## Illinois Journal of Mathematics

### Dressing orbits and a quantum Heisenberg group algebra

Byung-Jay Kahng

#### Abstract

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

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

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138402

Digital Object Identifier
doi:10.1215/ijm/1258138402

Mathematical Reviews number (MathSciNet)
MR2085430

Zentralblatt MATH identifier
1067.46065

#### Citation

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. https://projecteuclid.org/euclid.ijm/1258138402