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

In this paper we study a quadratic Poisson algebra structure on the space of bilinear forms on $\mathbb{C}^n$ with the property that for any $n, m \in\mathbb{N} $ such that $nm = N$, the restriction of the Poisson algebra to the space of bilinear forms with a 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 $\mathfrak{o}_n$ and for $m = 2$ is the twisted $q$-Yangian for $(\mathfrak{sp}_{2n})$. We describe the quantum braid group action in these two examples and conjecture the form of this action for any $m > 2$. Finally, we give an $R$-matrix interpretation of our results and discuss the relation with Poisson–Lie groups.