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

Given a standard complex semisimple Poisson Lie group $(G,\pi_\mathrm{st})$, generalised double Bruhat cells $G^\mathbf{u,v}$ and generalised Bruhat cells $\mathcal{O}^\mathbf{u}$ equipped with naturally defined holomorphic Poisson structures, where $\mathbf{u,v}$ are finite sequences of Weyl group elements, were defined and studied by Jiang-Hua Lu and the author. We prove in this paper that $G^\mathbf{u,u}$ is naturally a Poisson groupoid over $\mathcal{O}^\mathbf{u}$, extending a result from the aforementioned authors about double Bruhat cells in $(G,\pi_\mathrm{st})$.

Our result on $G^\mathbf{u,u}$ 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 $\mathcal{R}$‑matrix studied by Weinstein and Xu, to “twist” a direct product of Poisson groupoids.

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Received 18 September 2019

Accepted 30 October 2020

Published 8 December 2021