not. (1994), 53-59.

Miriam Cohen

Sara Westreich

Received by the editors in May 2016.

Communicated by S. Caenepeel.

2010 Mathematics Subject Classification : 16T05, 20C05.

Department of Mathematics

Ben Gurion University of the Negev, Beer Sheva, 8410501 Israel

email: mia@math.bgu.ac.il

Department of Management

Bar Ilan University, Ramat Gan, 529002 Israel

email: swestric@biu.ac.il

Finite-dimensional Hopf algebras H have an elusive nature. On one hand they resemble finite groups in many aspects, while on the other, elementary facts about finite groups are either hard to translate or even false. In fact, Kaplansky's famous conjectures from 1975[?], are all attempts to generalize results from groups to Hopf algebras. Some of his conjectures are still open.

Even the naive translation of normal subgroups to normal Hopf subalgebras is problematic. There exist semisimple Hopf algebras of dimension 36 that have no nontrivial normal Hopf subalgebras [?], yet Burnside's theorem on solvable groups assures the existence of normal subgroups for groups of order 36. A more suitable translation of subgroups are left coideal subalgebras. When N is a subgroup of a group G then kN can be considered as a left coideal subalgebra of the Hopf algebra kG. Normal left coideal subalgebras correspond to normal subgroups in this way.

Explicitly, a Hopf algebra H is an algebra over a field k endowed with a counital coassociative coalgebra structure map [DELTA] : H [right arrow]* H [cross product] H, compatible with the algebra structure of H, an augmentation algebra map [epsilon] : H [right arrow] k and an antipode S : H [right arrow]* H. A left coideal subalgebra N of H is a subalgebra of H such that [DELTA](N) [subset] H [cross product] N. There exists also an appropriate notion of normality.

Just as normal subgroups for groups, normal left coideal subalgebras give rise to Hopf quotients. Explicitly, if N is a normal left coideal subalgebra of H then the Hopf quotient related to N is H/H[N.sup.+] where [N.sup.+] = N [intersection] ker [epsilon] and H[N.sup.+] = [N.sup.+]H is then a Hopf ideal. On the level of groups, Hopf quotients and group algebras of quotient groups coincide. For if N is a normal subgroup of G then the group algebra k(G/N) is precisely the Hopf quotient kG/(kG) [(kN).sup.+].

However, while group quotients are always of the form G/N, N a normal subgroup, Hopf quotients arise from normal left coideal subalgebras which are not necessarily Hopf subalgebras. This is the cause for a major difference between general Hopf algebras and group algebras. We avoid this problem by using integrals.

The existence of integrals for finite dimensional Hopf algebras is one of the first major achievements in their structure theory, due to Larson and Sweedler from the late 60's [12]. Integrals play the role of the averaging elements [[summation].sub.g[member of]G] g for finite groups G. Since the 90's the existence of integrals for left coideal subalgebras was established as well (e.g [11, ?, 13, 14]). Using integrals for the Hopf algebra H, for its dual Hopf algebra H* and for left coideal subalgebras of both, enable us to generalize results from groups to Hopf algebras.

In this survey we describe conjugacy classes, character tables, commutators, nilpotency and solvability for semisimple Hopf algebras. At each stage we show how our general approach coincides with the classical notions for groups.

1 Preliminaries and notations

Throughout H is a d-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0 and H* is its dual which is a semisimple Hopf algebra as well. In several of the following we assume that k = [??]. Let [epsilon] : H [right arrow] k denote the augmentation map. We use the Sweedler notation for the coalgebra structure

[DELTA](h) =[summation][h.sub.1][cross product][h.sub.2] [member of] H [cross product] H

An element h [member of] H is cocommutative if [summation][h.sub.1] [cross product] [h.sub.2] = [summation] [h.sub.2] [cross product] [h.sub.1]. for all h [member of] H.

We denote by S and s the antipodes of H and H* respectively. It is known that when H is semisimple then [S.sup.2] = Id.

