Lie brackets on Hopf algebra cohomology

Note on Cohomology of Color Hopf and Lie Algebras

Let A be a (G, χ)-Hopf algebra with bijection 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)), where K is viewed as the trivial graded A-module via the counit of A, M is the adjoint A-module associated to the graded A-bimodule M and HHgr denotes the G-graded Hochschild cohomology. As an application, we deduce that the cohom...

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 HH∗gr(A,M)∼= Ext∗A-gr ( K,ad (M) ) , where K is viewed as the trivial graded A-module via the counit of A, adM 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 ...

Dirac Operators and Lie Algebra Cohomology

Dirac cohomology is a new tool to study unitary and admissible representations of semisimple Lie groups. It was introduced by Vogan and further studied by Kostant and ourselves [V2], [HP1], [K4]. The aim of this paper is to study the Dirac cohomology for the Kostant cubic Dirac operator and its relation to Lie algebra cohomology. We show that the Dirac cohomology coincides with the correspondin...

The Lie algebra cohomology of jets

Let g be a finite-dimensional complex reductive semi simple Lie algebra. We present a new calculation of the continuous cohomology of the Lie algebra zg[[z]]. In particular, we shall give an explicit formula for the Laplacian on the Lie algebra cochains, from which we can deduce that the cohomology in each dimension is a finite-dimensional representation of g which contains any irreducible repr...

Lie Algebra Cohomology and the Representations Ofsemisimple Lie Groups