Modular Quantizations of Lie Algebras of Cartan Type K via Drinfeld Twists of Jordanian Type

TL;DR

This paper develops explicit quantum deformations of Cartan type K Lie algebras using Drinfeld twists, leading to new modular Hopf algebras with specific dimensions and structures.

Contribution

It constructs explicit Jordanian-type Drinfeld twists for Lie algebras of Cartan type K and derives new modular quantizations with detailed algebraic properties.

Findings

Constructed explicit Drinfeld twists for Lie algebra of Cartan type K.

Derived new pointed Hopf algebras with specific dimensions.

Included Radford algebras as subalgebras within the new structures.

Abstract

We construct explicit Drinfel'd twists of Jordanian type for the generalized Cartan type K Lie algebras in characteristic 0 and obtain the corresponding quantizations, especially their integral forms. By making modular reductions including modulo p and modulo p-restrictedness reduction, and base changes, we derive certain modular quantizations of the restricted universal enveloping algebra $\mathbf u(\mathbf{K}(2n{+}1;\underline{1}))$ for the restricted simple Lie algebra of Cartan type K in characteristic p. They are new pointed Hopf algebras of noncommutative and noncocommutative and with dimension $p^{p^{2n+1}+1}$ (if $2n+4\not\equiv0 \; (\mod p)$) or $p^{p^{2n+1}}$ (if $2n+4\equiv0 \; (\mod p))$ over a truncated p-polynomials ring, which also contain the well-known Radford algebras as Hopf subalgebras. Some open questions are proposed.