Dihedral ∞-simplicial modules and dihedral homology of involutive homotopy unital A ∞-algebras

The notion of a dihedral {\infty} -simplicial module is introduced. The homotopy invariance of the structure of a dihedral {\infty} -simplicial module is proved. The concept of the dihedral homology of a dihedral {\infty} -simplicial module is developed. The notion of an involutive homotopy unital {A_{\infty}} -algebra is introduced. The dihedral {\infty} -simplicial module is constructed using an involutive homotopy unital {A_{\infty}} -algebra. The concept of the dihedral homology of an involutive homotopy unital {A_{\infty}} -algebra is developed. For the dihedral homology of involutive homotopy unital {A_{\infty}} -algebras, the analogue of the Krasauskas–Lapin–Solov’ev exact sequence is obtained.

Reflexive and dihedral (co)homology of a pre-additive category

The group dihedral homology of an algebra over a field with characteristic zero was introduced by Tsygan (1983). The dihedral homology and cohomology of an algebra with involution over commutative ring with identity, associated with the small category, were studied by Krasauskas et al. (1988), Loday (1987), and Lodder (1993). The aim of this work is concerned with dihedral and reflexive (co)homology of small pre-additive category. We also define the free product of involutive algebras associated with this category and study its dihedral homology group. Finally, following Perelygin (1990), we show that a small pre-additive category is Morita equivalence.

QUATERNION HOMOLOGY OF BANACH SPACE

In this article we are concerned with the quaternion Banach spaces and their homology. We obtain the relation between the quaternion and the dihedral homology for a unital involutive Banach algebra.

The relative dihedral homology of involutive algebras

Letf:A→Bbe a homomorphism of involutive algebrasA,B. The purpose of this paper is to define a free involutive algebra resolution of algebraBoverfand use it to define and study the relative dihedral homology.

Adams and Steenrod operators in dihedral homology

In this article, we define the Adam's and Steenrod's operators in the dihedral homology.