## Abstract

Let G be a reductive Lie group, g its Lie algebra, and M a G-manifold. Suppose A_{h}(M) is a U_{h} (g)-equivariant quantization of the function algebra A(M) on M. We develop a method of building U_{h} (g)-equivariant quantization on G-orbits in M as quotients of A_{h}(M). We are concerned with those quantizations that may be simultaneously represented as subalgebras in U*_{h}(g) and quotients of A_{h}(M). It turns out that they are in one-to-one correspondence with characters of the algebra A_{h}(M). We specialize our approach to the situation g = gl (n, ℂ), M = End(ℂ^{n}), and A_{h}(M) the so-called reflection equation algebra associated with the representation of ℂ_{h},(g) on ℂ^{n}. For this particular case, we present in an explicit form all possible quantizations of this type; they cover symmetric and bisymmetric orbits. We build a two-parameter deformation family and obtain, as a limit case, the U(g)-equivariant quantization of the Kirillov-Kostant-Souriau bracket on symmetric orbits.

Original language | English |
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Pages (from-to) | 533-555 |

Number of pages | 23 |

Journal | Communications in Mathematical Physics |

Volume | 234 |

Issue number | 3 |

State | Published - Mar 2003 |