Quadratic algebras related to elliptic curves

Details Quadratic algebras related to elliptic curves Quadratic algebras related to elliptic curves

2007-10-04

arxiv.org/abs/0710.1072v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

21 pages

Scientific paper

We construct quadratic finite-dimensional Poisson algebras and their quantum versions related to rank N and degree one vector bundles over elliptic curves with n marked points. The algebras are parameterized by the moduli of curves. For N=2 and n=1 they coincide with the Sklyanin algebras. We prove that the Poisson structure is compatible with the Lie-Poisson structure on the direct sum of n copies of sl(N). The derivation is based on the Poisson reduction from the canonical brackets on the affine space over the cotangent bundle to the groups of automorphisms of vector bundles.

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