Irreducible Representations of Crossed Products

Susan Montgomery^{1} Department of Mathematics, University of Southern California Los Angeles, California 90089, U. S. A.

S. J. Witherspoon^{2} Department of Mathematics, University of Toronto Toronto, Ontario M5S 3G3, Canada

Abstract We prove that if the dimension of any irreducible module for a finite dimensional algebra over an algebraically closed field divides the dimension of the algebra, then the same is true of any crossed product of that algebra with a group algebra or its dual, provided the characteristic of the field does not divide the order of the group. Kaplansky’s Conjecture regarding dimensions of irreducible modules for Hopf algebras then follows for those finite dimensional semisimple Hopf algebras constructed by a sequence of crossed products involving group algebras and their duals. We show that any semisimple Hopf algebra of prime power dimension in characteristic 0 is of this type, so that Kaplansky’s Conjecture holds for these Hopf algebras.

Introduction

In 1975, Kaplansky conjectured that the dimension of any irreducible H-module divides the dimension of H, where H is a finite dimensional semisimple Hopf algebra over an algebraically closed field [9]. This is well known for semisimple group algebras (a classical result of Frobenius), and for the dual of a group algebra, whose irreducible modules are all of dimension one. Recently, Nichols and Richmond have proven that if H has an irreducible module of dimension 2, then 2 divides the dimension of H [16]. Zhu has proven that the dimension of any irreducible D(H)-submodule of H divides the dimension of H, where D(H) is the quantum double of H and the field has characteristic 0 [22]. Aside from these cases and some examples however, the general case has remained unknown.

We prove this conjecture for those Hopf algebras which can be constructed by a sequence of crossed products involving group algebras and their duals. All finite dimensional semisimple Hopf algebras over algebraically closed fields known at this time are of this form, so Kaplansky’s Conjecture holds for all known examples. Further, we show that all semisimple Hopf algebras of prime power dimension over algebraically closed fields of characteristic 0 are of this form, establishing Kaplansky’s Conjecture for these Hopf algebras.

1Partially supported by NSF Grant DMS 95-00649.

2Partially supported by NSERC Grant OGP0170281.

More specifically, let G be a finite group, k an algebraically closed field whose characteristic does not divide the order of G, and A a finite dimensional k-algebra. We first show that if the divisibility property for dimensions of irreducible modules holds for A, then it holds for any crossed product A#_{σ}kG. This follows from an application to A#_{σ}kG of Clifford Theory for group-graded rings [7]. We give an explicit construction of A#_{σ}kG-modules from A-modules and modules for twisted group algebras of subgroups of G.

Next we consider a crossed product B = A#_{σ}(kG)^{∗} where (kG)^{∗} is the Hopf algebra dual of kG. By duality [2], the smash product B#kG is isomorphic to a matrix algebra M_{n}(A). If the divisibility property for dimensions of irreducible modules holds for A, then it holds for M_{n}(A), and our construction of B#kG-modules from B-modules then shows that it holds for B.

We define lower and upper semisolvable Hopf algebras to be those having certain normal series with factors either commutative or cocommutative. Our results involving kG and (kG)^{∗} show that Kaplansky’s Conjecture holds for a finite dimensional semisimple lower or upper semisolvable Hopf algebra H when the characteristic of k does not divide the dimension of H. We show that any semisimple Hopf algebra of prime power dimension over an algebraically closed field of characteristic 0 is upper and lower semisolvable, and thus Kaplansky’s Conjecture holds for these Hopf algebras.

All algebras and modules will be finite dimensional over the algebraically closed field k, and tensor products will be over k unless otherwise indicated.

1 Group crossed products

Let A be a finite dimensional algebra over an algebraically closed field k, G a finite group, and A#_{σ}kG a crossed product. That is, there is an invertible map σ : G × G → A and a k-linear map kG ⊗ A → A denoted g ⊗ a ↦ g · a satisfying

g · (h · a) = σ(g, h)(gh · a)σ(g, h)^{−1},
(g · σ(h, ℓ))σ(g, hℓ) = σ(g, h)σ(gh, ℓ),
σ(g, 1) = σ(1, g) = 1,
g · (ab) = (g · a)(g · b) , g · 1 = 1 , 1 · a = a,

for all g, h, ℓ ∈ G and a, b ∈ A. In particular, kG measures A and σ is a 2-cocycle. We denote the element a ⊗ g of A ⊗ kG by a#g, and define a product on this vector space by

(a#g)(b#h) = a(g · b)σ(g, h)#gh.

