Homologies diédrale et quaternionique. (Dihedral and quaternionic homology)
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.
homology of the Lie algebras of orthogonal groups
scientific article; zbMATH DE number 4019300
scientific article