We denote by [and] and [lambda] the integrals of H and H* respectively. Recall, h[and] = [and]h = ([epsilon],h) [and] for all h [member of] H and similarly p[lambda] = [lambda]p = (p, 1) [lambda] for all p [member of] H*. We choose [and] to be idempotent and [lambda] to satisfy ([lambda], [and]) = 1.

The Hopf algebra H* becomes a right and left H-module by the hit actions [??] [??] defined for all a [member of] H, p [member of] H*,

<p [??] a,a'> = <p,aa'> <a [??] p,a'> = <p,a' a>

H becomes a left and right H* module analogously.

It is known that [and] is a free generator for H as a left and a right H*-module. That is, H* [??] [and] = [and] [??] H* = H.

Denote by [.sub.ad] the left adjoint action of H on itself, that is, for all a,h [member of] H,

[h.sub.ad]a = [summation][h.sub.1]aS([h.sub.2]).

A left coideal subalgebra of H is a subalgebra N so that [DELTA](N) [subset] H [cross product] N. It is called normal if it is stable under the left adjoint action of H.

Every left coideal subalgebra of H is semisimple as an algebra and equipped with a 1-dimensional ideal of integrals, we denote by [[and].sub.N] the unique idempotent integral of N.

The most basic example of a semisimple Hopf algebra is H = kG, G a finite group, we keep this assumption throughout.

Example 1.1. Recall, kG is a Hopf algebra with [DELTA](g) = g [cross product] g, S(g) = [g.sub.-1] and <[epsilon], g> = 1 for all g [member of] G. Clearly kG is a cocommutative Hopf algebra. We have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is an idempotent integral for kG. The dual Hopf algebra kG* has a linear basis {pg} where <pg,g'> = [[delta].sub.g,g'], that is, a dual basis for the basis [{g}.sub.g[member of]G] of kG. We have [lambda] = [|G|p.sub.1].

It can be checked that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The semisimple Hopf algebra kG* is commutative and the elements {[p.sub.g]} form a set of primitive idempotents for it. We have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2 Conjugacy classes, Class sums and Character tables

The representation theories of semisimple Hopf algebras and of finite groups are closely related in the sense that they both give rise to fusion categories (that is, k -linear semisimple rigid tensor categories with finitely many simple objects and finite dimensional spaces of morphisms). These categories are tensor categories due to the fact that the tensor product of two representations is a representation as well via the coproduct structure. As a result their characters form an algebra.

Explicitly, for a finite-dimensional left H-module V denote its structure map H [right arrow]* En[d.sub.k](V) by [[rho].sub.V]. Then the character [x.sub.V] of V is defined by

<[x.sub.V],h> = Trace ([[rho].sub.V](h))

for all h [member of] H. Let d = dim H and {k = [V.sub.0], ... [V.sub.n-1]} be a complete set of nonisomorphic irreducible H-modules. We set Irr(H) = {[x.sub.i]}, the corresponding characters where [x.sub.i] = x[V.sub.i] and ([x.sub.1], 1) = [d.sub.i], the dimension of the irreducible module [V.sub.i]. Then we have

[x.sub.[V.sub.[x.sub.[W.sub. = [x.sub.[V.sub.[cross product]W,

where the product on the left coincide with the product in the Hopf algebra H*. We have also that [lambda] is the character of the left (right) regular representation of H, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Denote by R(H) [subset] H* the k-span of Irr(H). It is in fact an algebra which coincides with the algebra of cocommutative elements of H*. By ([?, 15]) it is a semisimple algebra. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a complete set of central primitive idempotents of R(H), and let {[f.sub.0], ... ,[f.sub.m-1]} be primitive orthogonal idempotents in R(H) so that [f.sub.i][F.sub.j] = [[delta].sub.ij] [f.sub.i].

