Prop profile of bi-Hamiltonian structures

TL;DR

This paper introduces a new algebraic framework linking bi-Hamiltonian structures with minimal resolutions of props, expanding the algebraic understanding of compatible Poisson structures in differential geometry.

Contribution

It defines a specific prop whose minimal resolution corresponds precisely to bi-Hamiltonian structures, bridging differential geometry and homological algebra.

Findings

Establishes a one-to-one correspondence between representations of the prop's minimal resolution and bi-Hamiltonian structures.

Provides a new algebraic tool for studying compatible Poisson structures.

Connects differential geometric concepts with homological algebra through prop theory.

Abstract

Recently S.A. Merkulov established a link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of minimal resolutions of props. In particular he described the prop profile of Poisson geometry. In this paper we define a prop such that representations of its minimal resolution in a vector space V are in a one-to-one correspondence with bi-Hamiltonian structures, i.e. pairs of compatible Poisson structures, on the formal manifold associated to V.