  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.
