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

TL;DR

This paper constructs a (2n+1)-dimensional quantum group from a skew-symmetric matrix by defining a Poisson-Lie group, analyzing its structure, and applying cocycle bicrossed product methods to achieve a deformation quantization.

Contribution

It introduces a new method to construct a quantum group from a skew-symmetric matrix using cocycle bicrossed products and deformation quantization techniques.

Findings

Successfully constructs a (2n+1)-dimensional quantum group

Shows the quantum group is a deformation quantization of a Poisson-Lie group

Provides a new approach linking skew-symmetric matrices to quantum groups

Abstract

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.