Hom-Akivis algebras

Abstract

Hom-Akivis algebras are introduced. The commutator-Hom-associator algebra of a non-Hom-associative algebra (i.e. a Hom-nonassociative algebra) is a Hom-Akivis algebra. It is shown that non-Hom-associative algebras can be obtained from nonassociative algebras by twisting along algebra automorphisms while Hom-Akivis algebras can be obtained from Akivis algebras by twisting along algebra endomorphisms. It is pointed out that a Hom-Akivis algebra associated to a Homalternative algebra is a Hom-Malcev algebra.

A twisted generalization of Lie-Yamaguti algebras

“…Subsequently, we show a way to construct Hom-LY algebras from LY algebras (or Malcev algebras) by twisting along self-morphisms (Corollary 3.2 and Corollary 3.3); this is an extension to binary-ternary algebras of a result due to D. Yau ([20], Theorem 2.3. Such an extension is first mentioned in [4], Corollary 4.6). In section 4 some relationships between Hom-LY algebras and Hom-Malcev algebras are considered.…”

“…A Hom-algebra in which the Hom-associativity is not assumed is called a nonassociative Hom-algebra [10] or a Hom-nonassociative algebra [19] (the expression of "non-Hom-associative" Hom-algebra is used in [4] for that type of Hom-algebras). With the notion of a Hom-triple system as above, we have the following (A, [, , ], α) is a Hom-triple system.…”

“…Remark. For α = Id (the identity map), we recover the triple system with ternary operation [[x, y], z] − (x, y, z) + (y, x, z) that is associated to each nonassociative algebra, since any nonassociative algebra has a natural Akivis algebra structure with respect to the commutator and associator operations [x, y] and (x, y, z), for all x, y, z (see, e.g., remarks in [4]).…”

“…Silvestrov [11], while D. Yau [20] gave a general construction method of Hom-type algebras starting from classical algebras and a twisting self-map. For information on various types of Hom-algebras, one may refer to [1], [4], [5], [9]- [11], [19]- [22].…”

On Identities in Hom-Malcev Algebras

“…Hom-Lie algebras were introduced in [3] as a tool in understanding the structure of some q-deformations of the Witt and the Virasoro algebras. Since then, the theory of Hom-type algebras began an intensive development (see, e.g., [2], [4], [6], [7], [8], [12], [13], [14], [15]). Hom-type algebras are defined by twisting the defining identities of some well-known algebras by a linear self-map, and when this twisting map is the identity map, one recovers the original type of considered algebras.…”