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

In [9], 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 [12], 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 [10] showed that the Drinfeld double of a quasi-Hopf algebra of dimension $n^3$ constructed in [9] is always twist equivalent to Lusztig small quantum group $\mathrm{u}_q(\mathfrak{sl}_2)$ 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 $Q \, \mathrm{u}_q(\mathfrak{sl}_2)$ of $\mathrm{u}_q(\mathfrak{sl}_2)$ is gotten.

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Published 13 October 2014