Quantum Divided Power Algebra, Q-Derivatives, and Some New Quantum Groups
Quantum Divided Power Algebra, Q-Derivatives, and Some New Quantum Groups
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DOI：
10.1006/jabr.2000.8385
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发表时间：
2000-10
影响因子：
0.9
通讯作者：
N. Hu
中科院分区：
文献类型：
--
作者：
N. Hu
本文的讨论是从量子n空间的结构本质出发的。建立了域k上Λ-分次θ-交换结合代数的一类辫子范畴G B.在G B中引入了与量子n-空间相关的k上量子除幂代数，并将其描述为辫状Hopf代数（根据它的2-上循环结构），在其上定义了所谓的特殊q-导数，使得几个新的有趣的量子群，特别是n元量子化多项式代数（作为n维交换李代数的量子化泛包络代数）和量子n空间的量子群，都是从我们的方法中独立于使用R矩阵导出的。作为对本文讨论的有效性的验证，通过某些q-微分算子的实现，量子分幂代数被赋予了Uq（sln）-模代数的结构.特别地，在Lusztig意义下，Uq（s l n）的四种根向量之一可以在该实现下精确地确定.
Abstract The discussions in the present paper arise from exploring intrinsically the structural nature of the quantum n-space. A kind of braided category G B of Λ-graded θ-commutative associative algebras over a field k is established. The quantum divided power algebra over k related to the quantum n-space is introduced and described as a braided Hopf algebra in G B (in terms of its 2-cocycle structure), over which the so-called special q-derivatives are defined so that several new interesting quantum groups, especially the quantized polynomial algebra in n variables (as the quantized universal enveloping algebra of the abelian Lie algebra of dimension n) and the quantum group associated to the quantum n-space, are derived from our approach independently of using the R-matrix. As a verification of its validity for our discussion, the quantum divided power algebra is equipped with the structure of a Uq( s l n)-module algebra via certain q-differential operators' realization. Particularly, one of the four kinds of root vectors of Uq( s l n) in the sense of Lusztig can be specified precisely under the realization.