Note on Cohomology of Color Hopf and Lie Algebras
نویسندگان
چکیده
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 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.
منابع مشابه
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 ...
متن کاملar X iv : m at h / 02 12 12 4 v 1 [ m at h . R A ] 9 D ec 2 00 2 COHOMOLOGY OF ABELIAN MATCHED PAIRS AND THE KAC SEQUENCE
The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general abelian matched pair of Hopf algebras, generalizing those of Kac and Masuoka for matched pairs of finite grou...
متن کاملA note on the new basis in the mod 2 Steenrod algebra
The Mod $2$ Steenrod algebra is a Hopf algebra that consists of the primary cohomology operations, denoted by $Sq^n$, between the cohomology groups with $mathbb{Z}_2$ coefficients of any topological space. Regarding to its vector space structure over $mathbb{Z}_2$, it has many base systems and some of the base systems can also be restricted to its sub algebras. On the contrary, in ...
متن کاملThe van Est spectral sequence for Hopf algebras
Various aspects of the de Rham cohomology of Hopf algebras are discussed. In particular, it is shown that the de Rham cohomology of an algebra with the differentiable coaction of a cosemisimple Hopf algebra with trivial 0-th cohomology group, reduces to the de Rham cohomology of (co)invariant forms. Spectral sequences are discussed and the van Est spectral sequence for Hopf algebras is introduc...
متن کاملOn Cohomology of Hopf Algebroids
Inspired by [3] we introduce the concept of extended Hopf algebra and consider their cyclic cohomology in the spirit of Connes-Moscovici [3, 4, 5]. Extended Hopf algebras are closely related, but different from, Hopf algebroids. Their definition is motivated by attempting to define cyclic cohomology of Hopf algebroids in general. Many of Hopf algebra like structures, including the Connes-Moscov...
متن کامل