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

TL;DR

This paper develops new quantizations of generalized Cartan type S Lie algebras in characteristic zero and positive characteristic, introducing novel Hopf algebra structures and Jordanian quantizations related to classical Lie algebras.

Contribution

It provides the first quantizations of these Lie algebras in modular settings and constructs new Hopf algebras with specific noncommutative and noncocommutative properties.

Findings

Quantized generalized Cartan type S Lie algebras in char 0.

Constructed integral forms and modular reductions for char p.

Identified new Hopf algebras containing Radford algebra as subalgebra.

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