Non-compact quantum groups arising from Heisenberg type Lie bialgebras

Details Non-compact quantum groups arising from Heisenberg type Lie bialgebras Non-compact quantum groups arising from Heisenberg type Lie bialgebras

1998-04-03

arxiv.org/abs/math/9804019v2

Mathematics

Operator Algebras

AMS LaTeX (36 pages), Revised version of previous math.OA9804019; to appear in J. Operator Theory

Scientific paper

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.

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