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

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