Poisson Structures and Birational Morphisms Associated with Bundles on Elliptic Curves

Let X be a complex elliptic curve. In this paper we define a natural Poisson structure on the moduli spaces of stable triples (E1, E2,Φ) where E1, E2 are algebraic vector bundles on X of fixed ranks (r1, r2) and degrees (d1, d2), Φ : E2 → E1 is a homomorphism. Such triples are considered up to an isomorphism and the stability condition depends on a real parameter τ . These moduli spaces were introduced by S. Bradlow and O. Garcia-Prada [5]. Our Poisson structure induces a Poisson structure on similar moduli spaces with fixed determinants of E1 and E2. For E2 = O and some values of parameters (r1, r2, d1, d2, τ) the latter moduli spaces are just the projective spaces. In particular, one of these moduli spaces is PExt(E,O), where E is a fixed stable bundle. The corresponding Poisson structures on PExt(E,O) were first defined and studied by B. Feigin and A. Odesskii [7]. Moreover, they constructed a flat family of quadratic algebras (Sklyanin algebras) Qd,r(x) where d = degE, r = rkE, parametrized by x ∈ X such that Qd,r(0) is the symmetric algebra in d variables and the quadratic Poisson bracket on the symmetric algebra associated with this deformation induces the above Poisson structure on P. The algebra Qd,r(x) is defined as the associative algebra over C with d generators ti, i ∈ Z/dZ and defining relations ∑