# Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells

## TL;DR

This paper demonstrates that certain generalized double Bruhat cells form natural Poisson groupoids over generalized Bruhat cells, extending previous results and introducing a novel local Poisson groupoid construction involving mixed product structures and symplectic groupoids.

## Contribution

It establishes that $G^{u,u}$ is a Poisson groupoid over $O^u$, extending prior work, and introduces a new local Poisson groupoid construction using double symplectic groupoids and R-matrix techniques.

## Key findings

- $G^{u,u}$ forms a Poisson groupoid over $O^u$.
- Introduces a local Poisson groupoid construction over mixed product structures.
- Connects the construction to the global R-matrix and symplectic groupoids.

## Abstract

Given a standard complex semisimple Poisson Lie group $(G, \pi_{st})$, generalised double Bruhat cells $G^{u, v}$ and generalised Bruhat cells $O^u$ equipped with naturally defined holomorphic Poisson structures, where u, v are finite sequences of Weyl group elements, were defined and studied by Jiang Hua Lu and the author. We prove in this paper that $G^{u,u}$ is naturally a Poisson groupoid over $O^u$, extending a result from the aforementioned authors about double Bruhat cells in $(G, \pi_{st})$. Our result on $G^{u,u}$ is obtained as an application of a construction interesting in its own right, of a local Poisson groupoid over a mixed product Poisson structure associated to the action of a pair of Lie bialgebras. This construction involves using a local Lagrangian bisection in a double symplectic groupoid closely related to the global R-matrix studied by Weinstein and Xu, to twist a direct product of Poisson groupoids.