Illinois Journal of Mathematics

Dressing orbits and a quantum Heisenberg group algebra

Byung-Jay Kahng

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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.

Article information

Illinois J. Math., Volume 48, Number 2 (2004), 609-634.

First available in Project Euclid: 13 November 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L65: Quantizations, deformations
Secondary: 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx]


Kahng, Byung-Jay. Dressing orbits and a quantum Heisenberg group algebra. Illinois J. Math. 48 (2004), no. 2, 609--634. doi:10.1215/ijm/1258138402.

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