Every multiplicative Hom-Malcev algebra has a natural multiplicative Hom-Lie triple system structure. Moreover, there is a natural Hom-Bol algebra structure on every multiplicative Hom-Malcev algebra and on every multiplicative right (or left) Homalternative algebra. 1
IntroductionThe study of Lie triple systems (Lts) on their own as algebraic objects started from Jacobson's work [11] and developed further by, e.g., Lister [13], Yamaguti [29] and other mathematicians. The interplay between Lts and the differential geometry of symmetric spaces is now folk (see, e.g., [12], [15]). Lts constitute examples of ternary algebras. If (g, [, ]) is a Lie algebra, then (g, [, , ]) is a Lts, where [x, y, z] := [[x, y], z] (see [11], [12], [15]). Another construction of Lts from binary algebras is the one from Malcev algebras found by Loos [14].Malcev algebras were introduced by Mal'tsev [20] in a study of commutator algebras of alternative algebras and also as a study of tangent algebras to local smooth Moufang loops. Mal'tsev used the name "Moufang-Lie algebras" for these nonassociative algebras while Sagle [25] introduced the term "Malcev algebras". Equivalent defining identities of Malcev algebras are pointed out in [25].Alternative algebras, Malcev algebras and Lts (among other algebras) received a twisted generalization in the development of the theory of Hom-algebras during these latest years. The forerunner of the theory of Hom-algebras is the Hom-Lie algebra introduced by Hartwig, Larsson and Silvestrov in [7] in order to describe the structure of some deformation of the Witt algebra and the Virasoro algebra. It is well-known that Lie algebras are related to associative algebras via the commutator bracket construction. In the search of a similar construction for Hom-Lie algebras, the notion of a Hom-associative algebra is introduced by Makhlouf and Silvestrov in [18], where it is proved that a Hom-associative algebra gives rise to a Hom-Lie algebra via the commutator bracket construction. Since then, various Hom-type structures are considered (see, e.g., [1], [2], [4]-[6], [9], [16]-[19], [32]-[35]). Roughly speaking, Hom-algebraic structures are corresponding ordinary algebraic structures whose defining identities are twisted by a linear self-map. A general method for constructing a Hom-type algebra from the ordinary type of algebra with a linear self-map is given by Yau in [31].In [1], [33], n-ary Hom-algebra structures generalizing n-ary algebras of Lie type or associative type were considered. In particular, generalizations of n-ary Nambu or Nambu-Lie algebras, called n-ary Hom-Nambu and Hom-Nambu-Lie algebras respectively, were introduced in [1] while Hom-Jordan were defined in [17] and Hom-Lie triple systems (Hom-Lts) were introduced in [33] (here, another definition of a Hom-Jordan algebra is given). It is shown [33] that Hom-Lts are ternary Hom-Nambu algebras with additional properties, that Hom-Lts arise also from Hom-Jordan triple systems or from other Hom-type algebras.
