Cohomologies and linear deformations of pre-Jacobi-Jordan algebras

Abstract: The purpose of this paper is to introduce an algebraic cohomology theory of left pre-Jacobi-Jordan algebras. We use the cohomological approach to study linear deformations of these algebras. We also introduce the notion of Nijenhuis operators on pre-Jacobi-Jordan algebras and we show that the deformation generated by a Nijenhuis operator is trivial.

• The paper introduces an algebraic cohomology framework for left pre-Jacobi-Jordan algebras, linking cohomological structures with deformation theory.
• It demonstrates that linear deformations induced by 2-cocycles, especially via Nijenhuis operators, are inherently trivial under specified conditions.
• The study details low-degree cohomologies with concrete algebra examples, offering insights into derivations and invariant algebraic structures.

Cohomologies and Linear Deformations of Pre-Jacobi-Jordan Algebras

This paper introduces an algebraic cohomology framework specifically designed for left pre-Jacobi-Jordan algebras and leverages this framework to investigate linear deformations within these algebraic structures. Furthermore, the concept of Nijenhuis operators on pre-Jacobi-Jordan algebras is introduced, with the application showing that deformations generated by such operators are inherently trivial.

Pre-Jacobi-Jordan Algebras

Pre-Jacobi-Jordan algebras are a category of non-associative algebras characterized by their left (or right) skew-symmetric property of the anti-associator, setting them apart from associative or purely anti-associative algebras. For a left pre-Jacobi-Jordan algebra, the anti-associator $Aasso(x, y, z)$ satisfies the condition $Aasso(x, y, z) + Aasso(y, x, z) = 0$ for all elements $x, y, z$ in the algebra. This property allows them to serve as a broader class within which Jacobi-Jordan algebras, requiring stricter conditions, reside.

Cohomology Theory Construction

The paper constructs a cohomology theory for pre-Jacobi-Jordan algebras. The approach integrates cochains, coboundaries, and cocycles tensorially, introducing complex $A_n$ and $C_n$ spaces which handle operations like the zigzag cohomology distinguishes. Differential operators $d^n$ and $\delta^n$ are introduced to define complex chains and ensure that the cohomological identity $d^{n+1} \circ \delta^n = 0$ holds, ensuring consistency in the cohomology structure.

Low-Degree Cohomologies

A significant aspect of the constructed cohomology is the interpretative analysis of low-degree cohomology groups:

• 0th Cohomology Group ($\mathcal{H}^0$): Reflects elements that remain invariant under the algebraic operations, encapsulating the algebra's central structure.
• 1st Cohomology Group ($\mathcal{H}^1$): Corresponds to derivations, specifically focusing on the outer antiderivations of the algebra, separating trivial derivations.

The paper provides examples, demonstrating the calculation and dimensional properties of cohomology groups $\mathcal{H}^1$ using specific algebraic structures $\mathcal{A}_1$ and $\mathcal{A}_2$, highlighting the applicability of these theoretical constructs.

Linear Deformations and Compatibility

Linear deformations within pre-Jacobi-Jordan algebras are shown to be generated by 2-cocycles. Concretely, given an algebra $(A, \cdot)$, a linear map $\omega: A \times A \to A$ induces a deformation if $\omega$ adheres to specific cocycle conditions, maintaining the algebra's defining properties under perturbation.

Furthermore, the triviality of linear deformations is discussed in the context of equivalence and cohomology classes, with an emphasis on transformations maintaining the core pre-Jacobi-Jordan attributes.

Nijenhuis Operators and Their Role

Nijenhuis operators, as defined in the framework of pre-Jacobi-Jordan algebras, serve as significant tools. These operators $N$ obey the condition $N(x) \cdot N(y) = N(x \cdot_N y)$, where the $N$-modified operation integrates the linear map directly into the algebraic structure. This insight generalizes the concept of Rota-Baxter operators when aligned with specific regular weights.

Notably, for a Nijenhuis operator $N$, any deformation generated is trivial. This provides a potent method for identifying redundancy within algebra perturbations, ensuring that algebraic modifications are meaningful and avoid redundancy.

Conclusion

The paper successfully extends algebraic cohomology to pre-Jacobi-Jordan algebras and aligns it with deformation theory, introducing a practical and theoretical framework for understanding algebraic structures subject to linear modifications. The introduction of Nijenhuis operators offers useful insights into identifying trivial deformations, enriching the structural analysis and manipulation of pre-Jacobi-Jordan algebras.