Cyclic homologies of crossed modules of algebras

Abstract

Abstract. The Hochschild and (cotriple) cyclic homologies of crossed modules of (notnecessarily-unital) associative algebras are investigated. Wodzicki's excision theorem is extended for inclusion crossed modules in the category of crossed modules of algebras. The cyclic and cotriple cyclic homologies of crossed modules are compared in terms of long exact homology sequence, generalising the relative cyclic homology exact sequence.

Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy

“…This is a working definition, completely bypassing its (unknown, and possibly not-existing) relation with simplicial algebras with Moore complex of length one. We note that crossed modules of associative algebras are defined in [Arv04,CIKL14,DIKL12], and our definition is more restrictive. Notice that, in the case of commutative algebras (where left and right actions are the same thing) our definition of crossed modules coincides with the one in [AP96].…”

Chevalley-Eilenberg homology of crossed modules of Lie algebras in lower dimensions

“…Denote by α the first homomorphism in the sequence (6). By Proposition 3.1 we have an epimorphism τ :…”

“…In particular, the existence of a five term exact sequence connecting the low-dimensional ChevalleyEilenberg homologies of crossed modules and their cokernel Lie algebras is proved. The analogous result for the cyclic and Hochschild homologies of crossed modules of associative algebras is given in [6], and its proof is based on using of the EilenbergZilber theorem and the Künneth formula, which are not valid in the case mentioned above. Moreover, a relationship between the homology of a crossed module of Lie algebras and the Chevalley-Eilenberg homology of Lie algebras with coefficients is established in the paper.…”

On Hopf 2-algebras

“…Indeed, the obstacle to define a crossed module of associative algebras corresponding to a crossed module of Lie algebras by taking the functor U term by term, is that the action is by derivations of the associative product (see section 4). Thus the term by term U -image of a crossed module of Lie algebras does not satisfy the condition of compatibility for a crossed module of associative algebras ρ : R → A, which reads for all a ∈ A and all r, r ′ ∈ R: a(rr ′ ) = (ar)r ′ , (rr ′ )a = r(r ′ a), see [DIK08]: the naive belief that a crossed module of Hopf algebras is in particular a crossed module of associative algebras which is simultaneously a crossed module of coassociative coalgebras is wrong.…”