In what follows we give the initial definitions that allow us to generalize results from group theory to the structure theory of Hopf algebras. We define a conjugacy class as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

We generalize also the notions of a Class sum and of a representative of a conjugacy class:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

We refer to [[eta].sub.i] as a normalized class sum. By results in [?], the elements [C.sub.i] are central in H.

When H = kG, G a finite group, the definition of conjugacy classes, class sums and representatives reduce to the usual definitions. To see this we continue with Example 1.1

Example 2.1. Let H = kG, G a finite group. Since kG* is a commutative algebra, so is R(H). It can be checked that if ([[member of].sub.i] is the ith conjugacy class of G, then the element [F.sub.i] [member of] kG*,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is cocommutative, hence belongs to R(H). In fact {[F.sub.i]} is the set of (central) primitive idempotents of R(H). In particular [F.sub.i] = [f.sub.i] in our notations.

The classical notions of conjugacy classes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and corresponding class sums [C.sub.i] are given respectively by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [g.sub.i] is an arbitrary representative of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By using the notations and the formulas in Example 1.1, they can be realized exactly as in (1) and (2). That is, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note also that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For general Hopf algebras we can not choose a representative [g.sub.i] from the conjugacy class [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We replace it by an "average" - the normalized class sum [[eta].sub.i] defined in (2). For groups it boils down to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The following result describes, in a certain sense, the "essence" of conjugacy classes. Given a Hopf algebra H, its Drinfel'd double D(H) defined in [9] is a Hopf algebra structure defined on H* [cross product] H, the tensor product of H and its dual H*. From a categorical point of view, the category of representations of D(H) is the center of the category of H-representations. We have shown in [6] that:

Proposition 2.2. Each conjugacy class [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an irreducible left D(H)-module and moreover,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as D(H)-modules, where [m.sub.i] = dim [f.sub.i]R(H).

We are ready now to define the character table for Hopf algebras. Recall, the (i,j)-th entry in the character table for G is ([x.sub.i],[g.sub.j]) where [g.sub.j] is an element of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since this value does not depend on the representative, we can replace [g.sub.j] by the 'average' representative [[eta].sub.j]. We thus define a generalized character table for H as follows:

Definition 2.3. The generalized character table ([[xi].sub.ij]) of a semisimple Hopf algebra H over [??] is given by:

[[xi].sub.ij] = <[x.sub.i], [[eta].sub.j]>,

for all irreducible characters [x.sub.i] and generalized class sums [[eta].sub.j] of H.

When R(H) is commutative then {[F.sub.i]} forms a basis of R(H) and [f.sub.i] = [F.sub.i] for all i. In this case, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the character table is a square matrix. Moreover, many other properties of character tables for groups can be generalized in this case.

The common family of Hopf algebras for which R(H) is a commutative algebra consists of quasitriangular Hopf algebras introduced by Drinfel'd [9] in the context of quantum groups. They can be described as quotients of Drinfel'd doubles. Alternatively, they can be described as Hopf algebras whose category of representations is also braided. Group algebras are always quasitriangular. Factorizable Hopf algebras are special kind of quasitriangular Hopf algebras. Drinfel'd doubles are the basic examples of factorizable Hopf algebras.

In [4] we listed properties of groups related to their character tables and showed how they can be generalized to Hopf algebras for which R(H) is a commutative algebra. We summarize:

Theorem 2.4. Let H be a semisimple Hopf algebra over [??] and assume R(H) is commutative. Then the following hold.

1. The entries of the character table [[xi].sub.ij] are algebraic integers.

2. In each row i, [[xi].sub.i0] = <[x.sub.i], 1> is maximal amongst all absolute values of the entries in this row. That is, |[[xi].sub.ij]| < <[x.sub.i], 1> for all 0 [less than or equal to] i [less than or equal to] n - 1.

3. For groups, the product of two class sums is an integral sum of class sums. The structure constants of this product are obtained from the character table.

