Note on the cohomology of color Hopf and Lie algebras

Abstract

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 HH gr ∗ ( A , M ) ≅ Ext A -gr ∗ ( K , ( M ) ad ) , where K is viewed as the trivial graded A-module via the counit of A, M ad is the adjoint A-module associated to the graded A-bimodule M and HH gr ∗ 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.