Generalized derivations of (color)n-ary algebras

Abstract

Abstract. We generalize the results of Leger and Luks about generalized derivations of Lie algebras to the case of color n-ary Ω-algebras. Particularly, we prove some properties of generalized derivations of color n-ary algebras; prove that a quasiderivation algebra of a color n-ary Ω-algebra can be embedded into the derivation algebra of a larger color n-ary Ω-algebra, and describe (anti)commutative n-ary algebras satisfying the condition QDer = End. §0 Introduction It is well known that the algebras of derivations and generalized derivations are very important in the study of Lie algebras and its generalizations. There are many generalizations of derivations (for example, Leibniz derivations [22] and Jordan derivations [13]). The notion of δ-derivations appeared in the paper of Filippov [8], he studied δ-derivations of prime Lie and Malcev algebras [9,10]. After that, δ-derivations of Jordan and Lie superalgebras were studied in [14][15][16][17]36] and many other works. The notion of generalized derivation is a generalization of δ-derivation. The most important and systematic research on the generalized derivations algebras of a Lie algebra and their subalgebras was due to Leger and Luks [24]. In their article, they studied properties of generalized derivation algebras and their subalgebras, for example, the quasiderivation algebras. They have determined the structure of algebras of quasiderivations and generalized derivations and proved that the quasiderivation algebra of a Lie algebra can be embedded into the derivation algebra of a larger Lie algebra. Their results were generalized by many authors. For example, Zhang and Zhang [34] generalized the above results to the case of Lie superalgebras; Chen, Ma, Ni and Zhou considered the generalized derivations of color Lie algebras, Hom-Lie superalgebras and Lie triple systems [3][4][5]. Generalized derivations of simple algebras and superalgebras were investigated in [11,23,31,32]. Perez-Izquierdo and Jimenez-Gestal used the generalized derivations to study non-associative algebras [12,26]. Derivations and generalized derivations of n-ary algebras were considered in [2, 18-21, 28, 33] and other. For example, Williams proved that, unlike the case of binary algebras, for any n ≥ 3 there exist a non-nilpotent n-Lie algebra with invertible derivation [33], Kaygorodov described (n + 1)-ary derivations of simple n-ary Malcev algebras [20] and generalized derivations algebra of semisimple Filippov algebras over an algebraically closed field of characteristic zero [21].The main purpose of our work is to generalize the results of Leger and Luks to the case of color n-ary algebras. Particularly, we prove some properties of generalized derivations of color n-ary algebras; prove that the quasiderivation algebra of a color n-ary Ω-algebra can be embedded into the derivation algebra of a large color n-ary Ω-algebra. We describe all nonabelian n-ary (anti)commutative algebras with the

A characterization of nilpotent nonassociative algebras by invertible Leibniz-derivations

“…To prove the next Lemma, we will need the notion of ternary derivation and generalized derivation. Generalized and ternary derivations of algebras and superalgebras were studied by authors in [23], other authors in [26], and in many other articles. This means that the triple (R x , −R x , R x ) is a ternary derivation of M 7 .…”

Higher Derivations of Finitary Incidence Algebras

“…Now take x ≤ z ≤ y. Then, e xy = e xz e zy in view of (16), so that by (15), (31) and (32) Proof. This follows from (ii) and (iii) of Lemma 1.3 and Lemmas 2.3 and 2.7.…”

n-Ary Generalized Lie-Type Color Algebras Admitting a Quasi-multiplicative Basis

“…Let ǫ be a bicharacter of G. Given two homogeneous elements x, y ∈ L we set ǫ(x, y) := ǫ(deg(x), deg(y)). Now we recall the notion of color n-ary Ω-algebra for an arbitrary family of polynomial identities Ω (see [6,27] for more details).…”