For Hopf algebras, if [C.sub.i], [C.sub.j] are any conjugacy sums, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where for any t,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If H is quasitriangular then we have integral coefficients {[c.sub.ijt]} up to a factor of [d.sup.-3]. If H is factorizable then these coefficients are integral up to a factor of [d.sup.-1].

4. For groups, different columns are orthogonal, while the square of the norm of each column j equals [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Different rows satisfy generalized orthogonality relations.

The same result for Hopf algebras takes the following form.

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As a result, the size of each conjugacy class can be determined from the norm of the corresponding column.

5. For groups, the kernel of the irreducible representation [V.sub.i] is the union of all conjugacy classes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for which <[x.sub.i],[[eta].sub.j]> = <[x.sub.i], 1> = dim [V.sub.i].

For Hopf algebras we use the term Left kernel defined in [1] that generalizes kernels of group representations to left kernels of H-representations. While kernels of group representations are normal subgroups, left kernels of H-representations are normal left coideal subalgebras. We have:

Let [V.sub.i] be an irreducible representation of H, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

6. For groups, a collection ofconjugacy classes forms a normal subgroup if and only if it is an intersection of kernels of some irreducible representations. As a result, the order of all the normal subgroups and the inclusion relations among them can be determined from their character tables.

Similar result holds for normal left coideal subalgebras. We have,

Let L = [[direct sum].sub.j[member of]J] [C.sub.j], for some set J [subset] {0, ... ,n - 1}. Then L is a subalgebra (necessarily normal left coideal) of H if and only if there exists I [subset] {0, ... , n - 1} so that J = [J.sub.[I.sub., where

JI = {j | [x.sub.i][F.sub.j] = <[x.sub.i], 1> [F.sub.j] for all i [member of] I}.

As a result, the dimensions of all the normal left coideal subalgebras of H, and the inclusion relations among them can be determined from its character table.

When R(H) is not commutative, generalizing results from groups is more complicated. The character tables are no longer square matrices and the central idempotents [F.sub.i] in R(H) are sums of primitive idempotents [fi.sub.j]. As a result, class sums [C.sub.i] do not belong to a unique conjugacy class, but to a sum of isomorphic conjugacy classes. Explicitly, set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we choose arbitrarily [f.sub.i] = [fi.sub.j], for some j, then we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [m.sub.i] = dim [f.sub.i]R(H).

In [5] we extended as possible the results in Theorem 2.4 to the general case where R(H) is not necessarily commutative. We summarize:

Theorem 2.5. Let H be a d-dimensional semisimple Hopf algebra over [??], then

* Integr ability: All entries of the character table are algebraic integers.

* Absolute maximality: (i) Each entry of the i-th row of the character table satisfies:

|<[x.sub.i],[[eta].sub.j]>| - [m.sub.j]<[x.sub.i],1>.

(ii) Equality holds if and only if right multiplication by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] acts on [f.sub.j]R(H) as [[alpha].sub.i] Id[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where |[[alpha].sub.i]| = <[x.sub.i],1>.

(iii) <[x.sub.i], [[eta].sub.j]> = [m.sub.j]<[x.sub.i], 1> if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] acts on [f.sub.j]R(H) as <[x.sub.i], 1> [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

* Orthogonality of columns:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

* Analogues of characterizations of kernels. Let {[F.sub.j]} be the set of central primitive idempotents of R(H) and let {[V.sub.i]} be the set of irreducible representations of H. Then the following are equivalent:

(i) [F.sub.j][x.sub.i] = <[x.sub.i],1> [F.sub.j]

(ii) {[x.sub.i],[[eta].sub.j]> = [m.sub.j] <[x.sub.i], 1>.

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3 Commutators, nilpotency and solvability

The commutator subalgebra H' of a semisimple Hopf algebra H was first defined in [1]. It is a normal left coideal subalgebra of H for which H/[HH'.sup.+] is commutative and is minimal with respect to this property.

