Exactly solvable models for $2+1 \mathrm{D}$ topological phases derived from crossed modules of semisimple Hopf algebras

When is a smash product semiprime? A partial answer

Let H be a finite dimensional, semisimple Hopf algebra over a field K and let A be an H- module algebra. Assume K is a splitting field for H and that H is strongly semiprime. If A is H- semiprime, we show the Connes spectrum of H acting on A consists of all of the irreducible representations of H is equivalent to every nonzero annihilator ideal of the smash product meets A nontrivially. If H is also cocommutative, we let I′ be the intersection of the annihilators of the modules in the Connes spectrum. We find some of the information encoded in the Hopf kernel of the natural map from H to H/I′.

Solvability for semisimple Hopf algebras via integrals

Self-dual modules of semisimple Hopf algebras

Kemer's theory for H-module algebras with application to the PI exponent

Irreducible representations of crossed products

Introduction. Suppose that A is a semisimple Hopf algebra over a field of characteristic 0, or that A is a semisimple cosemisimple involutory Hopf algebra over a field k of characteristic p > dim A. In this paper we prove that the group AutHopf(A) of Hopf algebra automorphisms of A is finite. We show that the semigroup EndHOpf(A) of Hopf algebra endomorphisms of A is also finite as a corollary. Generally the group of Hopf algebra automorphisms of a finite-dimensional Hopf algebra need not be finite. For any positive integer n and any field k, we construct a family of finite-dimensional Hopf algebras over k with automorphism group GLn(k). The fact that AutHopf(A) is finite is a consequence of a theorem of

For a semisimple quasi-triangular Hopf algebra [Formula: see text] over a field [Formula: see text] of characteristic zero, and a strongly separable quantum commutative [Formula: see text]-module algebra [Formula: see text], we show that [Formula: see text] is a weak Hopf algebra, and it can be embedded into a weak Hopf algebra [Formula: see text]. With these structures, [Formula: see text] is the monoidal category introduced by Cohen and Westreich, and [Formula: see text] is tensor equivalent to [Formula: see text]. If [Formula: see text] is in the Müger center of [Formula: see text], then the embedding is a quasi-triangular weak Hopf algebra morphism. This explains the presence of a subgroup inclusion in the characterization of irreducible Yetter–Drinfeld modules for a finite group algebra.

Let H be a weak Hopf algebra (quantum groupoid) in the sense of [1]. Then in this note we first introduce weak Drinfeld double D(H) and character algebra C(H) of H. Next we define a weak D(H)-module algebra and a C(H)-module on C H (H S ), the centralizer of H S . Finally we prove that if H is a semisimple weak Hopf algebra over an algebraically closed field k of characteristic 0, then the action of D(H) on C H (H S ) and the action of C(H) on C H (H S ) form a commuting pair, which generalizes the main result in [11].KeywordsWeak Hopf algebra (quantum groupoid)Weak Drinfeld doubleCharacter algebraCommuting pair

Let K be a field. Let H be a finite-dimensional K-Hopf algebra and D(H) be the Drinfel'd double of H. In this paper, we study Radford's induced module Hβ, where β is a group-like element in H∗. Using the commuting pair established in [7], we obtain an analogue of the class equation for [Formula: see text] when H is semisimple and cosemisimple. In case H is a finite group algebra or a factorizable semisimple cosemisimple Hopf algebra, we give an explicit decomposition of each Hβ into a direct sum of simple D(H)-modules.

This is a survey article on a question, posed in 1986 by M.Cohen and D.Fishman, whether the smash product $A\#H$ of a semisimple Hopf algebra and a semiprime left $H$-module algebra $A$ is itself semiprime.

Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dimV divides dimH). It was proved by Montgomery and Witherspoon that the conjecture is true for H of dimension p^n, p prime, and by Nichols and Richmond that if H has a 2-dimensional representation then dimH is even. In this paper we first prove that if V is an irreducible representation of D(H), the Drinfeld double of any finite-dimensional semisimple Hopf algebra H over k, then dimV divides dimH (not just dimD(H)=(dimH)^2). In doing this we use the theory of modular tensor categories (in particular Verlinde formula). We then use this statement to prove that Kaplansky's conjecture is true for finite-dimensional semisimple quasitriangular Hopf algebras over k. As a result we prove easily the result of Zhu that Kaplansky's conjecture on prime dimensional Hopf algebras over k is true, by passing to their Drinfeld doubles. Second, we use a theorem of Deligne on characterization of tannakian categories to prove that triangular semisimple Hopf algebras over k are equivalent to group algebras as quasi-Hopf algebras.

Invariants of the adjoint coaction and Yetter–Drinfeld categories

Given an action of a finite group G on a fusion category \({\mathcal{C}}\) we give a criterion for the category of G-equivariant objects in \({\mathcal{C}}\) to be group-theoretical, i.e., to be categorically Morita equivalent to a category of group-graded vector spaces. We use this criterion to answer affirmatively the question about existence of non-group-theoretical semisimple Hopf algebras asked by P. Etingof, V. Ostrik, and the author in [7]. Namely, we show that certain \({\mathbb{Z}}\)/2\({\mathbb{Z}}\)-equivariantizations of fusion categories constructed by D. Tambara and S. Yamagami [26] are equivalent to representation categories of non-group-theoretical semisimple Hopf algebras. We describe these Hopf algebras as extensions and show that they are upper and lower semisolvable.

In this paper we study the isotypic decomposition of the regular module of a finite-dimensional Hopf algebra over an algebraically closed field of characteristic zero. For a semisimple Hopf algebra, the idempotents realizing the isotypic decomposition can be explicitly expressed in terms of characters and the Haar integral. In this paper we investigate Hopf algebras with the Chevalley property, which are not necessarily semisimple. We find explicit expressions for idempotents in terms of Hopf-algebraic data, where the Haar integral is replaced by the regular character of the dual Hopf algebra. For a large class of Hopf algebras, these are shown to form a complete set of orthogonal idempotents. We give an example which illustrates that the Chevalley property is crucial.