The characteristic polynomial of an algebra and representations

TL;DR

This paper presents a new sufficient condition for certain linear maps to be algebra homomorphisms, with potential applications in studying representations of Clifford algebras and ring extensions.

Contribution

It introduces a novel criterion for linear maps to qualify as algebra homomorphisms, advancing the understanding of algebra representations.

Findings

New sufficient condition for algebra homomorphisms

Potential applications to Clifford algebra representations

Implications for finite ring extensions

Abstract

In this short note, we give a new sufficient condition for a linear map from a product of copies of a field to endomorphisms of a finite dimensional vector space over the same field to be an algebra homomorphism. We expect that this result can be applied to study representations of higher-degree Clifford algebras and finite extensions of commutative rings.