Symmetric Homology is Representation Homology

TL;DR

This paper demonstrates that symmetric homology for algebras over characteristic zero fields is equivalent to representation homology, enabling explicit computations and resolving key conjectures in the field.

Contribution

It establishes the equivalence between symmetric homology and representation homology for characteristic zero algebras and applies this to compute symmetric homology explicitly.

Findings

Symmetric homology is equivalent to representation homology in characteristic zero.

Explicit calculations of symmetric homology for polynomial and universal enveloping algebras.

Proof of conjectures relating symmetric homology to topological interpretations.

Abstract

Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz \cite{F} and was further developed in the work of S. Ault \cite{Au1, Au2}. In this paper, we show that, for algebras defined over a field of characteristic $0$, the symmetric homology theory is naturally equivalent to the (one-dimensional) representation homology theory introduced by the authors (jointly with G. Khachatryan) in \cite{BKR}. Using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz, including the main conjecture of \cite{AF07} on topological interpretation of symmetric homology of polynomial algebras.