Quantizations of generalized-Witt algebra and of Jacobson-Witt algebra in the modular case

JOURNAL OF ALGEBRA

卷:312

Article

Hu, Naihong ; Wang, Xiuling

关键词: quantization;    (basic) Drinfel'd twist;    Lie bialgebra;    generalized-Witt algebra;    Jacobson-Witt algebra;    Hopf algebra;   

DOI  :  10.1016/j.jalgebra.2007.02.019

来源: Elsevier

【 摘 要 】

We quantize the generalized-Witt algebra in charactefistic 0 with its Lie bialgebra structures discovered by Song-Su [G. Song, Y Su, Lie bialgebras of generalized-Witt type, arXiv: math. QA/05 04168, Sci. China Ser. A 49 (4) (2006) 533-544]. Via a modulo p reduction and a modulo p-restrictedness reduction process, we get 2(n) - 1 families of truncated p-polynomial noncocommutative deformations of the restricted universal enveloping algebra of the Jacobson-Witt algebra W(n; 1) (for the Cartan type simple modular restricted Lie algebra of W type). They are new families of noncommutative and noncocommutative Hopf algebras of dimension p(1 +npn) in characteristic p. Our results generalize a work of Grumspan [C. Grunspan, Quantizations of the Witt algebra and of simple Lie algebras in characteristic p, J. Algebra 280 (2004) 145161] in rank n = 1 case in characteristic 0. In the modular case, the argument for a refined version follows from the modular reduction approach (different from [C. Grunspan, Quantizations of the Witt algebra and of simple Lie algebras in characteristic p, J. Algebra 280 (2004) 145-161]) with some techniques from the modular Lie algebra theory. (c) 2007 Elsevier Inc. All rights reserved.

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