Alternative algebras admitting derivations with invertible values and invertible derivations

Abstract

We prove an analogue of the Bergen-Herstein-Lanski theorem for alternative algebras: describe all alternative algebras that admit derivations with invertible values. We also prove an analogue of Moens' theorem for alternative algebras (a finite-dimensional alternative algebra over a field of characteristic zero is nilpotent if and only if it admits an invertible Leibniz derivation).

Δ-Derivations OF SEMISIMPLE FINITE-DIMENSIONAL STRUCTURABLE ALGEBRAS

“…Using this fact and the multiplication in the algebra T (C), making direct substitution and verification we can easily see that the element e = − 1 16 8 i=2 e i ⊗ e i is the unit of the algebra. Let β = − α 16 .…”

“…The analogue of δderivation for n-ary algebras was investigated in [11,12]. We note that some generalizations of δ-derivations were proposed in articles [13]- [16] and δ-derivations were used for studying prime Lie algebras by V. Filippov [1], for construction of post-Lie structures by D. Burde and K. Dekimpe [17], for studying speciality of Jordan superalgebas by I. Kaygorodov and V. Zhelyabin [5], in analysis of the derivations of current Lie algebras by P. Zusmanovich [18], and for studying homotopes of Novikov algebras by V. Filippov and V. Sereda [19].…”

Generalized Derivations and Rota-Baxter Operators of $$\varvec{n}$$-ary Hom-Nambu Superalgebras

“…There are many generalizations of derivations. For example, Leibniz derivations [51] and δ-derivations of prime Lie and Malcev algebras [43][44][45]. The properties and structure of generalized derivations algebras of a Lie algebra and their subalgebras and quasi-derivation algebras were systematically studied in [66], where it was proved for example that the quasi-derivation algebra of a Lie algebra can be embedded into the derivation algebra of a larger Lie algebra.…”

Yau-type ternary Hom-Lie bialgebras

“…They have various applications such as relating algebra to geometry and also allow the construction of new algebraic structures. There are many generalizations of derivations (for example, Leibniz derivations [32]). The notion of δ -derivations appeared in the paper of Filippov [24], he studied δ -derivations of prime Lie and Malcev algebras [25,26].…”