This results in an associative algebra, denoted A#_{σ}kG [15, 18]. This algebra is a fully G-graded algebra, with g-component A#_{σ}kg for each g ∈ G [3, 7]. The algebra A embeds in A#_{σ}kG as the identity component. Clifford Theory for group-graded rings implies a correspondence between A#_{σ}kG-modules, and those of A and of twisted group algebras for subgroups of G [6, 7]. Alternatively, the correspondence may be generalized directly from Clifford Theory for group algebras [5] to crossed products A#_{σ}kG. We will describe this correspondence explicitly for a crossed product, in terms of the cocycle σ. These constructions do not require k to be algebraically closed.

We first introduce an action of G on the set of isomorphism classes of finite dimensional A-modules, the conjugation action of G. If T is an A-module with ρ : A → End_{k}(T) defining the action of A on T, and g ∈ G, the conjugate module g · T is defined to be the A-module with underlying vector space T and action g · ρ : A → End_{k}(T) defined by

g · ρ(a) = ρ(g^{−1} · a)

for all a ∈ A. We may also express the module g · T as (A#_{σ}kg) ⊗_{A} T ≅ (1#g^{−1})^{−1} ⊗_{A} T, where A acts by left multiplication on the first factor, as

(1#g^{−1})(a#1)(1#g^{−1})^{−1} = (g^{−1} · a)#1.

Here (1#g^{−1})^{−1} = σ(g, g^{−1})^{−1}#g. We note that dim(g · T) = dim(T), whereas in the general situation of a group-graded algebra, conjugate modules do not always have the same dimension [3]. If h ∈ G, then h · g · ρ is equivalent to hg · ρ as a representation, since g^{−1} · h^{−1} · a = σ(g^{−1}, h^{−1})((hg)^{−1} · a)σ(g^{−1}, h^{−1})^{−1}.

Let U be an A#_{σ}kG-module. We denote by U ↓_{A} the module U restricted to A. If U is irreducible, then U ↓_{A} is semisimple and isomorphic to a direct sum of copies of an irreducible A-module T and its distinct G-conjugates, each occurring the same number of times ([6, Exercises 18.9 and 18.10] or [7, Theorem 12.4]). That is, U ↓_{A}≅ (T_{1} ⊕ ⋯ ⊕ T_{k})^{⊕e} for some positive integer e, where T = T_{1}, …, T_{k} are the distinct (mutually nonisomorphic) G-conjugates of T.

Let G_{T} be the stabilizer in G of the irreducible A-module T:

G_{T} = {g ∈ G | g · T ≅ T }.

Then the A-submodule T^{⊕e} of U ↓_{A} may be considered to be an irreducible A#_{σ}kG_{T} - submodule of the restriction of U to A#_{σ}kG_{T}, as Proposition 1.1 below implies. The proposition follows from [6, Exercises 18.11 and 18.12] or [7, Corollary 11.16]. If V is an A#_{σ}kG_{T} -module, we will denote by V ↑^{G} the induced A#_{σ}kG-module (A#_{σ}kG)⊗_{A#_{σ}kG_{T}} V, where the action is given by left multiplication by the first factor. In this case, note that V ↑^{G} ≅ ∑_{g∈G/G_{T}} g · V as an A#_{σ}kG_{T} -module, since A#_{σ}kG is free as a right A#_{σ}kG_{T} - module. Therefore dim(V ↑^{G}) = |G : G_{T} | dim(V).

Proposition 1.1 Let T be an irreducible A-module, and G_{T} its stabilizer in G. Induction of modules provides an equivalence between the category of A#_{σ}kG_{T} -modules whose restriction to A is isomorphic to a direct sum of copies of T, and the category of A#_{σ}kG-modules whose restriction to A is isomorphic to a direct sum of copies of conjugates of T.

