Coalgebra measurings, cyclic theory and homologies of matrix algebras

TL;DR

This paper investigates how coalgebra measurings induce maps between Hochschild and cyclic homology of algebras, preserving various algebraic structures, and explores their effects on matrix algebra homologies.

Contribution

It establishes the well-behaved nature of these induced maps and relates homology theories of algebras to those of matrix algebras in multiple contexts.

Findings

Induced maps preserve $ ext{lambda}$-decomposition and product structures.

Relations between algebra homology maps and matrix algebra homology are established.

Connections between cyclic, Hochschild, Leibniz, and Lie algebra homologies are demonstrated.

Abstract

In this paper, we consider coalgebra measurings and the maps induced by them between Hochschild and cyclic homology of algebras. We show that these induced maps are well behaved with respect to the various structures appearing on Hochschild and cyclic homology, such as the $\lambda$-decomposition, the product structure, as well as the module structure of Hochschild homology over cyclic homology. Thereafter, we relate the maps between homology theories of algebras induced by a coalgebra measuring to those induced between homologies of matrix algebras. This is done in the following contexts: (a) cyclic homology and the primitive part of Lie algebra homology of the matrix algebra, (b) Hochschild homology and the primitive part of Leibniz homology of the matrix algebra, and (c) Dihedral homology of an involutive algebra and the primitive part of Lie algebra homology of symplectic or skew symmetric matrices.