Graded Poisson Algebras
Alberto S. Cattaneo, Domenico Fiorenza, Riccardo Longoni
TL;DR
The article surveys graded Poisson algebras and their BV counterparts, outlining degree conventions, the structure of $n$-Poisson and BV algebras, and a broad array of concrete realizations. It then connects these algebraic structures to geometry and topology via Lie algebroids, Lie–Rinehart algebras, Hochschild cohomology, graded symplectic manifolds, and shifted cotangent bundles, as well as to derived geometry through shifted Poisson structures on derived stacks. The text further discusses the role of these structures in quantization, including BRST and BV formalisms, and highlights field-theoretic realizations through AKSZ and higher homotopy generalizations ($P_\infty$). Overall, it presents graded Poisson theories as a unifying language for deformation, quantization, and topological field theories in both classical and derived settings.
Abstract
This note is an expanded and updated version of our entry with the same title for the 2006 Encyclopedia of Mathematical Physics. We give a brief overview of graded Poisson algebras, their main properties and their main applications, in the contexts of super differentiable and of derived algebraic geometry.