We next describe a construction of A#_{σ}kG_{T} -modules from modules for a twisted group algebra k^{β}G_{T} of G_{T}. Given a 2-cocycle β : G_{T} × G_{T} → k^{×}, where elements of G_{T} act trivially on k^{×}, the twisted group algebra k^{β}G_{T} is the crossed product k#_{β}kG_{T}.

Let ρ : A → End_{k}(T) express the action of A on the irreducible A-module T, and for each g ∈ G_{T}, let τ(g) ∈ End_{k}(T) be an isomorphism between g · T and T. That is,

ρ(a)τ(g) = τ(g)ρ(g^{−1} · a)

for all a ∈ A. Choose τ(1) to be the identity map. For each g, h ∈ G_{T}, it may be checked that the invertible map τ(g)τ(h)ρ(σ(h^{−1}, g^{−1}))τ(gh)^{−1} commutes with ρ(a) for all a ∈ A; that is, it is an A-homomorphism. As T is irreducible, Schur’s Lemma implies that there is a scalar α(g, h) ∈ k^{×} such that τ(g)τ(h)ρ(σ(h^{−1}, g^{−1}))τ(gh)^{−1} = α(g, h) ∘ id_{T}. So

τ(g)τ(h) = α(g, h)τ(gh)ρ(σ^{−1}(h^{−1}, g^{−1}))

for all g, h ∈ G_{T}. Using the 2-cocycle relation for σ and the definition of τ, we see that α is itself a 2-cocycle (with corresponding trivial action on k^{×}). We will be interested in the 2-cocycle α^{−1}.

Let M be a k^{α^{−1}}G_{T} -module. It may be checked that the vector space T ⊗ M has the structure of an A#_{σ}kG_{T} -module given by

(a#g) · (t ⊗ m) = ρ(a)τ(g)ρ(σ(g^{−1}, g))t ⊗ g · m

for all a ∈ A, g ∈ G_{T}, t ∈ T, and m ∈ M. Here g · m denotes the action of the image of g in k^{α^{−1}}G_{T} on m. The restriction (T ⊗ M)↓_{A} is a direct sum of copies of T. If f : M → N is a k^{α^{−1}}G_{T} -homomorphism, then id_{T} ⊗ f is an A#_{σ}kG_{T} -homomorphism from T ⊗ M to T ⊗ N.

On the other hand, if V is an A#_{σ}kG_{T} -module whose restriction to A is a direct sum of copies of T, the vector space Hom_{A}(T, V) becomes a k^{α^{−1}}G_{T} -module by defining

(g · ϕ)(t) = α^{−1}(g, g^{−1})#g · ϕ(τ(g^{−1})t)

for all g ∈ G_{T}, ϕ ∈ Hom_{A}(T, V), t ∈ T. If f : V → W is an A#_{σ}kG_{T} -homomorphism, then f_{∗} is a k^{α^{−1}}G_{T} -homomorphism from Hom_{A}(T, V) to Hom_{A}(T, W), where f_{∗}(ϕ) = f ∘ ϕ.

We have described functors between the categories of k^{α^{−1}}G_{T} -modules and of those A#_{σ}kG_{T} -modules whose restriction to A is a direct sum of copies of T. It may be checked that these functors provide a category equivalence, as stated in the following proposition. Alternatively, the proposition follows from a result of Dade about group-graded rings ([6, pp. 239–240] or [7, Theorem 10.6]).

Proposition 1.2 Let T be an irreducible A-module, G_{T} its stabilizer in G, and α the 2cocycle of G_{T} determined by T as above. There is an equivalence between the category of k^{α^{−1}}G_{T} -modules and the category of those A#_{σ}kG_{T} -modules whose restriction to A is isomorphic to a direct sum of copies of T.

Propositions 1.1 and 1.2, together with the above description of the category equivalences, immediately imply the following theorem.

