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

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.