[Paper Review] Cohomology and Deformations of left-symmetric Rinehart Algebras

Abdelkader Ben Hassine, Taoufik Chtioui+2 more|arXiv (Cornell University)|Sep 30, 2020

Advanced Topics in Algebra32 references4 citations

TL;DR

This paper introduces left-symmetric Rinehart algebras as a generalization of left-symmetric algebras and Lie-Rinehart algebras, establishing their cohomology theory and deformation theory. It constructs a graded Lie algebra whose Maurer-Cartan elements classify left-symmetric Rinehart algebras and shows that deformations are controlled by the second cohomology class. The work further links $ϴ$-operators and Nijenhuis operators, proving that compatible $ϴ$-operators arise from Nijenhuis operators via composition with invertible maps.

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.

Motivation & Objective

• To generalize left-symmetric algebras and Lie-Rinehart algebras by introducing a new algebraic structure: left-symmetric Rinehart algebras.
• To develop a cohomology theory for left-symmetric Rinehart algebras and study their representations.
• To construct a graded Lie algebra whose Maurer-Cartan elements characterize left-symmetric Rinehart algebras.
• To establish a deformation theory for left-symmetric Rinehart algebras controlled by second cohomology classes.
• To explore the relationship between $ϴ$-operators and Nijenhuis operators in this new algebraic framework.

Proposed method

• Define left-symmetric Rinehart algebras as a generalization of left-symmetric algebras with a compatible action of an algebra $A$.
• Construct the sub-adjacent Lie-Rinehart algebra using the commutator of the left multiplication.
• Define representations of left-symmetric Rinehart algebras and develop their cohomology via a coboundary operator.
• Construct a graded Lie algebra on the space of multi-derivations whose Maurer-Cartan elements correspond to left-symmetric Rinehart algebra structures.
• Define formal deformations and show that the second cohomology class controls obstructions to extending deformations.
• Introduce Nijenhuis operators and establish their role in generating trivial deformations, linking them to compatible $ϴ$-operators via composition with invertible maps.

Experimental results

Research questions

• RQ1 How can left-symmetric algebras be generalized to include an action of an associative algebra $A$, leading to a new class of algebras?
• RQ2 What is the cohomology theory for left-symmetric Rinehart algebras, and how does it relate to their representations?
• RQ3 How can a graded Lie algebra be constructed such that its Maurer-Cartan elements classify left-symmetric Rinehart algebra structures?
• RQ4 What is the role of the second cohomology class in controlling deformations of left-symmetric Rinehart algebras?
• RQ5 How are $ϴ$-operators and Nijenhuis operators related in the context of left-symmetric Rinehart algebras?

Key findings

• The left multiplication in a left-symmetric Rinehart algebra gives rise to a representation of the sub-adjacent Lie-Rinehart algebra.
• Left-symmetric Rinehart algebras can be constructed from $ϴ$-operators on Lie-Rinehart algebras via a specific algebraic construction.
• A graded Lie algebra is constructed on the space of multi-derivations, and its Maurer-Cartan elements precisely characterize left-symmetric Rinehart algebra structures.
• Deformations of left-symmetric Rinehart algebras are controlled by the second cohomology class in the deformation cohomology complex.
• A Nijenhuis operator $N$ on a left-symmetric Rinehart algebra induces a trivial deformation, and if $T_1$ and $T_2$ are compatible $ϴ$-operators with $T_2$ invertible, then $N = T_1 mid T_2^{-1}$ is a Nijenhuis operator.
• If $T$ is an $ϴ$-operator and $N$ is an invertible Nijenhuis operator such that $NT$ is also an $ϴ$-operator, then $T$ and $NT$ are compatible, establishing a direct link between the two concepts.