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 $\mathbf S(n;\underline1)$.