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

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dc.creatorHu, Naihong
dc.date2009-02-17
dc.date.accessioned2026-07-07T12:42:51Z
dc.date.available2026-07-07T12:42:51Z
dc.descriptionThe 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 $U_q(\frak {sl}_n)$-module algebra via a certain $q$-differential operators realization. Particularly, one of the four kinds of roots vectors of $U_q(\frak {sl}_n)$ in the sense of Lusztig can be specified precisely under the realization.
dc.description28 pages
dc.identifierhttps://arxiv.org/abs/0902.2858
dc.identifierhttp://arxiv.org/abs/0902.2858
dc.identifierJournal of Algebra 232, 507--540 (2000)
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/220224
dc.subjectQuantum Algebra
dc.subjectRepresentation Theory
dc.subject17B37, 81R50, 17B35
dc.titleQuantum Divided Power Algebra, q-Derivatives and Some New Quantum Groups
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