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Cohomology and Deformations of left-symmetric Rinehart Algebras

A. Ben Hassine, T. Chtioui, M. Elhamdadi, S. Mabrouk

Abstract

We introduce a notion of left-symmetric Rinehart algebras, which is a generalization of a left-symmetric algebras. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie-Rinehart algebra. We construct left-symmetric Rinehart algebra from O-operators on Lie-Rinehart algebra. We extensively investigate representations of a left-symmetric Rinehart algebras. Moreover, we study deformations of left-symmetric Rinehart algebras, which is controlled by the second cohomology class in the deformation cohomology. We also give the relationships between O-operators and Nijenhuis operators on left-symmetric Rinehart algebras.

Cohomology and Deformations of left-symmetric Rinehart Algebras

Abstract

We introduce a notion of left-symmetric Rinehart algebras, which is a generalization of a left-symmetric algebras. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie-Rinehart algebra. We construct left-symmetric Rinehart algebra from O-operators on Lie-Rinehart algebra. We extensively investigate representations of a left-symmetric Rinehart algebras. Moreover, we study deformations of left-symmetric Rinehart algebras, which is controlled by the second cohomology class in the deformation cohomology. We also give the relationships between O-operators and Nijenhuis operators on left-symmetric Rinehart algebras.

Paper Structure

This paper contains 12 sections, 31 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some basic properties of a left-symmetric Rinehart algebras
  4. Representations of left-symmetric Rinehart algebras
  5. The Matsushima-Nijenhuis bracket for left-symmetric Rinehart algebras
  6. Deformation of left-symmetric Rinehart algebra
  7. Formal deformations
  8. Obstructions to the extension theory of deformations
  9. Trivial deformation
  10. O-operators and Nijenhuis operators
  11. Relationships between O-operators and Nijenhuis operators
  12. Compatible O-operators and Nijenhuis operators

Key Result

lemma 1

Let $(L,\cdot)$ be a left-symmetric algebra. The commutator $[x,y]=x\cdot y-y\cdot x$ defines a Lie algebra $L$, which is called the sub-adjacent Lie algebra of $L$. The algebra $L$ is also called a compatible left-symmetric algebra on the Lie algebra $L$. Furthermore, the map $ad^L:L\rightarrow \ma

Theorems & Definitions (69)

  • definition 1
  • lemma 1
  • definition 2
  • lemma 2: See Burde2
  • definition 3
  • definition 4
  • lemma 3
  • definition 5
  • theorem 1
  • proof
  • ...and 59 more