Bicrossed products with the Taft algebra

Abstract

Let G be a group which admits a generating set consisting of finite order elements. We prove that any Hopf algebra which factorizes through the Taft algebra and the group Hopf algebra K[G] (equivalently, any bicrossed product between the aforementioned Hopf algebras) is isomorphic to a smash product between the same two Hopf algebras. The classification of these smash products is shown to be strongly linked to the problem of describing the group automorphisms of G. As an application, we completely describe by generators and relations and classify all bicrossed products between the Taft algebra and the group Hopf algebra K[D 2n ], where D 2n denotes the dihedral groups.2010 Mathematics Subject Classification. 16T05, 16S40.

The factorization problem for Jordan algebras. Applications

“…Therefore, the same strategy relying on matched pairs and bicrossed products was used to approach the factorization problem for various mathematical objects such as: (co)algebras [11,12,13], Lie algebras and Lie groups [21,24], Leibniz algebras [6], Hopf algebras [25], fusion categories [15] and so on. Furthermore, this new approach has also the advantage of opening the way to new classification methods as evidenced in [2,3,8,19] for Hopf algebras. Since their introduction by P. Jordan in 1933, Jordan algebras have appeared in various different fields of mathematics and mathematical physics such us the theory of superstrings, supersymmetry, projective geometry, Lie algebras and algebraic groups, representation theory or functional analysis [26].…”