Quantization of a Poisson structure on products of principal affine spaces

TL;DR

This paper constructs a deformation quantization of Poisson structures on products of principal affine spaces associated with complex semisimple Lie groups, extending Lie bialgebra techniques to Hopf algebras.

Contribution

It introduces a Hopf algebra analogue of polyuble Lie bialgebras and applies it to quantum groups, providing a new method for quantizing Poisson structures on principal affine spaces.

Findings

Deformation quantization $ abla$ of Poisson structures on $(N ackslash G)^m$

Quantization of graded Poisson algebras of line bundles on $(B ackslash G)^m$

Introduction of strongly coisotropic subalgebras ensuring quantization of homogeneous coordinate rings

Abstract

We give the analogue for Hopf algebras of the polyuble Lie bialgebra construction by Fock and Rosli. By applying this construction to the Drinfeld-Jimbo quantum group, we obtain a deformation quantization $\mathbb{C}_\hslash[(N \backslash G)^m]$ of a Poisson structure $\pi^{(m)}$ on products $(N \backslash G)^m$ of principal affine spaces of a connected and simply connected complex semisimple Lie group $G$. The Poisson structure $\pi^{(m)}$ descends to a Poisson structure $\pi_m$ on products $(B \backslash G)^m$ of the flag variety of $G$ which was introduced and studied by the Lu and the author. Any ample line bundle on $(B \backslash G)^m$ inherits a natural flat Poisson connection, and the corresponding graded Poisson algebra is quantized to a subalgebra of $\mathbb{C}_\hslash[(N \backslash G)^m]$. We define the notion of a strongly coisotropic subalgebra in a Hopf algebra, and explain how strong coisotropicity guarantees that any homogeneous coordinate ring of a homogeneous space of a Poisson Lie group can be quantized in the sense of Ciccoli, Fioresi, and Gavarini.