Classifying bicrossed products of two Taft algebras

Abstract

We classify all Hopf algebras which factor through two Taft algebras T n 2 (q) and respectively T m 2 (q). To start with, all possible matched pairs between the two Taft algebras are described: if q = q n−1 then the matched pairs are in bijection with the group of d-th roots of unity in k, where d = (m, n) while if q = q n−1 then besides the matched pairs above we obtain an additional family of matched pairs indexed by k * . The corresponding bicrossed products (double cross product in Majid's terminology) are explicitly described by generators and relations and classified. As a consequence of our approach, we are able to compute the number of isomorphism types of these bicrossed products as well as to describe their automorphism groups.2010 Mathematics Subject Classification. 16T05, 16S40.

Bicrossed products with the Taft algebra

“…We will see that actually the quest for classifying all bicrossed products between the Taft algebra and the group Hopf algebra K[G], where G is a group generated by a set of finite order elements, comes down to an old and notoriously difficult problem in group theory, namely that of describing the automorphisms of a certain given group. More precisely, our next result shows that any Hopf algebra isomorphism between two smash products as in Theorem 2.1 is in some sense induced by a group automorphism of G. ϕ (u, r, v) (a # t) = u(a)r(t (1) ) # ′ v(t (2) ).…”

“…Throughout K will be a field. Unless specified otherwise, all algebras, coalgebras, bialgebras, Hopf algebras, tensor products and homomorphisms are over K. For a coalgebra (C, ∆, ε) and y ∈ C, we use Sweedler's Σ-notation: ∆(y) = y (1) ⊗ y (2) , (I ⊗ ∆)∆(y) = y (1) ⊗ y (2) ⊗ y (3) , etc (summation understood). Let A and H be two Hopf algebras.…”

“…and v : K[G] → K[G] subject to the following compatibilities for all a, b ∈ T m 2 (q), t, w ∈ K[G]: u(a (1) ) ⊗ p(a (2) ) = u(a (2) ) ⊗ p(a (1) ) (16) u(ab) = u(a (1) ) p(a (2) ) ⊲ ′ u(b) (17) r(tw) = r(t (1) ) v(t (2) ) ⊲ ′ r(w) (18) r(t (1) ) v(t (2) ) ⊲ ′ u(a) = u(t (1) ⊲ a (1) ) p(t (2) ⊲ a (2)…”

“…The correspondence is such that the morphism ψ (u,p,r,v) associated to (u, p, r, v) is given as follows for all a ∈ T m 2 (q) and t ∈ K[G]: (1) ) p(a (2) ) ⊲ ′ r(t (1) ) # ′ p(a (3) )v(t (2) ).…”

“…The present study continues the work of [3] where a strategy for classifying bicrossed products of Hopf algebras was proposed. This avenue of investigation based on the description of all morphisms between two bicrossed products was previously pursued in [1], [2], [7], [17] and led to promising results. The paper is structured as follows.…”