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Hochschild cohomology of Lie-Rinehart algebras

Bjarne Kosmeijer, Hessel Posthuma

Abstract

We compute the Hochschild cohomology of universal enveloping algebras of Lie-Rinehart algebras in terms of the Poisson cohomology of the associated graded quotient algebras. Central in our approach are two cochain complexes of "nonlinear Chevalley-Eilenberg" cochains whose origins lie in Lie-Rinehart modules "up to homotopy", one on the Hochschild cochains of the base algebra, another related to the adjoint representation. The Poincare-Birkhoff-Witt isomorphism is then extended to a certain intertwiner between such modules. Finally, exploiting the twisted Calabi-Yau structure, we obtain results for the dual Hochschild and cyclic homology.

Hochschild cohomology of Lie-Rinehart algebras

Abstract

We compute the Hochschild cohomology of universal enveloping algebras of Lie-Rinehart algebras in terms of the Poisson cohomology of the associated graded quotient algebras. Central in our approach are two cochain complexes of "nonlinear Chevalley-Eilenberg" cochains whose origins lie in Lie-Rinehart modules "up to homotopy", one on the Hochschild cochains of the base algebra, another related to the adjoint representation. The Poincare-Birkhoff-Witt isomorphism is then extended to a certain intertwiner between such modules. Finally, exploiting the twisted Calabi-Yau structure, we obtain results for the dual Hochschild and cyclic homology.
Paper Structure (24 sections, 14 theorems, 157 equations)

This paper contains 24 sections, 14 theorems, 157 equations.

Table of Contents

  1. Preliminaries
  2. Lie--Rinehart algebras
  3. The adjoint representation
  4. Hochschild cohomology
  5. Quasi-Modules
  6. Example: the adjoint representation
  7. Example: Hochschild cochains
  8. Variations on PBW
  9. The Poincaré--Birkhoff--Witt isomorphism for Lie--Rinehart algebras
  10. Extension to the adjoint complex
  11. Extension to the nonlinear Chevalley--Eilenberg complex
  12. Hochschild cohomology
  13. A spectral sequence
  14. Reduction to the nonlinear Chevalley--Eilenberg complex
  15. Relation with the adjoint representation
  16. ...and 9 more sections

Key Result

Lemma 1.6

Let $L$ be projective as a module over $R$. There exists a short exact sequence of ${\rm Sym}_R(L)$-modules and choosing splitting is the same as choosing a connection $\nabla$ on $L$.

Theorems & Definitions (40)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Example 1.4: Lie algebras
  • Example 1.5: Differential operators on affine varieties
  • Lemma 1.6
  • proof
  • Remark 1.7
  • Definition 2.1
  • Remark 2.2
  • ...and 30 more