Quantizations of generalized Cartan type $S$ Lie algebras and of the special algebra $\mathbf{S}(n;\underline{1})$ in the modular case

Abstract

The generalized Cartan type $\mathbf{S}$ Lie algebras in char 0 with the Lie bialgebra structures involved are quantized, where the Drinfel'd twist we used is proved to be a variation of the Jordanian twist. As the passage from char 0 to char p, their quantization integral forms are given. By the modular reduction and base changes, we obtain certain quantizations of the restricted universal enveloping algebra $\mathbf{u}(\mathbf{S}(n;\underline{1}))$ (for the Cartan type simple modular restricted Lie algebra $\mathbf{S}(n;\underline{1})$ of $\mathbf{S}$ type). They are new Hopf algebras of truncated $p$-polynomial noncommutative and noncocommutative deformation of dimension $p^{1+(n-1)(p^n-1)}$, which contain the well-known Radford algebra (\cite{DR}) as a Hopf subalgebra. As a by-product, we also get some Jordanian quantizations for ${\mathfrak{sl}}_n$, which are induced from those horizontal quantizations of $\mathbf{S}(n;\underline{1})$.

33 pages

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• Some Twisted Results
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Cited by in corpus (4)

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• On non-connected pointed Hopf algebras of dimension 16 in characteristic 2
• Pointed Hopf algebras of dimension $p^{2}q$ in characteristic $p$
• Matching realization of $U_q(s{l}_{n+1})$ of higher rank in the quantum Weyl algebra ${\mathcal{W}}_q(2n)$