It is not hard to see that

H' = {h [member of] H|[sigma] [??] h = h[for all][sigma] [member of] G(H*)}. (3)

Here G(H*) denotes the group of group-like elements of H*, that is, the 1-dimensional (and thus multiplicative) characters on H.

Generalizing from groups we described H' in terms of Hopf algebraic commutators [6]. Let H be any Hopf algebra over k. For a,b [member of] H, define their commutator {a,b} as:

{a,b} = [summation][a.sub.1][b.sub.1]S[a.sub.2]S[b.sub.2]. (4)

Define

Com = spa[n.sub.k]{{a,b} | a,b [member of] H}. (5)

Then Com is a left coideal of H. In [6] We showed:

Proposition 3.1. Let H be a semisimple Hopf algebra then the commutator subalgebra H' of H is the algebra generated by Com.

Example 3.2. Let G be a finite group and G' its commutator subgroup. Then G/G' is an abelian group and G' is minimal with respect to this property. Moreover, all 1-dimensional characters of G, are trivial (that is, act as 1) when restricted to G'. This is the analogue of the commutator subalgebra H' of H and of (3).

Since [DELTA](g) = g [cross product] g, we find that the definition of the commutator given in (4), boils down to the group commutator [a,b] = ab[a.sup.-1][b.sup.-1] for all a, b [member of] G. The commutator group G' is the group generated by all commutators and is normal in G. Proposition 3.1 is a generalization of this fact.

Another concept which appears in the literature for group algebras is the commutator space of kG

K = S[p.sub.k]{xy - yx, x,y [member of] kG}.

The ideal generated by K is in fact kG[(kG').sup.+], which is the kernel of the algebra map kG [right arrow] kG'. Moreover, it is contained in the kernel of any algebra map kG [right arrow] k[bar.G] when [bar.G] is commutative. For general Hopf algebras, the corresponding Hopf ideal is [HH'.sup.+].

In [6] we introduced a family of elements in H' denoted by [z.sub.n], n > 1, which arise from the idempotent integral of H. This family consists of powers of the S-fixed central invertible element [z.sub.2],

[z.sub.2] = {[and], [and]'} = [summation][[and].sub.1][[and]'.sub.1]S[[and].sub.2]S[[and]'.sub.2]

where [and], [and]' are two copies of the idempotent integral of H. We refer to [z.sub.2] as an extensive commutator.

The extensive commutator [z.sub.2] has a very nice form which is the key for determining commutativity We showed

Theorem 3.3. Let H be a semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is commutative if and only if [z.sub.2] [member of] k.

Let {[E.sub.i]} be the set of central orthogonal idempotents of H. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The element [z.sub.2] generates H' in the following sense. H' is the algebra generated by the left coideal [z.sub.2] [??] H*.

What is the role played by the extensive commutator [z.sub.2] inside kG?

Example 3.4. Recall for groups [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where f is the Frobenius counting function. That is, f(g) counts the number of times g can be obtained as a commutator ab[a.sup.-1][b.sup.-1]. Clearly, if [[summation].sub.a,,b[member of]G] ab[a.sup.-1][b.sup.-1] [member of] k, then all commutators are trivial which means that G is abelian. This fact is expressed in the first statement of Theorem 3.3.

The group elements {g} that appear in {[and], [and]} with a nonzero coefficient, (that is f(g) [not equal to] 0), are precisely those that generate G'. This fact is expressed in the last part of Theorem 3.3 since [z.sub.2] [??] pg = f(g)g for all g [member of] G.

The fact that [d.sub.i]||G| is a famous property of groups. Frobenius has shown that the counting function f has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

meaning that f is a character on G. It can be seen that f and the extensive commutator [z.sub.2] are related by:

f = [lambda] [??] d[z.sub.2].

For general Hopf algebras, Kaplansky's 6th conjecture states that the dimension of any irreducible H-module divides dimH. This conjecture is still open. However, in [7] we refer to the function [f.sub.com] = [lambda] [??] d[z.sub.2] and to some other counting functions as distribution functions.

