Quadratic algebras related to elliptic curves

We construct quadratic finite-dimensional Poisson algebras corresponding to a rank-$N$ degree-one vector bundle over an elliptic curve with $n$ marked points and also construct the quantum version of the algebras. The algebras are parameterized by the moduli of curves. For $N=2$ and $n=1$, they coincide with Sklyanin algebras. We prove that the Poisson structure is compatible with the Lie–Poisson structure defined on the direct sum of $n$ copies of $sl(N)$. The origin of the algebras is related to the Poisson reduction of canonical brackets on an affine space over the bundle cotangent to automorphism groups of vector bundles.