Quantizations of generalized Cartan type S Lie algebras and of the special algebra S(n;\underline1) in the modular case

2009/02/17 by Naihong Hu, Xiuling Wang · 1 citation
Mathematics · #math.QA #math.RT #msc:17B37 #msc:17B62 #msc:17B50

paper · pdf · doi:10.1016/j.jpaa.2010.08.005

published as J. Pure and Applied Algebra 215 (6) (2011), 1205-1222 · 33 pages

arxiv created 2009/02/17 · arxiv updated 2014/10/06

Abstract

The generalized Cartan type 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(S(n;\underline1)) (for the Cartan type simple modular restricted Lie algebra S(n;\underline1) of S type). They are new Hopf algebras of truncated p-polynomial noncommutative and noncocommutative deformation of dimension p^{1+(n-1)(pn-1)}, which contain the well-known Radford algebra (\citeDR) 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).

Cited by

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

Related

• Quantizations of the Witt algebra and of simple Lie algebras in characteristic p
• Twisting of quantum (super)algebras. Connection of Drinfeld's and Cartan-Weyl realizations for quantum affine algebras
• Quantization of Schrödinger-Virasoro Lie algebra
• Quantizations of the extended affine Lie algebra \widetilde\fraksl₂(ℂ_q)
• Quantization of Lie bialgebras, part VI: quantization of generalized Kac-Moody algebras