Non-compact quantum groups arising from Heisenberg type Lie bialgebras

Abstract

The dual Lie bialgebra of a certain ``quasitriangular'' Lie bialgebra structure on the Heisenberg Lie 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 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.