Deformations of left-symmetric color algebras
Yin Chen, Runxuan Zhang
TL;DR
This work develops a deformation theory for finite-dimensional left-symmetric color algebras and ties deformations to the color cohomology spaces, notably $H^2_0(A,A)$ and $H^3_0(A,A)$. By formulating one-parameter deformations on the scalars extension $A_K$ and proving that $f_1$ must be a 2-cocycle, the authors connect deformations to degree-zero cohomology and establish equivalence criteria via coboundaries. A central result shows that vanishing of $H^3_0(A_K,A_K)$ eliminates obstructions to extending infinitesimal deformations, while the study of Nijenhuis and Rota-Baxter operators clarifies when deformations are trivial and how such operators structure the theory. The paper also provides concrete computations for 2-dimensional color algebras, offering explicit operator varieties and enriching the toolbox for constructing new left-symmetric color algebras and understanding their cohomology.
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.