Representations of Finite-Dimensional Hopf Algebras

Abstract

Let H denote a nite dimensional Hopf algebra with antipode S over a eld |. We give a new proof of the fact, due to Oberst and Schneider OS], that H is a symmetric algebra if and only if H is unimodular and S 2 is inner. If H is involutory and not semisimple, then the dimensions of all projective H-modules are shown to be divisible by char|. In the case where |is a splitting eld for H , we give a formula for the rank of the Cartan matrix of H , reduced mod char| , in terms of an integral for H. Explicit computations of the Cartan matrix, the ring structure of G 0 (H), and the structure of the principal indecomposable modules are carried out for certain speci c Hopf algebras, in particular for the restricted enveloping algebras of completely solvable p-Lie algebras and of sl(2; |).

On Kaplansky's Fifth Conjecture

“…(As we have already pointed out, similar statements also appear in [35,30,10,19,11].) Define B = G 0 H ⊗ R ⊂ A.…”

Higman Ideals and Verlinde-type Formulas for Hopf Algebras

“…Part (iii) of the lemma above implies now that the set {C i } as defined in (16), and thus the structure constants c i jt as defined in (20), are all invariants.…”

“…In [20] Lorenz gave a short proof of the class equation [10,29]. In what follows we basically use his arguments.…”

“…The study of Hopf algebras as Frobenius algebras and sometimes symmetric algebras has appeared for example in works of Oberst-Schneider [22], Schneider [25], Lorenz [20], Kadison-Stolin [17] and Cohen-Westreich [4,5]. A survey on the representation theory of semisimple Hopf algebras from this point of view appears in Montgomery's survey [21].…”

Finite-Dimensional Representations of a Quantum Double

“…In particular, it follows that P V * ∼ = P V * as (right) A-modules, where − * = Hom − k [1,5]. A finite-dimensional Hopf algebra H is a symmetric algebra if and only if H is unimodular and S 2 is inner, where S is the antipode of H [12,16]. It is well known [17] that the Drinfeld quantum double D H of any Hopf algebra H is unimodular, and S 2 is inner.…”