Hom-Lie Triple System and Hom-Bol Algebra Structures on Hom-Maltsev and Right Hom-Alternative Algebras

Every multiplicative Hom-Maltsev algebra has a natural multiplicative Hom-Lie triple system structure. Moreover, there is a natural Hom-Bol algebra structure on every multiplicative Hom-Maltsev algebra and on every multiplicative right (or left) Hom-alternative algebra.

Motivated by the relationships between some classes of binary algebras and some classes of binary-ternary algebras, a study of Hom-type generalization of binary-ternary algebras is initiated in [9] with the definition of Hom-Akivis algebras. Further, Hom-Lie-Yamaguti algebras are considered in [5] and Hom-Bol algebras [2] are defined as a twisted generalization of Bol algebras which are introduced and studied in [21], [26], [27] as infinitesimal structures tangent to smooth Bol loops (some aspects of the theory of Bol algebras are discussed in [3], [8] and [24]. In this paper, we will be concerned with right (or left) Hom-alternative algebras, Hom-Malcev algebras and Hom-Bol algebras. We extend the Loos' construction of Lts from Malcev algebras ( [14], Satz 1) to the Hom-algebra setting (Section 3). Specifically, we prove (Theorem 3.2) that every multiplicative Hom-Malcev algebra is naturally a multiplicative Hom-Lts by a suitable definition of the ternary operation. As a tool in the proof of this fact, we point out a kind of compatibility relation between the original binary operation of a given Hom-Malcev algebra and the ternary operation mentioned above (Lemma 3.1). Moreover, we obtain that every multiplicative Hom-Malcev algebra has a natural Hom-Bol algebra structure (Theorem 3.5). In [22] Mikheev proved that every right alternative algebra has a natural (left) Bol algebra structure. In [8] Hentzel and Peresi proved that not only a right alternative algebra but also a left alternative algebra have left Bol algebra structure. In Section 4 we prove that the Hom-analogue of these results hold. Specifically, every multiplicative right (or left) Homalternative algebra is a Hom-Bol algebra (Theorem 4.4). In Section 2 we recall some basic definitions and facts about Hom-algebras. We define the Hom-Jordan associator of a given Hom-algebra and point out that every Hom-algebra is a Hom-triple system with respect to the Hom-Jordan associator. This observation is used in the proof of Theorem 4.4.
All vector spaces and algebras are meant over an algebraically closed ground field K of characteristic 0.
, where x,y,z denotes the sum over cyclic permutation of x, y, z.
Remark 2.2. If α = id (the identity map), then a Hom-algebra (A, * , α) reduces to an ordinary algebra (A, * ), the Hom-Jacobian J α is the ordinary Jacobian J, and the Homassociator is the usual associator for the algebra (A, * ).
For our purpose, we make the following Definition 2.3. The Hom-Jordan associator of a Hom-algebra A := (A, * , α) is the trilinear map as J : is the Jordan product on A.
If α = id, the Hom-Jordan associator reduces to the usual Jordan associator.
Definition 2.4. (i) A Hom-Lie algebra is a Hom-algebra (A, * , α) such that the binary operation " * " is anticommutative and the Hom-Jacobi identity holds for all x, y, z in A ( [7]).
(ii) A Hom-Malcev algebra (or Hom-Maltsev algebra) is a Hom-algebra (A, * , α) such that the binary operation " * " is anticommutative and that the Hom-Malcev identity holds for all x, y, z in A ( [32]).
(iii) A Hom-Jordan algebra is a Hom-algebra (A, * , α) such that (A, * ) is a commutative algebra and the Hom-Jordan identity as(x * x, α(y), α(x)) = 0 is satisfied for all x, y in A ( [32]). Remark 2.5. When α = id, the Hom-Jacobi identity (2.1) is the usual Jacobi identity J(x, y, z) = 0. Likewise, for α = id, the Hom-Malcev identity (2.2) reduces to the Malcev identity J(x, y, x * z) = J(x, y, z) * x. Therefore a Lie (resp. Malcev) algebra (A, * ) may be seen as a Hom-Lie (resp. Hom-Malcev) algebra with the identity map as the twisting map. Also Hom-Malcev algebras generalize Hom-Lie algebras in the same way as Malcev algebras generalize Lie algebras. For α = id in the Hom-Jordan identity, we recover the usual Jordan identity. Observe that the definition of the Hom-Jordan identity in [32] is slightly different of the one formerly given in [17].