Theorem 1.3 Let T be an irreducible A-module, G_{T} its stabilizer in G, and α the 2-cocycle of G_{T} determined by T as above. There is an equivalence between the category of k^{α^{−1}}G_{T} - modules and the category of those A#_{σ}kG-modules whose restriction to A is isomorphic to a direct sum of copies of conjugates of T. This equivalence is given by sending a k^{α^{−1}}G_{T} - module M to the module (T ⊗ M)↑^{G} induced from the A#_{σ}kG_{T} -module T ⊗ M defined by (a#g) · (t ⊗ m) = ρ(a)τ(g)ρ(σ(g^{−1}, g))t ⊗ g · m for all a ∈ A, g ∈ G_{T}, t ∈ T, and m ∈ M.

As a consequence of the theorem, irreducible k^{α^{−1}}G_{T} -modules are in one-to-one correspondence with those irreducible A#_{σ}kG-modules having T as a direct summand. We also need the fact that the dimension of an irreducible module for a twisted group algebra over k divides the order of the group, provided the characteristic of k is relatively prime to the order of the group. This follows from the theory of Schur representation groups [8]. We note that Passman now has a simpler proof of Corollary 1.4 below that does not use Theorem 1.3.

Corollary 1.4 Let A be a finite dimensional k-algebra such that the dimension of any irreducible A-module divides the dimension of A. Let G be a finite group such that the characteristic of k does not divide |G|, and let B = A#_{σ}kG be a crossed product. Then the dimension of any irreducible B-module divides the dimension of B.

Proof: Let U be an irreducible B-module. By the discussion at the beginning of the section, U ↓_{A} is a direct sum of conjugates of an irreducible A-module T, each occurring the same number of times. Let G_{T} be the stabilizer of T, and α : G_{T} × G_{T} → k^{×} the 2-cocycle determined by T as above. By Theorem 1.3, there is an irreducible k^{α^{−1}}G_{T} -module M with U ≅ (T ⊗ M)↑^{G}, so that dim(U) = |G : G_{T} | dim(T) dim(M). Now dim(M) divides |G_{T} | as noted above, so dim(U) divides |G| dim(T). But dim(T) divides dim(A) by hypothesis, so dim(U) divides |G| dim(A) = dim(B). □

2 The dual of a group algebra

In this section we will consider A#_{σ}(kG)^{∗}-modules, where A is a finite dimensional kalgebra, G a finite group, and (kG)^{∗} the Hopf algebra dual of kG. We refer the reader to Section 3 for the definition of the general crossed product A#_{σ}H for a Hopf algebra H. Here we use duality to achieve an analogous result to Corollary 1.4 regarding dimensions of irreducible A#_{σ}(kG)^{∗}-modules. That is, G acts on B = A#_{σ}(kG)^{∗} by

g · (a#f) = a#(g · f),

where g · f(x) = f(xg^{−1}). Thus we may form the smash product B#kG, that is a crossed product with trivial cocycle. By duality B#kG ≅ M_{n}(A), where n is the order of G [2].

If the dimension of any irreducible A-module divides the dimension of A, then the same is true of M_{n}(A), as an irreducible M_{n}(A)-module is a direct sum of n copies of an irreducible A-module. In particular, the dimension of an irreducible M_{n}(A)-module is n times the dimension of the corresponding A-module.

At this point it is natural to ask if a converse to Corollary 1.4 is true. That is, suppose B is a finite dimensional k-algebra and the dimension of any irreducible B#_{σ}kG-module divides dim(B#_{σ}kG). Does it follow that the dimension of any irreducible B-module divides the dimension of B? The following example, which was pointed out to us by Guralnick, shows that the answer is no. Let B be an algebra for which there exists an irreducible module whose dimension does not divide dim(B), for example B = k ⊕ M_{2}(k). Then it is possible to choose an abelian group G of large enough order that irreducible B ⊗ kG-modules will have dimension dividing dim(B ⊗ kG). However duality provides a positive answer for the smash product of B = A#_{σ}(kG)^{∗} with kG.

Theorem 2.1 Let A be a finite dimensional k-algebra such that the dimension of any irreducible A-module divides the dimension of A. Let G be a finite group such that the characteristic of k does not divide |G|, and let B = A#_{σ}(kG)^{∗} be a crossed product. Then the dimension of any irreducible B-module divides the dimension of B.

