Derivations and the First Cohomology Group of Trivial Extension Algebras

Abstract

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.

Characterizations of Lie n -Centralizers on Certain Trivial Extension Algebras

“…Trivial extension algebras have been extensively studied in algebra and analysis (see [11][12][13][14][15][16]). In this paper, we will study the structure of Lie n-centralizers on a trivial extension algebra.…”

Triple Derivations of Trivial Extensions

“…Trivial extensions have been extensively studied in algebra and analysis (see, for instance, [1][2][3][4][5][6][7]).…”

Poisson Structures on Trivial Extension Algebras

“…In particular, they arise in the study of the infinitesimal geometry around Poisson submanifolds [12]. Recently, trivial extension algebras were characterized as some algebras on which every derivation is inner [1]. As is known, Poisson modules are closely related with the notion of flat contravariant derivative [2,3].…”