On identities in Hom-Malcev algebras

In a Hom-Malcev algebra an identity, equivalent to the Hom-Malcev identity, is found.


Introduction and statement of results
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.
In this setting, a Hom-type generalization of Malcev algebras (called Hom-Malcev algebras) is defined by D. Yau in [15]. Recall that a Malcev algebra is a nonassociative algebra (A, ·), where the binary operation "·" is anti-commutative, such that the identity holds for all x, y, z ∈ A (here J(x, y, z) denotes the Jacobian, i.e. J(x, y, z) = xy ·z+yz·x+zx·y). The identity (1.1) is known as the Malcev identity. Malcev algebras were introduced by A.I. Mal'tsev [9] (calling them Moufang-Lie algebras) as tangent algebras to local smooth loops, generalizing in this way a result in Lie theory stating that a Lie algebra is a tangent algebra to a local Lie group (in fact, Lie algebras are special case of Malcev algebras). Another approach to Malcev algebras is the one from alternative algebras: every alternative algebra is Malcev-admissible [9]. So one could say that the algebraic theory of Malcev algebras started from Malcev-admissibility of algebras. The foundations of the algebraic theory of Malcev algebras go back to E. Kleinfeld [5], A.A. Sagle [10] and, as mentioned in [10], to A.A. Albert and L.J. Paige. Some twisting of the Malcev identity (1.1) along any algebra self-map α of A gives rise to the notion of a Hom-Malcev algebra (A, ·, α) ( [15]; see definitions in section 2). Properties and constructions of Hom-Malcev algebras, as well as the relationships between these Homalgebras and Hom-alternative or Hom-Jordan algebras are investigated in [15]. In particular, it is shown that a Malcev algebra can be twisted into a Hom-Malcev algebra and that Hom-alternative algebras are Hom-Malcev admissible.
In [15], as for Malcev algebras (see [10], [11]), equivalent defining identities of a Hom-Malcev algebra are given. In this note, we mention another identity in a Hom-Malcev algebra that is equivalent to the ones found in [15]. Specifically, we shall prove the following Theorem. Let (A, ·, α) be a Hom-Malcev algebra. Then the identity holds for all w, x, y, z in A, where J α (x, y, z) = xy ·α(z)+yz ·α(x)+zx·α(y). Moreover, in any anti-commutative Hom-algebra (A, ·, α), the identity (1.2) is equivalent to the Hom-Malcev identity for all x, y, z in A.
In section 2 some instrumental lemmas are proved. Some results in these lemmas are a kind of the Hom-version of similar results by E. Kleinfeld [5] in case of Malcev algebras. The section 3 is devoted to the proof of the theorem.
Throughout this note we work over a ground field K of characteristic 0.
Definition 2.1. A multiplicative Hom-algebra is a triple (A, µ, α) , in which A is a K-module, µ : A × A → A is a bilinear map (the binary operation), and α : A → A is a linear map (the twisting map) such that α is an endomorphism of (A, µ). The Hom-algebra (A, µ, α) is said anticommutative if the operation µ is skew-symmetric, i.e. µ(x, y) = −µ(y, x), for all x, y ∈ A.
In the rest of this paper, we will use the abbreviation x · y = µ(x, y) in a Hom-algebra (A, µ, α).
The following simple lemma holds in any anticommutative Hom-algebra.
Lemma 2.4. In any anticommutative Hom-algebra (A, ·, α) the following holds: Proof. The skew-symmetry of J α (x, y, z) in w, x, y, z follows from the skew-symmetry of the operation "·".
In a Hom-Malcev (A, ·, α) we define the multilinear map G by for all w, x, y, z in A. i.e.

Proof
Relaying on the lemmas of section 2, we are now in position to prove the theorem.