模情形下广义Witt代数和Jacobson-Witt代数的量子化
Quantizations of generalized-Witt algebra and of Jacobson-Witt algebra in the modular case
摘要：
利用Song-Su（\cite{GY}）发现的特征为0的广义Witt代数的李双代数结构，构造了广义Witt代数.通过模p约化和模“p-限制性”约化过程，得到Jacobson-Witt代数\mathbf{W}（n; \underline{1}）（对于W型Cartan型单模限制李代数）的限制泛包络代数的2 ^n {-}1族截断p-多项式非余交换变形.我们的结果推广了Grunspan（J. Algebra 280（2004），145- 161，\cite{CG}）在秩n=1，特征为0的情形下的工作.在模化的情况下，一个改进版本的论点遵循模化约简方法（不同于\cite{CG}），并使用了模李代数理论中的一些技术。
英文摘要：
We quantize the generalized-Witt algebra in characteristic 0 with its Lie bialgebra structures discovered by Song-Su (\cite{GY}). 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 \mathbf{W}(n;\underline{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+np^n} in characteristic p. Our results generalize a work of Grunspan (J. Algebra 280 (2004), 145--161, \cite{CG}) 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 \cite{CG}) with some techniques from the modular Lie algebra theory.
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