Hom-Akivis algebras.
A Hom-Lie algebra is a Lie algebra with a linear endomorphism \(\alpha \) satisfying NEWLINE\[NEWLINE [[x,y],\alpha (z)]+[[y,z],\alpha (x)]+[[z,x],\alpha (y)]=0. NEWLINE\]NEWLINE Since Hom-Lie algebras were introduced in [\textit{J. Hartwig, D. Larsson} and \textit{S. Silvestrov}, J. Algebra 295, No. 2, 314--361 (2006; Zbl 1138.17012)] this notion has been generalized to other classes of algebraic structures.NEWLINENEWLINEAn Akivis algebra is an algebraic structure that formalizes the relationship between the commutators and the associators in a non-associative algebra; it has one bilinear antisymmetric operation \([\cdot ,\cdot ]\) and one trilinear operation \((\cdot ,\cdot ,\cdot )\) satisfying the relation NEWLINE\[NEWLINE [[x,y],z]+[[y,z],x]+[[z,x],y]=(a,b,c)+(b,c,a)+(c,a,b)- (a,c,b)-(c,b,a)-(b,a,c). NEWLINE\]NEWLINE These algebras were first studied by \textit{M. A. Akivis} [Sib. Math. J. 17, 3--8 (1976); translation from Sibir. Mat. Zh. 17, 5--11 (1976; Zbl 0337.53018)] under the name \(W\)-algebras. Later \textit{K. H. Hofmann} and \textit{K. Strambach} [Quasigroups and loops: theory and applications, Sigma Ser. Pure Math. 8, 205--262 (1990; Zbl 0747.22004)] called them Akivis algebras. An important subclass of the class of Akivis algebras are the Maltsev algebras, the tangent algebras of smooth local Moufang loops [\textit{A. I. Mal'tsev}, Mat. Sb., N. Ser. 36(78), 569--576 (1955; Zbl 0065.00702)]. More generally, the tangent space at the identity of any local loop has the structure of an Akivis algebra, though the correct notion of a tangent structure to a general local loop is that of a Sabinin algebra (for the relation between Sabinin algebras and Akivis algebras, see [\textit{I. P. Shestakov} and \textit{U. U. Umirbaev}, J. Algebra 250, No. 2, 533--548 (2002; Zbl 0993.17002)]).NEWLINENEWLINEIn the present paper the author defines Hom-Akivis algebras, gives low dimensional examples and discusses the influence of special properties of the mapping \(\alpha \). Finally, he considers Hom-Maltsev algebras, Hom-flexible algebras and Hom-alternative algebras.
scientific article; zbMATH DE number 6054237