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Quantum Divided Power Algebra, Q-Derivatives, and Some New Quantum Groups

Authors
Journal
Journal of Algebra
0021-8693
Publisher
Elsevier
Publication Date
Volume
232
Issue
2
Identifiers
DOI: 10.1006/jabr.2000.8385
Keywords
  • Quantum N-Space
  • Bicharacter
  • (Braided) Hopf Algebra
  • Quantum Divided Power (Restricted) Algebra
  • Q-Derivatives
  • (Hopf) Module Algebra
  • Quantum Root Vectors
Disciplines
  • Mathematics
  • Physics

Abstract

Abstract The discussions in the present paper arise from exploring intrinsically the structural nature of the quantum n-space. A kind of braided category GB 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 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 for our discussion, the quantum divided power algebra is equipped with the structure of a Uq(sln)-module algebra via certain q-differential operators' realization. Particularly, one of the four kinds of root vectors of Uq(sln) in the sense of Lusztig can be specified precisely under the realization.

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