Symmetric Homology of Algebras

Abstract

The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring $A=k[\Gamma ]$, the symmetric homology is related to stable homotopy theory via $H{S}_{\ast }(k[\Gamma ])\cong H_{\ast }(\Omega {\Omega }^{\infty }{S}^{\infty }(B\Gamma );k)$. Two chain complexes that compute $H{S}_{\ast }(A)$ are constructed, both making use of a symmetric monoidal category $\Delta S_{+}$ containing $\Delta S$. Two spectral sequences are found that aid in computing symmetric homology. The second spectral sequence is defined in terms of a family of complexes, $Sy{m}_{\ast }^{(p)}$. $Sy{m}^{(p)}$ is isomorphic to the suspension of the cycle-free chessboard complex ${\Omega }_{p+1}$ of Vrećica and Živaljević, and so recent results on the connectivity of ${\Omega }_n$ imply finite-dimensionality of the symmetric homology groups of finite-dimensional algebras. Some results about the $k{\Sigma }_{p+1}$--module structure of $Sy{m}^{(p)}$ are devloped. A partial resolution is found that allows computation of $H{S}_1(A)$ for finite-dimensional $A$ and some concrete computations are included.

This is a major revision of the previous paper

References in corpus (1)

• On the PROP corresponding to bialgebras

Cited by in corpus (8)

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• Derived Character Maps of Groups Representations
• Categorifying equivariant monoids