Nilpotent groups can be defined in terms of upper central series of normal subgroups,

1 = [Z.sub.0](G) [subset] [Z.sub.1](G) [subset] [Z.sub.2](G) [subset] ... [subset] [Z.sub.t](G) = G,

where [Z.sub.1] (G) = Z(G), the center of the group G, and

[Z.sub.i+1](G)/[Z.sub.i](G) = Z(G/[Z.sub.i](G)).

Equivalently, it is defined in terms of lower central series of commutators,

G [contains] [G.sub.1] [contains] [G.sub.2] [contains] ... [contains] 1,

where [G.sub.1] = [G, G], [G.sub.i] = [G, [G.sub.i-1]], the ith commutator subgroup.

Hopf commutators on one hand and integrals of certain left coideal subalgebras enable us to define nilpotent Hopf algebras as a natural generalization of nilpotent groups. The generalization of upper central series is not trivial though. While the set {[Z.sub.i](G)} are all normal subgroups of G, it makes sense to consider quotients of successive subgroups, this is no longer the case for Hopf algebras. Though we can replace normal subgroups with normal left coideal subalgebras, we do not have a meaningful replacement for quotients.

We overcome this problem by using integrals for left coideal subalgebras. Observe that [Z.sub.i+1](G)/[Z.sub.i](G) is the image of [Z.sub.i+1](G) under the group projection [[pi].sub.i] : G [right arrow] G/[Z.sub.i](G). In [8], we showed that if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a Hopf algebra projection, and [[and].sub.Zi] is the integral of Zi, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an algebra (and H-module) isomorphism. In particular, if [Z.sub.i+1] [contains] [Z.sub.i], then we have an algebra isomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This observation enables us to replace quotients with subalgebras and thus suggest a definition for an upper central series for Hopf algebras as follows:

Definition 3.5. An ascending (upper) series for H is a series of normal left coideal subalgebras

k = [Z.sub.0] [subset] [Z.sub.1] [subset] ... [subset] [Z.sub.t] [subset]

so that

[Z.sub.i+1][and][Z.sub.i] = Z(H[[and].sub.Zi]),

for all 1 [less than or equal to] i [less than or equal to] t, where [[and].sub.Zi] is the integral of [Z.sub.i] and Z(H[[and].sub.Zi]) is the center of H[[and].sub.Zi].

The lower series for H are described via Hopf commutators as follows.

Definition 3.6. A descending lower series for H is a series of iterated commutators,

H = [N.sub.0] [contains] [N.sub.1] [contains] ... [contains] [N.sub.t] [contains],

so that [N.sub.0] = H, [N.sub.i] = the normal left coideal subalgebra generated by {[N.sub.i-1], [and]}.

As for groups, when H is nilpotent the first series ends with H while the second one ends with k. They are also interrelated. Based on [7, Th. 3.7] and [8, Prop. 3.8] we have,

Theorem 3.7. Let H be a semisimple Hopf algebra over an algebraically closed field of characteristic 0, let {[Z.sub.i]} be an ascending central series and {[N.sub.i]} be a descending central series. Then

[N.sub.t] = [k.sub.1] [??] [Z.sub.t] = H.

In this case H is nilpotent and we have: [N.sub.t-i] [bar.[subset]] [Z.sub.i] for all 0 [less than or equal to] i [less than or equal to] t.

Iterated commutators of the extensive commutator z2 yield another criterion for nilpotency of H. Define an operator T : Z (H) [??] Z (H) by

T (z) = {z, [and]}.

We have:

Proposition 3.8. (i) The matrix of T with respect to the basis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is A, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

(ii) A has non-negative rational entries and the first column of A has all entries equal 1.

(iii) The first row of [A.sup.m] is (1,0, ... , 0) for all m [greater than or equal to] 0.

