Modular quantizations of Lie algebras of Cartan type $H$ via Drinfeld Twists

TL;DR

This paper constructs explicit Drinfeld twists for Lie algebras of Cartan type H, leading to new modular quantizations and non-pointed Hopf algebras with applications to Jordanian quantizations.

Contribution

It introduces explicit Drinfeld twists for Cartan type H Lie algebras and derives new modular quantizations and non-pointed Hopf algebras in characteristic p.

Findings

New non-pointed Hopf algebras of prime-power dimension

Explicit Drinfeld twists for Lie algebras of Cartan type H

Jordanian quantizations for sp_{2n}

Abstract

We construct explicit Drinfel'd twists for the generalized Cartan type $H$ Lie algebras in characteristic $0$ and obtain the corresponding quantizations and their integral forms. Via making modular reductions including modulo $p$ reduction and modulo $p$-restrictedness reduction, and base changes, we derive certain modular quantizations of the restricted universal enveloping algebra $\mathbf u(\mathbf{H}(2n;\underline{1}))$ in characteristic $p$. They are new non-pointed Hopf algebras of truncated $p$-polynomial noncommutative and noncocommutative deformation of prime-power dimension $p^{p^{2n}-1}$, which contain the well-known Radford algebras as Hopf subalgebras. As a by-product, we also get some Jordanian quantizations for $\mathfrak {sp}_{2n}$.