Note on the cohomology of color Hopf and Lie algebras

Let A be a (G,χ)-Hopf algebra with bijective antipode and let M be a G-graded A-bimodule. We prove that there exists an isomorphism where is viewed as the trivial graded A-module via the counit of A, is the adjoint A-module associated to the graded A-bimodule M and denotes the G-graded Hochschild cohomology. As an application, we deduce that the graded cohomology of color Lie algebra L is isomorphic to the graded Hochschild cohomology of its universal enveloping algebra U(L), solving a question of M. Scheunert.