Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coalgebra measurings, cyclic theory and homologies of matrix algebras

Abhishek Banerjee, Surjeet Kour

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 $λ$-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.

Coalgebra measurings, cyclic theory and homologies of matrix algebras

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 -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.
Paper Structure (7 sections, 40 theorems, 151 equations)

This paper contains 7 sections, 40 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Measurings and morphisms on cyclic homology
  3. Measurings and the product in cyclic homology
  4. Measurings and $\lambda$-decomposition on Hochschild and cyclic homology
  5. Measurings and maps in Lie algebra homology and Leibniz homology
  6. Measurings and product structures on Lie homology and Leibniz homology
  7. Involutive algebras, dihedral homology and measurings

Key Result

Theorem 1.1

(see P1.1 and P2.2) Let $C$ be a cocommutative $K$-coalgebra and let $\Phi:C\longrightarrow Hom_K(A,A')$ be a measuring between algebras. For each $x\in C$, we have induced maps on Hochschild homology and cyclic homology groups respectively. Further, the maps $HH^\Phi_\bullet(x)$ and $HC^\Phi_\bullet(x)$ in 1.4ctr are compatible with the operators appearing in the Connes periodicity sequences for

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 67 more