Braid group symmetries on Poisson algebras arising from quantum symmetric pairs

TL;DR

This paper explores braid group symmetries on Poisson algebras derived from quantum symmetric pairs, providing new algebraic structures and explicit Poisson bracket descriptions in the semi-classical limit.

Contribution

It introduces braid group symmetries and PBW bases for the integral form of $ ext{U}^ ext{i}$, extending previous work to a broader class of quantum symmetric pairs.

Findings

Established braid group symmetries on the integral form of $ ext{U}^ ext{i}$

Derived explicit Poisson brackets for the associated Poisson algebra

Provided examples including Dubrovin-Ugaglia Poisson brackets

Abstract

Let $(\mathrm{U},\mathrm{U}^\imath)$ be the quantum symmetric pair of arbitrary finite type and $G^*$ be the associated dual Poisson-Lie group. Generalizing the work of De Concini and Procesi, the first author introduced an integral form for the $\imath$quantum group $\mathrm{U}^\imath$ and its semi-classical limit was shown to be the coordinate algebra for a Poisson homogeneous space of $G^*$. In this paper, we establish (relative) braid group symmetries and PBW bases on this integral form of $\mathrm{U}^\imath$. By taking the semi-classical limit, we obtain braid group symmetries and polynomial generators on the associated Poisson algebra. These symmetries further allow us to describe the Poisson brackets explicitly. Examples of such Poisson structures include Dubrovin-Ugaglia Poisson brackets.