Proof: Let T be an irreducible B-module. The action of G on B described above yields an action of G on the isomorphism classes of B-modules, as described in Section 1. Let G_{T} be the subgroup of all g ∈ G such that g · T ≅ T, and let α : G_{T} × G_{T} → k^{×} be the 2-cocycle determined by T as in Section 1. Let M be any irreducible k^{α^{−1}}G_{T} -module. By Theorem 1.3, there is a corresponding irreducible B#kG-module U = (T ⊗ M)↑^{G}, so that

dim(U) = |G : G_{T} | dim(T) dim(M).

By duality, we have B#kG ≅ M_{n}(A), where n = |G| [2]. Let X be an irreducible Amodule corresponding to the M_{n}(A)-module U, so that dim(U) = |G| dim(X). Comparing to the above expression for dim(U), we have

dim(T) = |G_{T} | dim(X)/ dim(M).

As M is an irreducible k^{α^{−1}}G_{T} -module, dim(M) divides |G_{T} |, and so |G_{T} |/ dim(M) divides |G_{T} |. By hypothesis, dim(X) divides dim(A). Therefore dim(T) divides |G_{T} | dim(A), which divides |G| dim(A) = dim(B). □

An inductive argument using Corollary 1.4 and Theorem 2.1 yields the following consequence.

Corollary 2.2 Let A be a finite dimensional algebra with a sequence of subalgebras A_{n+1} = k ⊂ A_{n} ⊂ ⋯ ⊂ A_{2} ⊂ A_{1} = A such that for each i, A_{i} is isomorphic to A_{i+1}#_{σ}kG or A_{i+1}#_{σ}(kG)^{∗} for some group G and cocycle σ. Suppose further that the characteristic of k does not divide the order of any of the groups G involved. Then the dimension of any irreducible A-module divides the dimension of A.

3 Semisolvable Hopf algebras

Let H be a finite dimensional Hopf algebra over the algebraically closed field k. (The assumption that k be algebraically closed is not needed until Theorem 3.4.) A lower normal series for H is a series of proper subHopf algebras

H_{n+1} = k ⊂ H_{n} ⊂ ⋯ ⊂ H_{2} ⊂ H_{1} = H,

where H_{i+1} is normal in H_{i} for each i. That is, (ad_{ℓ}H_{i})(H_{i+1}) ⊆ H_{i+1} and (ad_{r}H_{i})(H_{i+1}) ⊆ H_{i+1}, where ad_{ℓ} and ad_{r} are the left and right adjoint actions [15]. As H_{i+1} is normal in H_{i}, H_{i}H_{i+1}^{+} = H_{i+1}^{+}H_{i} is a Hopf ideal of H_{i}, where H_{i+1}^{+} is the augmentation ideal of H_{i+1}. The factors of this normal series are the quotients \overline{H_{i}} = H_{i}/H_{i}H_{i+1}^{+}. An upper normal series for H is a series defined inductively as follows. Let H_{(0)} = H. Let H_{i} be a normal subHopf algebra of H_{(i−1)}, and define H_{(i)} = H_{(i−1)}/H_{(i−1)}H_{i}^{+}. We assume H_{n} = H_{(n−1)} for some positive integer n so that H_{(n)} = k. The factors are the subHopf algebras H_{i} of the quotients H_{(i−1)}. We use the following lemma to relate normal series of H and H^{∗}.

Lemma 3.1 Let H be a finite dimensional Hopf algebra and K a normal subHopf algebra of H. Then J = (H/HK^{+})^{∗} is isomorphic to a normal subHopf algebra of H^{∗}, and H^{∗}/H^{∗}J^{+} is isomorphic to K^{∗}.

Proof: By the Nichols-Zoeller Theorem [17], H is free over K. As K is normal in H, it follows from [20, Lemma 1.3 (2)] that HK^{+} is a normal Hopf ideal of H. Equivalently, J = (H/HK^{+})^{∗} is isomorphic to a normal subHopf algebra of H^{∗} as H is finite dimensional [15, p. 36].

