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 of a Poisson structure on products of principal affine spaces of a connected and simply connected complex semisimple Lie group . The Poisson structure descends to a Poisson structure on products of the flag variety of which was introduced and studied by the Lu and the author. Any ample line bundle on inherits a natural flat Poisson connection, and the corresponding graded Poisson algebra is quantized to a subalgebra of .
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. 1049–1074DOI 10.4171/JNCG/386