Poisson algebras of block-upper-triangular bilinear forms and braid group action

TL;DR

This paper introduces a quadratic Poisson algebra on bilinear forms with block-upper-triangular matrices, classifies its central elements, and constructs a braid group action, with implications for quantisation and quantum groups.

Contribution

It develops a new Poisson algebra structure on block-upper-triangular bilinear forms, classifies central elements, and constructs a braid group action, extending to quantum groups for specific cases.

Findings

Classified all central elements of the Poisson algebra.

Constructed the groupoid of morphisms preserving the Poisson structure.

Established quantum braid group actions for specific cases.

Abstract

In this paper we study a quadratic Poisson algebra structure on the space of bilinear forms on $C^{N}$ with the property that for any $n,m\in N$ such that $n m =N$, the restriction of the Poisson algebra to the space of bilinear forms with block-upper-triangular matrix composed from blocks of size $m\times m$ is Poisson. We classify all central elements and characterise the Lie algebroid structure compatible with the Poisson algebra. We integrate this algebroid obtaining the corresponding groupoid of morphisms of block-upper-triangular bilinear forms. The groupoid elements automatically preserve the Poisson algebra. We then obtain the braid group action on the Poisson algebra as elementary generators within the groupoid. We discuss the affinisation and quantisation of this Poisson algebra, showing that in the case $m=1$ the quantum affine algebra is the twisted $q$-Yangian for ${o}_n$ and for $m=2$ is the twisted $q$-Yangian for ${sp}_{2n}$. We describe the quantum braid group action in these two examples and conjecture the form of this action for any $m>2$.