A (2n+1)(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∗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.

► We consider a Poisson–Lie group GG, equipped with a non-linear Poisson bracket.
► The Poisson data guides us to construct a twisted crossed product C∗C∗-algebra SS.
► The C∗C∗-algebra SS is shown to be a strict deformation quantization of the group GG.
► The Poisson data also suggests comultiplication, antipode, and Haar weight on SS.
► SS is in fact a locally compact quantum group of “cocycle bicrossed product” type.