Quantizations of generalized-Witt algebra and of Jacobson–Witt algebra in the modular case
Quantizations of generalized-Witt algebra and of Jacobson–Witt algebra in the modular case
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DOI：
10.1016/j.jalgebra.2007.02.019
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发表时间：
2006-02
影响因子：
0.9
通讯作者：
N. Hu;Xiuling Wang
中科院分区：
文献类型：
--
作者：
N. Hu;Xiuling Wang
利用Song-Su [G.宋，Y。苏，广义Witt型李双代数，arXiv：数学QA/0504168，科学。中国系列A 49（4）（2006）533-544]。通过模p约化和模“p-限制性”约化过程，得到Jacobson-Witt代数W（n;1 <$）（对于W型Cartan型单模限制李代数）的限制泛包络代数的2n−1族截断p-多项式非余交换变形.它们是特征p维的非交换和非余交换Hopf代数的新族。Grunspan，Quantizations of the Witt algebra and of simple Lie algebras in characteristic p，J. Algebra 280（2004）145-161] in rank n=1 case in characteristic 0.在模块化的情况下，一个精炼版本的论点来自模块化约简方法（不同于[C。Grunspan，Quantizations of the Witt algebra and of simple Lie algebras in characteristic p，J. Algebra 280（2004）145-161]）。
We quantize the generalized-Witt algebra in characteristic 0 with its Lie bialgebra structures discovered by Song–Su [G. Song, Y. Su, Lie bialgebras of generalized-Witt type, arXiv: math.QA/0504168, Sci. China Ser. A 49 (4) (2006) 533–544]. Via a modulo p reduction and a modulo “p-restrictedness” reduction process, we get 2n−1 families of truncated p-polynomial noncocommutative deformations of the restricted universal enveloping algebra of the Jacobson–Witt algebra W(n;1̲) (for the Cartan type simple modular restricted Lie algebra of W type). They are new families of noncommutative and noncocommutative Hopf algebras of dimension [Formula: see text] in characteristic p. Our results generalize a work of Grunspan [C. Grunspan, Quantizations of the Witt algebra and of simple Lie algebras in characteristic p, J. Algebra 280 (2004) 145–161] 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.