Deformations of left-symmetric color algebras

TL;DR

This paper develops a deformation theory for finite-dimensional left-symmetric color algebras, introducing new methods to construct algebraic structures and analyze their cohomology, equivalence, and extendability.

Contribution

It presents a novel deformation framework for left-symmetric color algebras, including criteria for extendability and the role of Nijenhuis and Rota-Baxter operators.

Findings

Infinitesimal deformations are nontrivially extendable when third cohomology is trivial.

Provides a cohomology interpretation for left-symmetric color algebras.

Analyzes the impact of Nijenhuis and Rota-Baxter operators on deformation classes.

Abstract

We develop a deformation theory for finite-dimensional left-symmetric color algebras, which can be used to construct new algebraic structures and interpret left-symmetric color cohomology spaces of lower degrees. We explore equivalence classes and extendability of deformations for a fixed left-symmetric color algebra, demonstrating that each infinitesimal deformation is nontrivially extendable if the third cohomology subspace of degree zero is trivial. We also study Nijenhuis operators and Rota-Baxter operators on a left-symmetric color algebra, providing a better understanding of the equivalence class of the trivial infinitesimal deformation.