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Classical and variational Poisson cohomology

Bojko Bakalov, Alberto De Sole, Reimundo Heluani, Victor G. Kac, Veronica Vignoli

Abstract

We prove that, for a Poisson vertex algebra V, the canonical injective homomorphism of the variational cohomology of V to its classical cohomology is an isomorphism, provided that V, viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables.

Classical and variational Poisson cohomology

Abstract

We prove that, for a Poisson vertex algebra V, the canonical injective homomorphism of the variational cohomology of V to its classical cohomology is an isomorphism, provided that V, viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables.

Paper Structure

This paper contains 30 sections, 22 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Variational PVA cohomology
  3. Poisson vertex algebras
  4. Variational PVA complex
  5. Preliminaries on the symmetric group and on graphs
  6. Shuffles
  7. Monotone permutations
  8. Graphs
  9. Graphs of type $\ul k$ and proper $\ul k$-lines
  10. Cycle relations on graphs
  11. Harrison relations
  12. Notation for subgraphs and collapsed graphs
  13. Classical PVA cohomology
  14. Space of classical cochains
  15. Differential
  16. ...and 15 more sections

Key Result

Theorem 1.1

Provided that, as a differential algebra, the PVA $\mathcal{V}$ is a finitely-generated algebra of differential polynomials, the map $\varphi^*$ is an isomorphism.

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Theorem 3.5: BDSHK19
  • ...and 40 more