Twisted tensor products: Alexander-Whitney and Eilenberg-Zilber maps

TL;DR

This paper extends classical Alexander-Whitney and Eilenberg-Zilber maps to twisted tensor products, enabling transfer of homological information and aiding deformation theory of complex algebraic structures.

Contribution

It introduces chain maps for twisted tensor products, broadening the applicability of classical maps to include skew group algebras and Hopf algebra smash products.

Findings

Extended chain maps for twisted tensor products.

Facilitated transfer of homological data between resolutions.

Accelerated deformation theory analysis of algebraic structures.

Abstract

Alexander-Whitney and Eilenberg-Zilber maps traditionally convert between the tensor product of standard resolutions and the standard resolution of a tensor product of algebras. We examine Alexander-Whitney and Eilenberg-Zilber maps for twisted tensor products, which include skew group algebras, smash products of Hopf algebras, Ore extensions, and universal enveloping algebras. These maps convert between the twist of standard resolutions and the standard resolution of a twist. We extend these to chain maps to and from twists of other resolutions. This allows one to transfer homological information between various resolutions of algebras and to expedite results on the deformation theory of twisted tensor product algebras.