# Quantization of a Poisson structure on products of principal affine spaces

### Victor Mouquin

Shanghai Jiaotong University, China

## 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 $C_{ℏ}[(N\G)_{m}]$ of a Poisson structure $π_{(m)}$ on products $(N\G)_{m}$ of principal affine spaces of a connected and simply connected complex semisimple Lie group $G$. The Poisson structure $π_{(m)}$ descends to a Poisson structure $π_{m}$ on products $(B\G)_{m}$ of the flag variety of $G$ which was introduced and studied by the Lu and the author. Any ample line bundle on $(B\G)_{m}$ inherits a natural flat Poisson connection, and the corresponding graded Poisson algebra is quantized to a subalgebra of $C_{ℏ}[(N\G)_{m}]$.

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.

## Cite this article

Victor Mouquin, Quantization of a Poisson structure on products of principal affine spaces. J. Noncommut. Geom. 14 (2020), no. 3, pp. 1049–1074

DOI 10.4171/JNCG/386