Let \overline{H} = H/HK^{+} and π : H → \overline{H} the natural projection. By [19, Lemma 1.3] or [15, Proposition 3.4.3] we have K = H^{co\overline{H}}, the coinvariants in H of the \overline{H}-comodule given by (id⊗π)∘Δ, where Δ is the coproduct on H. Thus the following equalizer diagram is exact:

K →^{i} H →^{f} ; →_{g} H ⊗ \overline{H},

where i is inclusion, f(h) = h ⊗ \overline{1} and g(h) = (id ⊗ π) ∘ Δ(h) = ∑ h_{1} ⊗ \overline{h}_{2}, as in [15, pp. 34–35]. Dualizing this diagram as in [15, p. 143] or [19, Theorem 2.4 (2)] gives the coequalizer diagram

H^{∗} ⊗ J →^{f^{∗}} ; →_{g^{∗}} H^{∗} →^{i^{∗}} K^{∗},

as J = \overline{H}^{∗}. Thus K^{∗} can be defined by this exact sequence. Now Ker(i^{∗}) = Im(f^{∗} −g^{∗}) = H^{∗}J^{+}, and so K^{∗} ≅ H^{∗}/H^{∗}J^{+}. □

Theorem 3.2 Let H be a finite dimensional Hopf algebra. If H has a lower normal series with factors \overline{H_{i}}, then H^{∗} has an upper normal series with factors (H^{∗})_{i} ≅ (\overline{H_{i}})^{∗}. If H has an upper normal series with factors H_{i}, then H^{∗} has a lower normal series with factors \overline{(H^{∗})_{i}} ≅ (H_{i})^{∗}.

Proof: Let H_{n+1} = k ⊂ H_{n} ⊂ … ⊂ H_{2} ⊂ H_{1} = H be a lower normal series for H with factors \overline{H_{i}} = H_{i}/H_{i}H_{i+1}^{+}. If n ≤ 1, there is nothing to prove. If n > 1, let J = (H/HH_{2}^{+})^{∗}, a normal subHopf algebra of H^{∗} with H^{∗}/H^{∗}J^{+} ≅ (H_{2})^{∗} by Lemma 3.1. By induction (H_{2})^{∗} has an upper normal series with factors isomorphic to (\overline{H_{i}})^{∗} for i = 2, …, n. Letting (H^{∗})_{1} = J, H^{∗} has an upper normal series with factors (H^{∗})_{i} ≅ (\overline{H_{i}})^{∗} for i = 1, …, n.

Let H_{(0)} = H , H_{(1)} = H/HH_{1}^{+} , …, H_{(n)} = H_{(n−1)}/H_{(n−1)}H_{n}^{+} = k be an upper normal series for H. If n ≤ 1, there is nothing to prove. If n > 1, let J = (H_{(1)})^{∗} = (H/HH_{1}^{+})^{∗}. By Lemma 3.1, J is a normal subHopf algebra of H^{∗}, and H^{∗}/H^{∗}J^{+} ≅ (H_{1})^{∗}. By induction J = (H_{(1)})^{∗} has a lower normal series with factors isomorphic to (H_{i})^{∗} for i = 2, …, n. Letting (H^{∗})_{2} = J, H^{∗} has a lower normal series with factors \overline{(H^{∗})_{i}} ≅ (H_{i})^{∗} for i = 1, …, n. □

We say that H is lower solvable (respectively lower cosolvable) if there is a lower normal series for H all factors of which are commutative (respectively cocommutative), and that H is lower semisolvable if there is a lower normal series for H each factor of which is either commutative or cocommutative. We say that H is upper solvable (respectively upper cosolvable, upper semisolvable) if H has an upper normal series each factor of which is commutative (respectively cocommutative, either commutative or cocommutative). As the dual of a commutative Hopf algebra is cocommutative and vice versa, the next result follows immediately from Theorem 3.2.

Corollary 3.3 Let H be a finite dimensional Hopf algebra. Then H is lower (respectively upper) solvable if and only if H^{∗} is upper (respectively lower) cosolvable, and H is lower semisolvable if and only if H^{∗} is upper semisolvable.