Hom-Malcev algebras are introduced in [32] in connection with a study of Hom-alternative algebras introduced in [17]. In fact it is proved ( [32], Theorem 3.8) that every Hom-alternative algebra is Hom-Malcev admissible, i.e. the commutator Hom-algebra of any Hom-alternative algebra is a Hom-Malcev algebra (this is the Hom-analogue of Mal'tsev's construction of Malcev algebras as commutator algebras of alternative algebras [20]). This result is also mentioned in [9], section 4, using an approach via Hom-Akivis algebras (this approach is close to the one of Mal'tsev in [20]). Also, every Hom-alternative algebra is Hom-Jordan admissible, i.e. its plus Hom-algebra is a Hom-Jordan algebra ( [32]). Examples of Hom-alternative algebras and Hom-Jordan algebras could be found in [17] and [32]. An example of a right Hom-alternative algebra that is not left Hom-alternative is given in [35].
The Hom-algebras mentioned above are binary Hom-algebras. The first generalization of binary algebras was the ternary algebras introduced in [11]. Ternary algebraic structures also appeared in various domains of theoretical and mathematical physics (see, e.g., [23]). Likewise, binary Hom-algebras are generalized to n-ary Hom-algebra structures in [1] (see also [33]).
holds for all u, v, w, x, y in A. The identity (2.5) is called the ternary Hom-Nambu identity.
Remark 2.7. When α 1 = id = α 2 one recovers the usual ternary Nambu algebra. One may refer to [23] for the origins of Nambu algebras. In [1], examples of n-ary Hom-Nambu algebras that are not Nambu algebras are provided.
One notes that when the twisting maps α 1 , α 2 are both equal to the identity map id, then we recover the usual notion of a Lie triple system [13], [29]. Examples of Hom-Lts could be found in [33].
A particular situation , interesting for our setting, occurs when the twisting maps α i are all equal, is then said to be multiplicative [33]. In case of multiplicativity, the ternary Hom-Nambu identity (2.5) then reads In [5] a (multiplicative) Hom-triple system is defined as a (multiplicative) ternary Homalgebra (A, [, , ], α) such that (2.6) and (2.7) are satisfied (thus a multiplicative Hom-Lts is seen as a Hom-triple system in which the identity (2.8) holds; observe that this definition of a Hom-triple system is different from the one formerly given in [33], where a Hom-triple system is just the Hom-algebra (A, [, , ], α)). With this vision of a Hom-triple system, it is shown ( [5]) that every multiplicative non-Hom-associative algebra (i.e. not necessarily Hom-associative algebra) has a natural Hom-triple system structure if define [x, y, z] := [[x, y], α(z)] − as(x, y, z) + as(y, x, z). We note here that we get the same result if define another ternary operation on a given Hom-algebra. Specifically, we have the following Proposition 2.9. Let A = (A, * , α) be a multiplicative Hom-algebra. Define on A the ternary operation for all x, y, z ∈ A. Then (A, (, , ), α) is a multiplicative Hom-triple system. Proof. A proof follows from the straightforward checking of the identities (2.6) and (2.7) for "(, , )" using the commutativity of the Jordan product "•".
Since our results here depend on multiplicativity, in the rest of this paper we assume that all Hom-algebras (binary or ternary) are multiplicative and while dealing with the binary operation " * " and where there is no danger of confusion, we will use juxtaposition in order to reduce the number of braces i.e., e.g., xy * α(z) means (x * y) * α(z).
Various results and constructions related to Hom-Lts are given in [33]. In particular, it is shown that every Lts L can be twisted along any self-morphism of L into a multiplicative Hom-Lts. For our purpose we just mention the following result.
