Dressing orbits and a quantum Heisenberg group algebra

Details Dressing orbits and a quantum Heisenberg group algebra Dressing orbits and a quantum Heisenberg group algebra

2002-10-31

arxiv.org/abs/math/0211003v3

Mathematics

Operator Algebras

25 pages, 1 figure

Scientific paper

In this paper, as a generalization of Kirillov's orbit theory, we explore the relationship between the dressing orbits and irreducible *-representations of the Hopf C*-algebras (A,\Delta) and (\tilde{A}, \tilde{\Delta}) we constructed earlier. We discuss the one-to-one correspondence between them, including their topological aspects. On each dressing orbit (which are symplectic leaves of the underlying Poisson structure), one can define a Moyal-type deformed product at the function level. The deformation is more or less modeled by the irreducible representation corresponding to the orbit. We point out that the problem of finding a direct integral decomposition of the regular representation into irreducibles (Plancherel theorem) has an interesting interpretation in terms of these deformed products.

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