2009/02/17 by Naihong Hu · 1 citation
Mathematics · #math.QA #math.RT #msc:17B37 #msc:81R50 #msc:17B35
published as Journal of Algebra 232, 507--540 (2000) · 28 pages
arxiv created 2009/02/17 · arxiv updated 2009/12/01
The discussions in the present paper arise from exploring intrinsically the structure nature of the quantum n-space. A kind of braided category \Cal GB of \La-graded þ-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 \Cal GB (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 of our discussion, the quantum divided power algebra is equipped with a structure of Uq(\frak sln)-module algebra via a certain q-differential operators realization. Particularly, one of the four kinds of roots vectors of Uq(\frak sln) in the sense of Lusztig can be specified precisely under the realization.