One observes that the product [x, y, z] = 2xy * z − yz * x − zx * y is the one defined in [14] providing a Malcev algebra (A, * ) with a Lts structure. A construction describing another view of Proposition 2.10 above, will be given in Section 3 (see Proposition 3.4) via Hom-Malcev algebras. For the time being, we point out the following slight generalization of the result above, producing a sequence of multiplicative Hom-Lts from a given Malcev algebra.
Construction results and some examples of Hom-Bol algebras are given in [2]. In particular, Hom-Bol algebras can be constructed from Malcev algebras. The Hom-analogues of the construction of Bol algebras from Malcev algebras [21] or from right alternative algebras [22] (see also [8]) are considered in this paper.

Hom-Lts and Hom-Bol algebras from Hom-Malcev algebras
In this section, we prove that every multiplicative Hom-Malcev algebra has a natural multiplicative Hom-Lts structure (Theorem 3.2) and, moreover, a natural Hom-Bol algebra structure (Theorem 3.5). Theorem 3.2 could be seen as the Hom-analogue of the Loos' result ( [14], Satz 1). Besides the identities (2.3) and (2.4), Lemma 3.1 below is a tool in the proof of this result. Theorem 3.5 could be seen as the Hom-analogue of a construction by Mikheev [21] of Bol algebras from Malcev algebras. Proposition 3.4 is another view of a result in [33] (see Proposition 2.10 above).
Then (A, {, , }) turns out to be a Lts. This result, in the Hom-algebra setting, looks as in Theorem 3.2 below. Similarly as in the Loos construction, our investigations are based on the following ternary operation in a Hom-Malcev algebra (A, * , α): From (3.2) it clearly follows that {, , } α can also be written as (3.3) {x, y, z} α = −J α (x, y, z) + 3xy * α(z).
One observes that when α = id, we recover the product (3.1). This agrees with the reduction of the Hom-Malcev algebra (A, * , α) to the Malcev algebra (A, * ). First, we prove the following for all u, v, x, y in A.
The aim of Section 4 is a generalization of Corollary 3.6 to multiplicative right (or left) Hom-alternative algebras.
Various constructions of Hom-Lts are offered in [33] starting from either Hom-associative algebras, Hom-Lie algebras, Hom-Jordan triple systems, ternary totally Hom-associative algebras, Malcev algebras or alternative algebras. In practice, it is easier to construct Hom-Lts or Hom-Bol algebras from well-known (binary) algebras such as, e.g., alternative algebras or Malcev algebras. From this point of view, our construction results (Theorem 3.2, Proposition 3.4 and Theorem 3.5) have rather a theoretical feature (the extension to Hom-algebra setting of the Loos' result [14] and a result by Mikheev [21]) than a practical method for constructing Hom-Lts or Hom-Bol algebras. However, it could be of some interest to get a Hom-Lts or a Hom-Bol algebra from a given Hom-Malcev algebra without resorting to the corresponding Malcev algebra.

Hom-Lts and Hom-Bol algebras from right (or left) Hom-alternative algebras
In this section we prove that every multiplicative right (or left) Hom-alternative algebra has a natural Hom-Bol algebra structure (and, subsequently, a natural Hom-Lts structure). This is the Hom-analogue of a result by Mikheev [22] and by Hentzel and Peresi [8].
First we recall some few basic properties of right Hom-alternative algebras that could be found in [17], [35].
The linearized form of the right Hom-alternative identity as(x, y, y) = 0 is given by the following result. for all x, y, z ∈ A.
Note that in case when (A, * , α) is a left Hom-alternative algebra, the identity (4.5) reads as In any multiplicative right (or left) Hom-alternative algebra (A, * , α) we consider the ternary operation defined by (2.9), i.e. where as J is the Hom-Jordan associator defined in Section 2. Observe that for α = id the ternary operation "(, , )" is precisely the one defined in [8] (see also [22], Remark 2) and that makes any right (or left) alternative algebra into a left Bol algebra. In [8], Hentzel and Peresi used the approach of Mikheev [22] who formerly proved that the commutator algebra of any right alternative algebra has a left Bol algebra structure.  for all x, y, z ∈ A.
We are now in position to prove the main result of this section. Proof. We prove the theorem for a multiplicative right Hom-alternative algebra (A, * , α) (the proof of the left case is the mirror of the right one).