Poisson homotopy algebra: An idiosyncratic survey of homotopy algebraic topics related to Alan’s interests

Abstract

Homotopy algebra is playing an increasing role in mathematical physics. Especially in the Hamiltonian and Lagrangian settings, it is intimately related to some of Alan’s interests, e.g., Courant and Lie algebroids. There is a comparatively long history of such structure in cohomological physics in terms of equations that hold mod exact terms (typically, divergences) or only “on shell,’’meaning modulo the Euler-Lagrange equations of “motion’’; more recently, higher homotopies have come into prominence. Higher homotopies were developed first within algebraic topology and may not yet be commonly available tools for symplectic geometers and mathematical physicists. This is an expanded version of my talk at Alanfest, planned as a gentle introduction to the basic point of view with a variety of applications to substantiate its relevance. Most technical details are supplied by references to the original work or to [MSS02].