On the Symmetric Homology of Algebras

TL;DR

This paper develops the foundations of symmetric homology for algebras, relating it to stable homotopy theory, and provides computational tools and results for finite-dimensional algebras.

Contribution

It introduces a new framework for symmetric homology using crossed simplicial groups and constructs chain complexes and spectral sequences for computation.

Findings

Constructed two chain complexes for symmetric homology

Related symmetric homology of group rings to stable homotopy theory

Provided explicit partial resolutions for finite-dimensional algebras

Abstract

Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by symmetric groups. The foundations for the theory of symmetric homology of algebras are developed in the context of crossed simplicial groups using derived functors and the symmetric bar construction of Fiedorowicz. The symmetric homology of group rings is related to stable homotopy theory. Two chain complexes are constructed that compute symmetric homology, as well as two spectral sequences. In the setup of the second spectral sequence, a complex isomorphic to the suspension of the cycle-free chessboard complex of Vrecica and Zivaljevic appears. Homology operations are defined on the symmetric homology groups over Z/p, p a prime. Finally, an explicit partial resolution is presented, permitting the computation of the zeroth and first symmetric homology groups of finite-dimensional algebras.