Some remarks on symmetric linear functions and pseudotrace maps

TL;DR

This paper explores the relationship between symmetric linear functions, pseudotrace maps, and projective modules in finite-dimensional associative algebras, providing new insights and proofs for their properties.

Contribution

It demonstrates that pseudotrace maps are special cases of symmetric linear functions and characterizes modules interlocked with these functions as projective, offering new proofs and perspectives.

Findings

Pseudotrace maps are special symmetric linear functions.

Modules interlocked with a symmetric linear function are projective.

Provides new proofs for properties of pseudotrace maps in finite-dimensional algebras.

Abstract

Let A be a finite-dimensional associative algebra and $\phi$ a symmetric linear function on $A$. In this note, we will show that the pseudotrace maps are obtained as special cases of well-known symmetric linear functions on the endomorphism rings of projective modules. We also prove that modules are interlocked with $\phi$ if and only if they are projective. As an application of our approach, we will give proofs of several propositions and theorems for pseudotrace maps for an arbitrary finite-dimensional associative algebra.