The quasi-Hopf analogue of $\mathrm{u}_q(\mathfrak{sl}_2)$

Abstract

In [4], some quasi-Hopf algebras of dimension n 3 , which can be understood as the quasi-Hopf analogues of Taft algebras, are constructed. Moreover, the quasi-Hopf analogues of generalized Taft algebras are considered in [7], where the language of the dual of a quasi-Hopf algebra is used. The Drinfeld doubles of such quasi-Hopf algebras are computed in this paper. The authors in [5] shew that the Drinfeld double of a quasi-Hopf algebra of dimension n 3 constructed in [4] is always twist equivalent to Lusztig's small quantum group uq(sl 2 ) if n is odd. Based on computations and analysis, we show that this is not the case if n is even. That is, the quasi-Hopf analogue Q uq(sl 2 ) of uq(sl 2 ) is gotten.

Finite-dimensional quasi-Hopf algebras of Cartan type

“…The above example provides us some non-radically graded quasi-Hopf algebras associated to A 1 × A 1 . We point out that there is a similar notion of a quasi-version of u q (sl 2 ) in [24], where Liu defined a quasi-Hopf analogue of u q (sl 2 ) as a quantum double of a quasi-Hopf algebra associated to A 1 . It is obvious that these two definitions are different, since the dimension of a quasi-version of u q (sl 2 ) is not a square in general.…”