(iv) In the first column of [A.sup.m] we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

For all m > 0, 0 [less than or equal to] i [less than or equal to] n - 1. In particular, The first column of [A.sup.m] consists of positive rational numbers.

We refer to A as the Commutator matrix of H. Observe that this matrix depends only on the structure constants of the ring of characters on H. Eigenvalues of A are related to nilpotancy of H.

Theorem 3.9. Let H be a semisimple Hopf algebra over an algebraically closed field of characteristic 0 and assume [x.sub.i]s([x.sub.i]) [member of] Z(R(H)) for each irreducible character [x.sub.i]. Then H is nilpotent if and only if its commutator matrix A has eigenvalues {1,0} where the algebraic multiplicity of 1 is 1.

The condition [x.sub.i]s([x.sub.i]) [member of] Z(R(H)) is satisfied by an abundance of semisimple H. In particular by all quasitriangular Hopf algebras, but also by H = kG*, even though R(H) = kG is not necessarily commutative.

We next define an important family of central iterated commutators .

[[gamma].sub.0] = [and], [[gamma].sub.1] = T([and]) = {[and], [and]}, ... , [[gamma].sub.m] = [T.sub.m]([and]) = {[[gamma].sub.m-1], [and]}. (7)

We can describe the central iterated commutators [[gamma].sub.n] in terms of the coefficients of A. In particular, they are all positive rational numbers.

Proposition 3.10. Let [[gamma].sub.m] be defined as in (7) and the matrix A be defined as in Proposition 3.8. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all m [greater than or equal to] 1. Moreover, the coefficient of each [E.sub.i] in [[gamma].sub.m] is a non-zero rational number, in particular [[gamma].sub.m] is invertible.

Proposition 3.10 yields another criterion for nilpotency of H.

Theorem 3.11. Let H be a semisimple Hopfalgebra over an algebraically closed field of characteristic 0, and let [[gamma].sub.t] be defined as in (7). Assume [x.sub.i]s([x.sub.i]) [member of] Z(R(H)) for each irreducible character [x.sub.i]. Then H is nilpotent if and only if [[gamma].sub.m] = 1for some m [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Its index of nilpotency is the least integer m so that [[gamma].sub.m] = 1.

The last concept we wish to discuss is solvability. On the level of category theory there exists a notion of solvability [10] and it is customary to define solvable Hopf algebras H as those for which Rep(H) is a solvable category. However, this non-intrinsic definition is unsatisfactory as it contradicts our intuition from group theory. Commutative or nilpotent Hopf algebras are not always solvable in this sense [10, Prop. 4.5(ii),Remark 4.6(i)]. We suggested in [8] an intrinsic definition of solvability which is consistent with solvability for finite groups and as desired, commutative or nilpotent Hopf algebras are indeed solvable.

A solvable group has a subnormal series of subgroups. That is,

1 = [G.sub.0] [subset] [G.sub.1] [subset] ... [subset] G,

so that [G.sub.i] is normal in [G.sub.i+1]. G is solvable if it has a subnormal series so that [G.sub.i+1]/[G.sub.i] is an abelian group. Generalizing this definition to Hopf algebras requires an appropriate translation for subnormal series in addition to abelian quotients. This is done using integrals as follows:

Definition 3.12. Let H be a semisimple Hopf algebra. A subnormal series of left coideal subalgebras of H

k = [N.sub.0] [subset] [N.sub.1] [subset] ... [subset] [N.sub.t] = H

is a series satisfying:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all 0 [less than or equal to] i [less than or equal to] t - 1, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the integral of [N.sub.i]. H is solvable if for all a,b [member of] [N.sub.i+1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 3.13. For finite groups, the definition of solvable Hopf algebras boils down to the usual definition of solvability for groups.

To see this, let [G.sub.1] [subset] G and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] x. If [and][G.sub.1] is central in G, then for g [member of] G we have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It follows that for x[member of] [G.sub.1], xg = gy for some y [member of] [G.sub.1]. Thus [G.sub.1] is normal in G.

