Cyclic homology of braided Hopf crossed products

TL;DR

This paper introduces a simplified mixed complex for computing various homologies of braided Hopf crossed products, extending to relative cyclic homology and providing more accessible tools for algebraic analysis.

Contribution

It develops a simpler mixed complex for Hochschild and cyclic homology of crossed products, including relative cases and cleft braided Hopf structures, improving computational approaches.

Findings

Simpler mixed complex for Hochschild and cyclic homology

Extension to relative cyclic homology with subalgebra K

Application to cleft braided Hopf crossed products

Abstract

Let k be a field, A a unitary associative k-algebra and V a k-vector space endowed with a distinguished element 1_V. We obtain a mixed complex, simpler that the canonical one, that gives the Hochschild, cyclic, negative and periodic homology of a crossed product E:=A#_f V, in the sense of Brzezinski. We actually work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra K of A that satisfies suitable hypothesis and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homology of E relative to K. Then, when E is a cleft braided Hopf crossed product, we obtain a simpler mixed complex, that also gives the Hochschild, cyclic, negative and periodic homology of E.