Springer Online Journal Archives 1860-2000
Abstract: Let ? be the function algebra on a semisimple orbit, M, in the coadjoint representation of a simple Lie group, g, with the Lie algebra ?. We study one and two parameter quantizations ? h and ? t,h of ? such that the multiplication on the quantized algebra is invariant under action of the Drinfeld–Jimbo quantum group, U h (?). In particular, the algebra ? t,h specializes at h= 0 to a U(?)-invariant ($G$-invariant) quantization, %Ascr; t ,0. We prove that the Poisson bracket corresponding to ? h 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 H 2(M) parameter family, then we construct their quantizations. A two parameter (or double) quantization, $? t,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.
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