Cohomology of 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.