Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras

TL;DR

This paper introduces mixed product Poisson structures linked to Poisson Lie groups and Lie bialgebras, providing new geometric constructions and examples on flag varieties and complex Lie groups.

Contribution

It develops a novel framework for mixed product Poisson structures, connecting to fusion products and extending to new examples on flag varieties.

Findings

Equivalent to fusion products of quasi-Poisson G-manifolds for quasitriangular Lie bialgebras

Constructs four series of holomorphic Poisson structures on product spaces

Provides new geometric examples related to complex semi-simple Lie groups

Abstract

We introduce and study some mixed product Poisson structures on product manifolds associated to Poisson Lie groups and Lie bialgebras. For quasitriangular Lie bialgebras, our construction is equivalent to that of fusion products of quasi-Poisson G-manifolds introduced by Alekseev, Kosmann- Schwarzbach, and Meinrenken. Our primary examples include four series of holomorphic Poisson structures on products of flag varieties and related spaces of complex semi-simple Lie groups.