Derivations and the first cohomology group of trivial extension algebras

TL;DR

This paper provides a detailed study of derivations on trivial extension algebras, generalizing known results and characterizing when all derivations are inner, with implications for their matrix representations.

Contribution

It extends existing results on derivations and cohomology of trivial extension algebras, offering new characterizations and structural insights.

Findings

Generalized known results on derivations

Characterized trivial extension algebras with all derivations inner

Linked trivial extension algebras to triangular matrix representations

Abstract

In this paper we investigate in details derivations on trivial extension algebras. We obtain generalizations of both known results on derivations on triangular matrix algebras and a known result on first cohomology group of trivial extension algebras. As a consequence we get the characterization of trivial extension algebras on which every derivation is inner. We show that, under some conditions, a trivial extension algebra on which every derivation is inner has necessarily a triangular matrix representation. The paper starts with detailed study (with examples) of the relation between the trivial extension algebras and the triangular matrix algebras.