Homologies diédrale et quaternionique. (Dihedral and quaternionic homology)
type: Item
instance of: scholarly article
title: Homologies diédrale et quaternionique. (Dihedral and quaternionic homology)
zbMATH Open document ID: 0627.18006
DOI: 10.1016/0001-8708(87)90032-6
author: J.-L. Loday
published in: Advances in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: This paper introduces dihedral and quaternionic homology, which are theories analogous to cyclic homology. A dihedral module M over some ground ring is a simplicial module \((M_ 0,M_ 1,...)\) together with suitable actions of the dihedral groups \(D_ n\) on the \(M_ n\). One can also regard M as a quaternionic module or cyclic module by using induced actions of the quaternionic groups \(Q_ n\) or cyclic groups \(C_ n\). One gets dihedral homology groups \(HD_ n(M)\) and quaternionic homology groups \(HQ_ n(M)\) as derived functors analogous to the cyclic homology groups \(HC_ n(M)\). One can also get \(HQ_ n(M)\) as the total homology of a fairly small concrete bicomplex. If 2 is invertible in the ground ring, then \(HD_ n(M)=HQ_ n(M)\) and \(HD_ n(M)\) is a direct summand of \(HC_ n(M).\) An important special case is given by \(M_ n=A^{\otimes n+1}\) for some k-algebra A with involution. In particular A can be a group algebra, and the paper gives calculations for this case. The author announces a connection with the homology of the Lie algebras of orthogonal groups.
reviewed by: Richard John Steiner
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 17B55
zbMATH DE Number: 4019300
zbMATH Keywords: quaternionic homology
zbMATH Keywords: cyclic homology
zbMATH Keywords: simplicial module
zbMATH Keywords: dihedral homology
zbMATH Keywords: derived functors
zbMATH Keywords: homology of the Lie algebras of orthogonal groups
MaRDI profile type: Publication
full work available at URL: 0001-8708(87)90032-6
OpenAlex ID: W1996816288
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description: scientific article; zbMATH DE number 4019300