Functional realization of some elliptic Hamiltonian structures and bosonization of the corresponding quantum algebras

Let p be the parabolic subalgebra of some semisimple Lie algebra g and P the corresponding group. Let M(E , p) be the moduli space of P -bundles on the elliptic curve E . In [5] we define the Hamiltonian structure on the manifold M(E , p). There is a natural problem: to quantize the coordinate ring of each connected component of M(E , p). We denote by Qn(E , τ) the corresponding quantum algebras in the case g = sl2. Here τ ∈ E is a parameter of quantization, n ∈ N is a number of the connected component of M(E , p). This component is isomorphic to P in this case. So its coordinate ring is isomorphic to the polynomial ring in n variables and the algebra Qn(E , τ) is a graded deformation of this polynomial ring. We denote the corresponding Poisson algebra by qn(E). More generally, we denote by Qn,k(E , τ) the corresponding quantum algebras in the case g = slk+1 and p is a parabolic subalgebra for the flag 0 ⊂ V ⊂ C , dimV = 1. Here τ ∈ E is a parameter of quantization, n ∈ N is a number of the connected component of M(E , p). If n and k have no common divisors, then this component is isomorphic to P. So its coordinate ring is isomorphic to the polynomial ring in n variables and the algebra Qn,k(E , τ) is a graded deformation of this polynomial ring. We denote the corresponding Poisson algebra by qn,k(E). We have Qn(E , τ) = Qn,1(E , τ), qn(E) = qn,1(E). In the papers [1,2] we constructed the family of associative algebras Qn(E , τ). The algebra Qn(E , τ) is Z-graded and depends on 2 continuous parameters: an elliptic curve E = C Γ and a point τ ∈ E . We have Qn(E , τ) = C⊕ F1 ⊕ F2 ⊕ . . . and Fα ∗ Fβ ⊂ Fα+β . The Hilbert function is 1 + ∑