Factorizable quasi-Hopf algebras—applications

Abstract

We deÿne the notion of factorizable quasi-Hopf algebra by using a categorical point of view. We show that the Drinfeld double D(H ) of any ÿnite dimensional quasi-Hopf algebra H is factorizable, and we characterize D(H ) when H itself is factorizable. Finally, we prove that any ÿnite dimensional factorizable quasi-Hopf algebra is unimodular. In particular, we obtain that the Drinfeld double D(H ) is a unimodular quasi-Hopf algebra.

Involutory Quasi-Hopf Algebras

“…First we will show that if k contains a primitive fourth root of unity then there are exactly two quasi-triangular structures on H (2). In what follows, p ± = 1 2 (1 ± g).…”

“…The relation in Eq. 2.31 is automatically satisfied because of the commutativity and cocommutativity of H (2). Thus the R-matrices for H (2) are in bijective correspondence with the solutions of the equation…”

“…As for Hopf algebras we denote (h) = h 1 ⊗ h 2 , but since is only quasi-coassociative we adopt the further convention (summation understood): (2,2) , for all h ∈ H. We will denote the tensor components of by capital letters, and the ones of −1 by small letters, namely…”

“…We will now study further properties of H (2). First we will show that if k contains a primitive fourth root of unity then there are exactly two quasi-triangular structures on H (2).…”

“…One can easily check that { p ± } is a basis for H(2) consisting of orthogonal idempotents, and that p + + p − = 1, p + − p − = g. Proposition 3.6 Suppose that k is a field of characteristic different from 2 containing a primitive fourth root of unity i. Then there are exactly two different R-matrices for H (2), namely,…”