Shifted Poisson geometry and meromorphic matrix algebras over an elliptic curve

TL;DR

This paper classifies symplectic leaves of meromorphic endomorphisms of stable vector bundles on elliptic curves using shifted Poisson structures, linking algebraic geometry with Poisson-Lie groups.

Contribution

It provides a classification of symplectic leaves in meromorphic matrix algebras over elliptic curves via shifted Poisson geometry, connecting to ind Poisson-Lie groups.

Findings

Classification of symplectic leaves on the regular part of meromorphic matrix algebras.

Decomposition of the product of leaves under multiplication.

Poisson structures are invariant under derived category autoequivalences.

Abstract

In this paper we classify symplectic leaves of the regular part of the projectivization of the space of meromorphic endomorphisms of a stable vector bundle on an elliptic curve, using the study of shifted Poisson structures on the moduli of complexes from our previous work \cite{HP17}. This Poisson ind-scheme is closely related to the ind Poisson-Lie group associated to Belavin's elliptic $r$-matrix, studied by Sklyanin, Cherednik and Reyman and Semenov-Tian-Shansky. Our result leads to a classification of symplectic leaves on the regular part of meromorphic matrix algebras over an elliptic curve, which can be viewed as the Lie algebra of the above-mentioned ind Poisson-Lie group. We also describe the decomposition of the product of leaves under the multiplication morphism and show the invariance of Poisson structures under autoequivalences of the derived category of coherent sheaves on an elliptic curve.