Journal of Operator Theory

Volume 44, Issue 2, Fall 2000 pp. 303-334.

Non-compact quantum groups arising from Heisenberg type Lie bialgebras

Authors: Byung-Jay Kahng
Author institution: Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA

Summary: The dual Lie bialgebra of a certain {\it quasitriangular} Lie bialgebra structure on the Heisenberg algebra determines a (non-compact) Poisson-Lie group $G$. The compatible Poisson bracket on $G$ is non-linear, but it can still be realized as a ``cocycle perturbation'' of the linear Poisson bracket. We construct a certain twisted group $C^*$-algebra $A$, which is shown to be a strict deformation quantization of $G$. Motivated by the data at the Poisson (classical) level, we then construct on $A$ its {\it locally compact quantum group} structures: comultiplication, counit, antipode and Haar weight, as well as its associated multiplicative unitary operator. We also find a quasitriangular ``quantum universal $R$-matrix'' type operator for $A$, which agrees well with the quasitriangularity at the Lie bialgebra level.

Keywords: Deformation quantization, twisted group $C^*$-algebra, Poisson-Lie group, quantum group

Contents Full-Text PDF