Nonlinear Lie-Type Derivations of finitary Incidence Algebras and Related Topics

Abstract

This is a continuation of our earlier works [29,51,52] with respect to (non-)linear Lie-type derivations of finitary incidence algebras. Let X be a pre-ordered set, R be a 2-torsionfree and (n − 1)torsionfree commutative ring with identity, where n ≥ 2 is an integer. Let F I(X, R) be the finitary incidence algebra of X over R. In this paper, a complete clarification is obtained for the structure of nonlinear Lie-type derivations of F I(X, R). We introduce a new class of derivations on F I(X, R) named inner-like derivations, and prove that each nonlinear Lie n-derivation on F I(X, R) is the sum of an inner-like derivation, a transitive induced derivation and a quasi-additive induced Lie n-derivation. Furthermore, if X is finite, we show that a quasi-additive induced Lie n-derivation can be expressed as the sum of an additive induced Lie derivation and a central-valued map annihilating all (n − 1)-th commutators. We also provide a sufficient and necessary condition such that every nonlinear Lie n-derivation of F I(X, R) is of proper form. Some related topics for further research are proposed in the last section of this article.Let A be an associative algebra over a commutative ring R. Let [ , ] and • be the Lie product and Jordan product respectively, i.e., [x, y] = xy − yx and x • y = xy + yx for all x, y ∈ A. Then (A, [ , ]) is a Lie algebra and (A, •) is a Jordan algebra. It is a fascinating topic to study the connection between the associative, Lie and Jordan structures of A. In this field, there are two classes of important algebraic maps: algebra homomorphisms and differential operators. For example Jordan homomorphisms, Lie homomorphisms, Jordan derivations and Lie derivations. In the AMS Hour Talk of 1961, Herstein proposed many problems

Derivations of Incidence Algebras

“…Much attention has been paid to the derivations of incidence rings, including ordinary and Lie derivations and Jordan derivations (see [1][2][3][4][5][6][7][8][9]). We point to a very informative introduction in [9] with a history of derivation studies and an extensive list of references.…”

“…Much attention has been paid to the derivations of incidence rings, including ordinary and Lie derivations and Jordan derivations (see [1][2][3][4][5][6][7][8][9]). We point to a very informative introduction in [9] with a history of derivation studies and an extensive list of references. Note that other important linear maps of incidence rings are also systematically studied: automorphisms, anti-automorphisms, and involutions (see [10][11][12][13][14]).…”