Note on The Cohomology of Color Hopf and Lie Algebras

TL;DR

This paper establishes a fundamental isomorphism between graded Hochschild cohomology of certain Hopf algebras and Ext groups, and applies this to relate the cohomology of color Lie algebras to their universal enveloping algebras.

Contribution

It proves a new isomorphism linking graded Hochschild cohomology with Ext groups for $(G, ext{chi})$-Hopf algebras and applies it to color Lie algebras, solving an open problem.

Findings

Established an isomorphism between HH^*_{gr}(A, M) and Ext^*_{A-gr}( ext{K}, ^{ad} M)

Showed the cohomology of color Lie algebra L is isomorphic to the graded Hochschild cohomology of U(L)

Solved a question posed by M. Scheunert regarding cohomology of color Lie algebras.

Abstract

Let $A$ be a $(G, \chi)$-Hopf algebra with bijection antipode and let $M$ be a $G$-graded $A$-bimodule. We prove that there exists an isomorphism \mathrm{HH}^*_{\rm gr}(A, M)\cong{\rm Ext}^*_{A{-}{\rm gr}} (\K, {^{ad}(M)}), where $\K$ is viewed as the trivial graded $A$-module via the counit of $A$, $^{ad} M$ is the adjoint $A$-module associated to the graded $A$-bimodule $M$ and $\mathrm{HH}_{\rm gr}$ denotes the $G$-graded Hochschild cohomology. As an application, we deduce that the cohomology of color Lie algebra $L$ is isomorphic to the graded Hochschild cohomology of the universal enveloping algebra $U(L)$, solving a question of M. Scheunert.