  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
$m>2$.
