Cohomology of left-symmetric color algebras

TL;DR

This paper introduces a new cohomology theory for finite-dimensional left-symmetric color algebras, linking it to Lie color cohomology and exploring low-dimensional algebra varieties.

Contribution

It develops a novel cohomology framework for left-symmetric color algebras and connects it to existing Lie color cohomology, extending previous results.

Findings

Cohomology of left-symmetric color algebras can be computed via Lie color algebra cohomology.

Established a connection between left-symmetric and Lie color cohomology.

Explored the structure of 2D and 3D left-symmetric color algebras.

Abstract

We develop a new cohomology theory for finite-dimensional left-symmetric color algebras and their finite-dimensional bimodules, establishing a connection between Lie color cohomology and left-symmetric color cohomology. We prove that the cohomology of a left-symmetric color algebra $A$ with coefficients in a bimodule $V$ can be computed by a lower degree cohomology of the corresponding Lie color algebra with coefficients in Hom$(A,V)$, generalizing a result of Dzhumadil'daev in right-symmetric cohomology. We also explore the varieties of two-dimensional and three-dimensional left-symmetric color algebras.