TY - JOUR

T1 - Double quantization on some orbits in the coadjoint representations of simple Lie groups

AU - Donin, J.

AU - Gurevich, D.

AU - Shnider, S.

PY - 1999

Y1 - 1999

N2 - Let A be the function algebra on a semisimple orbit, M, in the coadjoint representation of a simple Lie group, G, with the Lie algebra g-fraktur sign. We study one and two parameter quantizations Ah and At,h of A such that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group, Uh(g-fraktur sign). In particular, the algebra At,h specializes at h = 0 to a U(g-fraktur sign)-invariant (G-invariant) quantization, At,0. We prove that the Poisson bracket corresponding to Ah must be the sum of the so-called r-matrix and an invariant bracket. We classify such brackets for all semisimple orbits, M, and show that they form a dim H2(M) parameter family, then we construct their quantizations. A two parameter (or double) quantization, At,h, corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on M. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization of these pairs in some of the cases.

AB - Let A be the function algebra on a semisimple orbit, M, in the coadjoint representation of a simple Lie group, G, with the Lie algebra g-fraktur sign. We study one and two parameter quantizations Ah and At,h of A such that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group, Uh(g-fraktur sign). In particular, the algebra At,h specializes at h = 0 to a U(g-fraktur sign)-invariant (G-invariant) quantization, At,0. We prove that the Poisson bracket corresponding to Ah must be the sum of the so-called r-matrix and an invariant bracket. We classify such brackets for all semisimple orbits, M, and show that they form a dim H2(M) parameter family, then we construct their quantizations. A two parameter (or double) quantization, At,h, corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on M. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization of these pairs in some of the cases.

UR - http://www.scopus.com/inward/record.url?scp=0033473949&partnerID=8YFLogxK

U2 - 10.1007/s002200050636

DO - 10.1007/s002200050636

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AN - SCOPUS:0033473949

SN - 0010-3616

VL - 204

SP - 39

EP - 60

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 1

ER -