A (2n+1)-dimensional quantum group constructed from a skew-symmetric matrix

Beginning with a skew-symmetric matrix, we define a certain Poisson--Lie group. Its Poisson bracket can be viewed as a cocycle perturbation of the linear (or "Lie-Poisson") Poisson bracket. By analyzing this Poisson structure, we gather enough information to construct a C*-algebraic locally compact quantum group, via the "cocycle bicrossed product construction" method. The quantum group thus obtained is shown to be a deformation quantization of the Poisson-Lie group, in the sense of Rieffel

Quantizations of some Poisson-Lie groups: The bicrossed product construction

By working with several specific Poisson-Lie groups arising from Heisenberg Lie bialgebras and by carrying out their quantizations, a case is made for a useful but simple method of constructing locally compact quantum groups. The strategy is to analyze and collect enough information from a Poisson-Lie group, and using it to carry out a ``cocycle bicrossed product construction''. Constructions are done using multiplicative unitary operators, obtaining C*-algebraic, locally compact quantum (semi-)groups.Comment: 26 page