We next define the crossed product of an algebra A with a Hopf algebra H. We use summation notation for the coproduct of H [15, 1.4.2]. Suppose there is a convolutioninvertible k-linear map σ : H ⊗H → A and a k-linear map H ⊗A → A denoted h⊗a ↦ h·a satisfying

h · (ℓ · a) = ∑ σ(h_{1}, ℓ_{1})(h_{2}ℓ_{2} · a)σ^{−1}(h_{3}, ℓ_{3}),
∑[h_{1} · σ(ℓ_{1}, m_{1})]σ(h_{2}, ℓ_{2}m_{2}) = ∑ σ(h_{1}, ℓ_{1})σ(h_{2}ℓ_{2}, m),
σ(h, 1) = σ(1, h) = ϵ(h)1,
h · (ab) = ∑(h_{1} · a)(h_{2} · b) , h · 1 = ϵ(h)1 , 1 · a = a,

for all h, ℓ, m ∈ H and a, b ∈ A. In particular, H measures A and σ is a 2-cocycle. We denote the element a ⊗ h of A ⊗ H by a#h, and define a product on this vector space by

(a#h)(b#ℓ) = ∑ a(h_{1} · b)σ(h_{2}, ℓ_{1})#h_{3}ℓ_{2}.

This results in an associative algebra, denoted A#_{σ}H [15].

We will need some more results from [19]. First, if K is a normal subHopf algebra of the finite dimensional Hopf algebra H, then H ≅ K#_{σ}(H/HK^{+}) for some 2-cocycle σ. It follows that if H has a lower normal series H_{n+1} = k ⊂ H_{n} ⊂ ⋯ ⊂ H_{1} = H, then H ≅ (((H_{n}#_{σ_{n−1}} \overline{H_{n−1}}) #_{σ_{n−2}} \overline{H_{n−2}}) #_{σ_{n−3}} ⋯) #_{σ_{1}} \overline{H_{1}}. The second result we will need is transitivity of crossed products: Let A be a finite dimensional algebra, H a finite dimensional Hopf algebra, B = A#_{σ}H a crossed product, and \overline{H} a quotient of H. Then B ≅ B^{co\overline{H}}#_{τ} \overline{H} for some 2-cocycle τ. In particular, if \overline{H} = H/HK^{+} for a normal subHopf algebra K of H, then B^{co\overline{H}} = A#_{σ}K, and so A#_{σ}H ≅ (A#_{σ}K)#_{τ} \overline{H}.

We next generalize Corollary 1.4 and Theorem 2.1 to crossed products involving upper or lower semisolvable Hopf algebras. The following theorem in particular implies that the dimension of any irreducible H-module divides the dimension of H when H is a finite dimensional semisimple upper or lower semisolvable Hopf algebra such that the characteristic of k does not divide the dimension of H.

Theorem 3.4 Let A be a finite dimensional algebra over an algebraically closed field k such that the dimension of any irreducible A-module divides the dimension of A. Let H be a finite dimensional semisimple lower or upper semisolvable Hopf algebra over k such that the characteristic of k does not divide the dimension of H, and let B = A#_{σ}H be a crossed product. Then the dimension of any irreducible B-module divides the dimension of B.

Proof: First assume H is lower semisolvable, with lower normal series

H_{n+1} = k ⊂ H_{n} ⊂ ⋯ ⊂ H_{2} ⊂ H_{1} = H,

each factor of which is either commutative or cocommutative. If n ≤ 1, then H is either commutative or cocommutative. If H is commutative, then H ≅ (kG)^{∗} for some finite group G ([4, 10] or [15, Theorem 2.3.1]). Therefore by Theorem 2.1, the dimension of any irreducible B-module divides dim(B). If H is cocommutative, we consider the cases of prime characteristic and characteristic 0 separately. If k is of characteristic 0, then consider H^{∗}, a finite dimensional commutative semisimple Hopf algebra by [11]. By the above argument, H^{∗} ≅ (kG)^{∗} and so H ≅ kG for some finite group G. If k is of prime characteristic p, then H ≅ (kL)^{∗}#kG for a finite group G and a finite abelian p-group L by [15, Corollary 5.6.4] and [15, Theorem 5.7.1], due independently to Sweedler and Demazure-Gabriel. As p does not divide the dimension of H, L is in fact trivial, so H ≅ kG in this case as well. By Corollary 1.4, the dimension of any irreducible B-module divides dim(B).

