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

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 ${\mathfrak{sp}}_{2n}$.

33 pages

Cited by in corpus (2)

• Modular Quantizations of Lie Algebras of Cartan Type K via Drinfeld Twists of Jordanian Type
• On non-connected pointed Hopf algebras of dimension 16 in characteristic 2