Hom-alternative Algebras and Hom-jordan Algebras

The purpose of this paper is to introduce Hom-alternative algebras and Hom-Jordan algebras. We discuss some of their properties and provide construction procedures using ordinary alternative algebras or Jordan algebras. Also, we show that a polarization of Hom-associative algebra leads to Hom-Jordan algebra. INTRODUCTION Hom-algebraic structures are algebras where the identities defining the structure are twisted by a homomorphism. They have been intensively investigated in the literature recently. The Hom-Lie algebras were introduced and discussed in [12, 15, 16, 17], motivated by quasi-deformations of Lie algebras of vector fields, in particular q-deformations of Witt and Virasoro algebras. Hom-associative algebras were introduced in [18], where it is shown that the commutator bracket of a Hom-associative algebra gives rise to a Hom-Lie algebra and where a classification of Hom-Lie admissible algebras is established. Given a Hom-Lie algebra, there is a universal enveloping Hom-associative algebra (see [29]). Dualizing Hom-associative algebras, one can define Hom-coassociative coalgebras, Hom-bialgebras and Hom-Hopf algebras which were introduced in ( [21, 19]), see also [5, 32, 33, 34, 35, 36]. It is shown in [31] that the universal enveloping Hom-associative algebra carries a structure of Hom-bialgebra. See also [2, 3, 8, 9, 10, 20, 30] for other works on twisted algebraic structures. The purpose of this paper is to introduce Hom-alternative algebras and Hom-Jordan algebras which are twisted version of the ordinary alternative algebras and Jordan algebras. We discuss some of their properties and provide construction procedures using ordinary alternative algebras or Jordan algebras. Also, we show that a polarization of Hom-associative algebra leads to Hom-Jordan algebra. In the first Section of this paper we introduce Hom-alternative algebras and study their properties. In particular, we define a twisted version of the associator and show that it is an alternating function of its arguments. The second Section is devoted to construction of Hom-alternative algebras. We show that an ordinary alternative algebra and one of its algebra endomorphisms lead to a Hom-alternative algebra where the twisting map is actually the algebra endomorphism. This process was introduced in [30] for Lie and associative algebras and more generally to G-associative algebras (see [18] for this class of algebras) and generalized to coalgebras in [21], [19] and to n-ary algebras of Lie and associative types in [3]. We derive examples of Hom-alternative algebras from 4-dimensional alternative algebras which are not associative and from algebra of octonions. The last Section is dedicated to Jordan algebras. We introduce a notion of Hom-Jordan algebras and show that it fits with the Hom-associative structure, that is a Hom-associative algebra leads to Hom-Jordan algebra by polarization. Also, we provide a way to construct a Hom-Jordan algebra starting from an ordinary Jordan algebra and an algebra endomorphism. 2000 Mathematics Subject Classification. 17D05,17C10,17A30.