Lie Derivations on Trivial Extension Algebras

Abstract

In this paper we provide some conditions under which a Lie derivation on a trivial extension algebra is proper, that is, it can be expressed as a sum of a derivation and a center valued map vanishing at commutators. We then apply our results for triangular algebras. Some illuminating examples are also included.2010 Mathematics Subject Classification. 16W25, 15A78, 47B47.

Derivations and the First Cohomology Group of Trivial Extension Algebras

“…Moreover, if such an idempotent exists, say e (f Se = 0, where f = 1 − e), then, for M := eSf , emf = m for every m ∈ M . This property on M and its consequences (see Lemma 2.5 below) play a crucial role in proving some interesting results (see, for instance, [3,11,13]). In [11,Example 3.13] and [13,Example 2.6], examples of trivial extension algebras A ⋉ M with the suitable idempotent of A without having a triangular matrix representation are given.…”

“…This property on M and its consequences (see Lemma 2.5 below) play a crucial role in proving some interesting results (see, for instance, [3,11,13]). In [11,Example 3.13] and [13,Example 2.6], examples of trivial extension algebras A ⋉ M with the suitable idempotent of A without having a triangular matrix representation are given. A deep observation of these examples leads to consider the following particular case of Proposition 2.1.…”

“…For every m ∈ N and a ∈ S, am = eaem and ma = mf af .In this paper, we sometimes consider a trivial extension A ⋉ M which satisfies the following property (satisfied by the ring of Example 2.4): There is a non trivial idempotent e of A such that eAf = 0 (where f = 1 − e) and emf = m for every m ∈ M (thus A ⋉ M satisfies all the equivalent properties of Proposition 2.5). We refer to such an algebra by a trivial extension of type (⋆) (this trivial extension algebras were first considered in[13]). Note that in this case, we have (e, 0)(a,m)(e, 0) = (eae, 0) and (f, 0)(a, m)(f, 0) = (f af, 0).…”

More on Lie derivations of a generalized matrix algebra

“…Cheung [4] initiated the study of various mappings on triangular algebras; in particular, he studied the Lie derivation property for triangular algebras in [3]. Following his work [3], Lie derivations on a wide variety of algebras have been studied by many authors (see [1,2,5,6,7,8,10,11,13,14] and the references therein). The main result of Cheung [3] has recently extended by Du and Wang [5] for a generalized matrix algebra.…”

Jordan Higher Derivations on Trivial Extension Algebras

“…In [10] (see also [8]), Lie derivations of A ⋉ M are discussed. In this paper, our main aim is to provide some sufficient conditions under which a Jordan higher derivation on A ⋉ M become a higher derivation.…”