Cohomology of left-symmetric color algebras

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.

💡 Research Summary

This paper develops a cohomology theory for finite‑dimensional left‑symmetric color algebras (LSCAs) and their finite‑dimensional bimodules, establishing a precise link with the cohomology of the associated Lie color algebras. After recalling the necessary background on abelian groups, skew‑symmetric bicharacters ϵ, and the definition of a left‑symmetric color algebra (an algebra A graded by a group G with a product satisfying the ϵ‑left‑symmetric identity), the authors introduce bimodules and several constructions: the tensor product of bimodules (Proposition 2.14) and a right‑trivial bimodule structure on the first cochain space (Proposition 2.16).

The central innovation lies in the definition of cochain spaces Cⁿ(A,V) as the G‑graded vector space of homogeneous linear maps from A⊗∧ⁿ_ϵ A to V, where ∧ⁿ_ϵ A denotes the ϵ‑exterior power. This choice mirrors the cochain space used in Dzhumadil’daev’s right‑symmetric cohomology and is more natural for the graded setting than the multilinear maps used in earlier work (e.g., NBN09). The coboundary operator δⁿ is defined by a formula that incorporates the ϵ‑signs dictated by the left‑symmetric identity. Low‑degree cohomology groups receive concrete interpretations: H⁰(A,V) consists of invariants, while H¹(A,V) classifies outer ϵ‑derivations.

The main theorem (Theorem 1.1) proves that for every n≥1 there is a natural isomorphism of G‑graded vector spaces \