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

DOI10.4310/JSG.2021.V19.N4.A4zbMATH Open1485.53102arXiv1908.04044MaRDI QIDQ2063700

Victor Mouquin

Publication date: 11 January 2022

Published in: The Journal of Symplectic Geometry (Search for Journal in Brave)

Abstract: Given a standard complex semisimple Poisson Lie group $(G,p{i}_{st})$, 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 $(G,p{i}_{st})$. 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.

Full work available at URL: https://arxiv.org/abs/1908.04044

zbMATH Keywords

double Bruhat cellsLie bialgebraPoisson groupoidsPoisson Lie groupsemi simple

Mathematics Subject Classification ID

Poisson manifolds; Poisson groupoids and algebroids (53D17)

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