Double quantization on coadjoint representations of simple Lie groups and its orbits

Details Double quantization on coadjoint representations of simple Lie groups and its orbits Double quantization on coadjoint representations of simple Lie groups and its orbits

1999-09-27

arxiv.org/abs/math/9909160v1

Mathematics

Quantum Algebra

Latex2e, 32 pp

Scientific paper

Let M be a manifold with an action of a Lie group G, $\A$ the function algebra on M. The first problem we consider is to construct a $U_h(\g)$ invariant quantization, $\A_h$, of $\A$, where $U_h(\g)$ is a quantum group corresponding to G. Let s be a G invariant Poisson bracket on M. The second problem we consider is to construct a $U_h(\g)$ invariant two parameter (double) quantization, $\A_{t,h}$, of $\A$ such that $\A_{t,0}$ is a G invariant quantization of $s$. We call $\A_{t,h}$ a $U_h(\g)$ invariant quantization of the Poisson bracket s. In the paper we study the cases when G is a simple Lie group and $M$ is the coadjoint representation $\g^*$ of G or a semisimple orbit in this representation. The paper is founded on the papers: J.Donin, Double quantization on the coadjoint representation of $sl(n)^*$, Czechoslovak J. of Physics, 47 (1997), no 11, 1115-1122, q-alg/9707031, and J.Donin, D.Gurevich, and S.Shnider, Double Quantization on Some Orbits in the Coadjoint Representations of Simple Lie Groups, Com. Math. Phys., 204 (1999), no. 1, 39-60, math/9807159, and contains some additional results.

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