## Illinois Journal of Mathematics

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

### Dressing orbits and a quantum Heisenberg group algebra

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/ijm/1258138402

**Mathematical Reviews number (MathSciNet)**

MR2085430

**Zentralblatt MATH identifier**

1067.46065

**Subjects**

Primary: 46L65: Quantizations, deformations

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

#### 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