We next assume n > 1. Then H ≅ H_{2}#_{τ} \overline{H_{1}} for some cocycle τ, and by transitivity of crossed products,

B ≅ (A#_{σ}H_{2})#_{μ}\overline{H_{1}}

for some cocycle μ [19]. As H = H_{1} is semisimple, both \overline{H_{1}} and H_{2} are semisimple; this second fact follows from the Nichols-Zoeller Theorem [17] and [15, Corollary 2.2.2]. Clearly H_{2} is lower semisolvable while \overline{H_{1}} is either commutative or cocommutative. The characteristic of k cannot divide dim(H_{2}) or dim(\overline{H_{1}}) by the Nichols-Zoeller Theorem [17]. By induction, the dimension of any irreducible A#_{σ}H_{2}-module divides dim(A#_{σ}H_{2}). As \overline{H_{1}} is either commutative or cocommutative, the arguments above now show that the dimension of any irreducible B-module divides dim(B).

Next assume H is upper semisolvable, and let

H_{(0)} = H , H_{(1)} = H/HH_{1}^{+} , …, H_{(n)} = H_{(n−1)}/H_{(n−1)}H_{n}^{+} = k

be an upper normal series for H with each H_{i} commutative or cocommutative. If n ≤ 1, then H is commutative or cocommutative and the first paragraph above applies. Assume n > 1. Then H ≅ H_{1}#_{τ}H_{(1)} for some cocycle τ, and by transitivity of crossed products

B ≅ (A#_{σ}H_{1})#_{μ}H_{(1)}

for some cocycle μ [19]. As H is semisimple, both H_{1} and H_{(1)} are semisimple. Clearly H_{(1)} is upper semisolvable while H_{1} is either commutative or cocommutative. By the arguments in the first paragraph above, the dimension of any irreducible A#_{σ}H_{1}-module divides dim(A#_{σ}H_{1}). By induction, the dimension of any irreducible B-module divides dim(B). □

Now we assume that k has characteristic 0. The next theorem in particular implies that all semisimple Hopf algebras of prime power dimension over k are both upper and lower solvable and cosolvable.

Theorem 3.5 Let H be a semisimple Hopf algebra of dimension p^{n} for a prime p over an algebraically closed field k of characteristic 0. Then H has both an upper and a lower normal series each factor of which is isomorphic to k[Z/pZ].

Proof: First we will prove by induction that H has an upper normal series each factor of which is isomorphic to k[Z/pZ]. If dim(H) = p, then H ≅ k[Z/pZ] by [21], so assume dim(H) = p^{n} with n > 1. By [14], H has a central group-like element g. We may assume g has order p, for if not, g generates a group which contains an element of order p. Let H_{1} ≅ k[Z/pZ] be the subHopf algebra of H generated by g, and let H_{(1)} = H/HH_{1}^{+}. Now H_{(1)} has dimension p^{n−1} and is semisimple, so by induction, H_{(1)} has an upper normal series each factor of which is isomorphic to k[Z/pZ]. Therefore H does as well.

As H is semisimple and k has characteristic 0, H^{∗} is also semisimple [11]. The above procedure applied to H^{∗}, also of dimension p^{n}, shows that H^{∗} has an upper normal series each factor of which is isomorphic to k[Z/pZ]. By Theorem 3.2, H has a lower normal series each factor of which is isomorphic to (k[Z/pZ])^{∗} ≅ k[Z/pZ]. □

Corollary 3.6 Let H be a semisimple Hopf algebra of dimension p^{n} for a prime p over an algebraically closed field k of characteristic 0. Then the dimension of any irreducible H-module divides the dimension of H.

Proof: By Theorem 3.5, H is both upper and lower semisolvable. By Theorem 3.4 with A = k and σ trivial, the dimension of any irreducible H-module divides dim(H). □

Masuoka has classified semisimple Hopf algebras of dimension p^{3} over k [12, 13]. In particular, for odd p there are p+1 such Hopf algebras not isomorphic to group algebras or their duals, and for p = 2 there is one such. Masuoka has also shown that a Hopf algebra of dimension p^{2} is isomorphic to a group algebra [14].

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