Block triangular bilinear forms and braid group action

In this paper we review some recent results about a quadratic Poisson algebra structure on the space of bilinear forms on C-N with the property that for any n, m is an element of N such that nm = N, the restriction of the Poisson algebra to the space of bilinear forms with block-upper-triangular matrix composed from blocks of size m x m is Poisson. We give some examples and classify all central elements. We also obtain the braid group action on the Poisson algebra. Finally, we discuss quantisation. All proofs can be found in [1, 2, 3].