Shifted Poisson geometry and Sklyanin algebra

Abstract

The Feigin-Odesskii Poisson structures are semiclassical limits of the FeiginOdesskii elliptic algebras. These noncommutative algebras are vast generalization of Sklyanin algebras. It is an open problem that how to classify the symplectic leaves of these Poisson structures. With Sasha Polischuk, we construct a (1-d) shifted Poisson structure on the moduli stack of bounded complexes of vector bundles on projective Calabi-Yau d-folds. When d=1, our Poisson structure descends to Feigin-Odesskii’s Poisson structure on certain components of the moduli stack. The derived geometry of the moduli stack leads to a geometric description of the symplectic leaves. Using algebraic geometry, we give an explicit classification of symplectic leaves for those Poisson structures of “endomorphism” type. This is a joint work with Sasha Polishchuk.