Braided Symmetric Algebras of Simple $U_q(sl_2)$-Modules and Their Geometry

Details Braided Symmetric Algebras of Simple $U_q(sl_2)$-Modules and Their Geometry Braided Symmetric Algebras of Simple $U_q(sl_2)$-Modules and Their Geometry

2010-09-15

arxiv.org/abs/1009.2931v2

Mathematics

Quantum Algebra

19 pages,relationship to non--commutative geometry expanded

Scientific paper

In the present paper we prove decomposition formulae for the braided symmetric powers of simple modules over the quantized enveloping algebra $U_q(sl_2)$; natural quantum analogues of the classical symmetric powers of a module over a complex semisimple Lie algebra. We show that their point modules form natural non-commutative curves and surfaces and conjecture that braided symmetric algebras give rise to an interesting non-commutative geometry, which can be viewed as a flat deformation of the geometry associated to their classical limits.

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