The GL-module structure of the Hochschild homology of truncated tensor algebras.
type: Item
instance of: scholarly article
title: The GL-module structure of the Hochschild homology of truncated tensor algebras.
zbMATH Open document ID: 1060.16007
author: Guillermo Ames
author: Leandro Cagliero
author: Paulo Tirao
published in: Journal of Pure and Applied Algebra
publication date: 2004-08-19T00:00:00Z
review text: Let \(V\) be a finite dimensional complex vector space and \(A\) be a truncated tensor algebra of \(V\). The natural action of \(\text{GL}(V)\) on \(A\) extends to an action on the Hochschild homology of \(A\), \(HH(A)\). The goal of this paper is to determine the \(\text{GL}(V)\)-module structure of \(HH(A)\). Using work of \textit{M. J. Bardzell} [J. Algebra 188, No. 1, 69-89 (1997; Zbl 0885.16011)], the authors first construct a projective resolution of \(A\) over the enveloping algebra of \(A\). By decomposing this into subcomplexes, they are able to decompose \(HH(A)\) into a sum of eigenspaces. Knowing from Schur duality that \(V^{\otimes n}\) is completely reducible, the authors are then able to obtain formulas for the multiplicities of simple \(\text{GL}(V)\)-modules in a decomposition of \(HH(A)\). The simple \(\text{GL}(V)\)-modules are parameterized by Young diagrams and the authors then proceed to make specific computations of multiplicities for some ``standard'' Young diagrams and diagrams of depth two. The latter results make use of a hook-like formula due to \textit{V. M. Zhuravlev} [Sb. Math. 187, No. 2, 215-236 (1996); translation from Mat. Sb. 187, No. 2, 59-80 (1996; Zbl 0873.17008)]. Finally, the authors present some computations for truncation at low heights and results of some computer computations.
reviewed by: Christopher P. Bendel
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20J99
zbMATH DE Number: 2092903
zbMATH Keywords: Hochschild homology
zbMATH Keywords: tensor algebras
zbMATH Keywords: GL-modules
zbMATH Keywords: simple modules
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.02.006
OpenAlex ID: W2075869460
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DOI: 10.1016/J.JPAA.2004.02.006
description: scientific article; zbMATH DE number 2092903