Quasitriangular structures of the double of a finite group

Abstract

We give a classification of all quasitriangular structures and ribbon elements of D(G) explicitly in terms of group homomorphisms and central subgroups. This can equivalently be interpreted as an explicit description of all braidings with which the tensor category Rep(D(G)) can be endowed. We also characterize their equivalence classes under the action of Aut(D(G)) and determine when they are factorizable.

Classifying bicrossed products of two Taft algebras

“…The line of inquiry presented in [2] was followed in [5] where the Hopf algebras which factor through two Sweedler's Hopf algebras are described and classified as well as in [11] where the automorphism group of the Drinfel'd double of a purely non-abelian finite group is completely described. Similar ideas are also employed in [12] in order to determine all quasitriangular structures and ribbon elements on the Drinfel'd double of a finite group over an arbitrary field.…”

Braid gaugings and categorical invariants

“…A complete, group-theoretical description of the quasitriangular structures for D(G), the Drinfeld double of a finite group G, was obtained by the author in [14]. This is equivalently a description of all possible braidings with which the tensor category Rep(D(G)) of finite dimensional representations of D(G) can be endowed.…”

“…Nikshych [23] has more recently obtained an alternative description from a more general categorical perspective. While many of the classical applications of the classification from [14] were discussed therein-such as a description of the ribbon elements and a discussion of when there were isomorphisms of ribbon Hopf algebras-most of the applications to the categorical side, such as categorical invariants, were left open. This paper arose out of the explorations of such applications.…”

“…Part of [14] was spent investigating and classifying what were called "central weak R-matrices" of D(G). While these seemed to carry a lot of structure, and seemed closely related in form to the quasitriangular structures, it was suggested to the author that these had no categorical interpretation, and so the similarity was likely a coincidence of limited use outside of the very specific category under consideration.…”

Correction to: Automorphisms of the Doubles of Purely Non-Abelian Finite Groups

“…So let u, v be as in the statement. By definition u * v is normal if and only if (u * v)(g x ) = (u * v)(g) x for all g, x ∈ G. As in [5,Lemma 3.2] (see also [4,Section 3]) u * v is normal if and only if (Su * v) * id is an algebra morphism, if and only if ((Su * v) * id) (u * v), if and only if (u * v)(g)g −1 ∈ C kG (Im(u * v)) for all g ∈ G. Since we always have C kG (Im(u * )) ⊆ C kG (Im(u * v)), it follows that Eq. 1 implies that u * v is normal, as desired.…”