We claim now that G/[G.sub.1] is abelian if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all a,b [member of] G. Indeed, assume [bar.a][bar.b] = [bar.b][bar.a] for all [bar.a],[bar.b] [member of] G/[G.sub.1]. Then ab[a.sup.-1] = by for some y [member of] [G.sub.1]. This implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Conversely, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then ab[a.sup.-1] = by for some y [member of] [G.sub.1]. This shows our claim.

We showed in [8] that many properties of solvable groups can be generalized to solvable Hopf algebras. In particular, any nilpotent Hopf algebra is solvable. We proved also an analogue of Burnside's [p.sup.a][q.sup.b] theorem for semisimple quasi triangular Hopf algebras.

Theorem 3.14. Let H be a quasitriangular semisimple Hopf algebra of dimension [p.sup.a][q.sup.b] over afield k of characteristic 0, p, q primes. Then H is solvable.

References

[1] S. Burciu, Kernel of representations and coideal subalgebras for Hopf algebras, Glasg. Math. J. 54 (2012), no. 1,107-119.

[2] C.W Curtis and I.W Reiner, Methods of representation theory, Vol I, John Wiley and Sons, Inc. New York, 1990.

[3] M. Cohen and S. Westreich, Conjugacy Classes, Class Sums and Character Tables for Hopf Algebras, Comm. Algebra, 39, (2011), no. 12,4618-4633.

[4] M.Cohen and S. Westreich, Recovering information from the character tables of Hopf algebras - normality, dimensions and quotients, Contemporary Math (585), Amer. Math. Soc , (2013), 213-227.

[5] M. Cohen and S. Westreich, Character tables and normal left coideal subalgebras, J. Pure Appl. Algebra 218 (2014), 1845-1866.

[6] M. Cohen and S. Westreich, Are we counting or measuring something? J. Algebra 398 (2014),111-130.

[7] M. Cohen and S. Westreich, Probabalistically nilpotent Hopf algebras, T. Am. Math. Soc, 368, No. 6, (2016), 4295-4314.

[8] M. Cohen and S. Westreich, Solvability for semisimple Hopf algebras via integrals, J. Algebra 472 (2017), 67-94.

[9] V. G. Drinfel'd, On almost cocommutative Hopf algebras, Leningrad Math. J. 1 (1990), 321-342.

[10] P. Etingof, D. Nikshych and V. Ostrik, Weakly group-theoretical and solvable fusion categories, Adv. Math. 226, (2011), 176-205.

[11] M. Kopinnen, Coideal subalgebras in Hopf algebras: freeness, integrals, smash products, Comm. Algebra 21(2), (1993), 427-444.

[12] R.G. Larson, M.E. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969) 75-93.

[13] S. Skriyabin, Projectivity and freeness over comodule algebras, Trans. of AMS, 359, no 6, (2007), 2597-2623

[14] Takeuchi. Quotient Spaces for Hopf Algebras. Comm. Alg., 22(7), (1995), 2503-2523.

[15] Y. Zhu, Hopf algebras of prime dimension, Int. Math. Res. Not. (1994), 53-59.

Miriam Cohen

Sara Westreich

Received by the editors in May 2016.

Communicated by S. Caenepeel.

2010 Mathematics Subject Classification : 16T05, 20C05.

Department of Mathematics

Ben Gurion University of the Negev, Beer Sheva, 8410501 Israel

email: mia@math.bgu.ac.il

Department of Management

Bar Ilan University, Ramat Gan, 529002 Israel

email: swestric@biu.ac.il

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Author: | Cohen, Miriam; Westreich, Sara |
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Publication: | Bulletin of the Belgian Mathematical Society - Simon Stevin |

Article Type: | Report |

Date: | Jan 1, 2017 |

Words: | 5773 |

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