Actions of pointed Hopf algebras with reduced pi invariants

2000

Let H denote a semisimple Hopf algebra over an algebraically closed field k of characteristic 0. We show that the degree of any irreducible representation of H whose character belongs to the center of H * must be a divisor of dim k H.

Let $H$ be a finite dimensional pointed rank one Hopf algebra of nilpotent type. We first determine all finite dimensional indecomposable $H$-modules up to isomorphism, and then establish the Clebsch-Gordan formulas for the decompositions of the tensor products of indecomposable $H$-modules by virtue of almost split sequences. The Green ring $r(H)$ of $H$ will be presented in terms of generators and relations. It turns out that the Green ring $r(H)$ is commutative and is generated by one variable over the Grothendieck ring $G_0(H)$ of $H$ modulo one relation. Moreover, $r(H)$ is Frobenius and symmetric with dual bases associated to almost split sequences, and its Jacobson radical is a principal ideal. Finally, we show that the stable Green ring, the Green ring of the stable module category, is isomorphic to the quotient ring of $r(H)$ modulo the principal ideal generated by the projective cover of the trivial module. It turns out that the complexified stable Green algebra is a group...

I will survey recent developments on the classification of finite-dimensional pointed Hopf algebras with non-abelian group. Concretely, I will describe two main topics: i) How to attach a generalized root system, or equivalently a Weyl groupoid, to a completely reducible Yetter-Drinfeld module W. Reference:[AHS]. This is done by looking at the Nichols algebra B (W); I will try to avoid technicalities and concentrate on some applications.

Mathematical Research Letters, 1999

2006

Given a Hopf algebra H and an H-module algebra A it is often important to be able of extending the H-action from A to a localisation of A. In this paper we are going to discuss sucient conditions to extend Hopf actions as well as to characterise those localisations where the action can be extended. Our particular interest lies in

Journal of Algebra, 1998

Journal of Algebra, 1998

This paper makes a contribution to the problem of classifying finite-dimensional semisimple Hopf algebras H over an algebraically closed field k of characteristic 0. Specifically, we show that if H has dimension pq for primes p and q then H is trivial, that is, H is either a group algebra or the dual of a group algebra. Previously known cases include dimension 2p [M1], dimension p 2 [M2], and dimensions 3p, 5p and 7p [GW]. Westreich and the second author also obtained the same result for H which is, along with its dual H * , of Frobenius type (i.e. the dimensions of their irreducible representations divide the dimension of H) [GW, Theorem 3.5]. They concluded with the conjecture that any semisimple Hopf algebra H of dimension pq over k is trivial. In this paper we use Theorem 1.4 in [EG] to prove that both H and H * are of Frobenius type, and hence prove this conjecture.

Proceedings of the American Mathematical Society

Let k be an algebraically closed field. The main goal of this paper is to classify the finite-dimensional pointed Hopf algebras over k of finite corep-resentation type. To do so, we give a necessary and sufficient condition for a basic Hopf algebra over k to be of finite representation type firstly. Explicitly, we prove that a basic Hopf algebra over k is of finite representation type if and only if it is Nakayama. By this conclusion, we classify all finite-dimensional pointed Hopf algebras over k of finite corepresentation type.

Algebras and Representation Theory, 2021

This note discusses a framework for the investigation of the prime spectrum of an associative algebra A that is equipped with an action of a Hopf algebra H . In particular, we study a notion of H -rationality for ideals of A and comment on a possible Dixmier-Moeglin equivalence for H -prime ideals of A .

Journal of Algebra, 2011

In this paper, we prove that a non-semisimple Hopf algebra H of dimension 4p with p an odd prime over an algebraically closed field of characteristic zero is pointed provided H contains more than two group-like elements. In particular, we prove that non-semisimple Hopf algebras of dimensions 20, 28 and 44 are pointed or their duals are pointed, and this completes the classification of Hopf algebras in these dimensions.