TY - JOUR
T1 - Method of quantum characters in equivariant quantization
AU - Donin, J.
AU - Mudrov, A.
PY - 2003/3
Y1 - 2003/3
N2 - Let G be a reductive Lie group, g its Lie algebra, and M a G-manifold. Suppose Ah(M) is a Uh (g)-equivariant quantization of the function algebra A(M) on M. We develop a method of building Uh (g)-equivariant quantization on G-orbits in M as quotients of Ah(M). We are concerned with those quantizations that may be simultaneously represented as subalgebras in U*h(g) and quotients of Ah(M). It turns out that they are in one-to-one correspondence with characters of the algebra Ah(M). We specialize our approach to the situation g = gl (n, ℂ), M = End(ℂn), and Ah(M) the so-called reflection equation algebra associated with the representation of ℂh,(g) on ℂn. 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.
AB - Let G be a reductive Lie group, g its Lie algebra, and M a G-manifold. Suppose Ah(M) is a Uh (g)-equivariant quantization of the function algebra A(M) on M. We develop a method of building Uh (g)-equivariant quantization on G-orbits in M as quotients of Ah(M). We are concerned with those quantizations that may be simultaneously represented as subalgebras in U*h(g) and quotients of Ah(M). It turns out that they are in one-to-one correspondence with characters of the algebra Ah(M). We specialize our approach to the situation g = gl (n, ℂ), M = End(ℂn), and Ah(M) the so-called reflection equation algebra associated with the representation of ℂh,(g) on ℂn. 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.
UR - http://www.scopus.com/inward/record.url?scp=0037359518&partnerID=8YFLogxK
U2 - 10.1007/s00220-002-0771-7
DO - 10.1007/s00220-002-0771-7
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AN - SCOPUS:0037359518
SN - 0010-3616
VL - 234
SP - 533
EP - 555
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
IS - 3
ER -