Commutators in the Endomorphism Ring of a Complex

TL;DR

This paper extends classical results on commutators in endomorphism rings from finite dimensional vector spaces to complexes, providing new characterizations using generalized trace concepts.

Contribution

It introduces a novel framework for understanding commutators in the endomorphism ring of complexes, generalizing trace-based characterizations from finite dimensional spaces.

Findings

Characterization of commutators in the endomorphism ring of complexes

Introduction of generalized trace concepts for complexes

Extension of classical trace-zero commutator results

Abstract

According to an old result of Albert and Muckenhoupt, the commutators in the endomorphism ring of a finite dimensional vector space are precisely the elements of trace zero. We replace the finite dimensional vector space with a complex of finite dimensional vector spaces, and characterize commutators and other elements with commutator-like properties in terms of appropriately defined traces.