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In this paper we study a quadratic Poisson algebra structure on the space ofbilinear 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 formswith block-upper-triangular matrix composed from blocks of size $m\times m$ isPoisson. We classify all central elements and characterise the Lie algebroidstructure compatible with the Poisson algebra. We integrate this algebroidobtaining the corresponding groupoid of morphisms of block-upper-triangularbilinear forms. The groupoid elements automatically preserve the Poissonalgebra. We then obtain the braid group action on the Poisson algebra aselementary generators within the groupoid. We discuss the affinisation andquantisation of this Poisson algebra, showing that in the case $m=1$ thequantum affine algebra is the twisted $q$-Yangian for ${o}_n$ and for $m=2$ isthe twisted $q$-Yangian for ${sp}_{2n}$. We describe the quantum braid groupaction in these two examples and conjecture the form of this action for any$m>2$.

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