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

1. Introduction

The study of Lie triple systems (Lts) on their own as algebraic objects started from Jacobson's work [1] and developed further by, for example, Lister [2], Yamaguti [3], and other mathematicians. 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 [1, 4, 5]). Another construction of Lts from binary algebras is the one from Maltsev algebras found by Loos [6].

Maltsev algebras were introduced by Maltsev [7] in a study of commutator algebras of alternative algebras and also as a study of tangent algebras to local smooth Moufang loops. Maltsev used the name "Moufang-Lie algebras" for these nonassociative algebras while Sagle [8] introduced the term "Malcev algebras." Equivalent defining identities of Maltsev algebras are pointed out in [8].

Alternative algebras, Maltsev 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 et al. in [9] 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 [10], 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., [11-23]). 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 [24].

In [11, 21], n-ary Hom-algebra structures generalizing nary 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 [11] while Hom-Jordan algebras were defined in [18] and Hom-Lie triple systems (Hom-Lts) were introduced in [21] (another definition of a Hom-Jordan algebra is given in [20]). It is shown [21] that Hom-Lts are ternary Hom-Nambu algebras with additional properties and that Hom-Lts arise also from Hom-Jordan triple systems or from other Hom-type algebras.

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 [16] with the definition of Hom-Akivis algebras. Further, Hom-Lie-Yamaguti algebras are considered in [14] and Hom-Bol algebras [12] are defined as a twisted generalization of Bol algebras which are introduced and studied in [25-27] as infinitesimal structures tangent to smooth Bol loops (some aspects of the theory of Bol algebras are discussed in [28-30]).

In this paper, we will be concerned with right (or left) Hom-alternative algebras, Hom-Maltsev algebras, Hom-Lts, and Hom-Bol algebras. We extend Loos' construction of Lts from Maltsev algebras ([6], Satz 1) to the Hom-algebra setting (Section 3). Specifically, we prove (Theorem 14) that every multiplicative Hom-Maltsev 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-Maltsev algebra and the ternary operation mentioned above (Lemma 13). Moreover, we obtain that every multiplicative Hom-Maltsev algebra has a natural Hom-Bol algebra structure (Theorem 17). In [31] Mikheev proved that every right alternative algebra has a natural (left) Bol algebra structure. In [29] Hentzel and Peresi proved that not only a right alternative algebra but also a left alternative algebra has left Bol algebra structure. In Section 4 we prove that the Hom-analogue of these results holds. Specifically, every multiplicative right (or left) Hom-alternative algebra is a Hom-Bol algebra (Theorem 23). It could be observed that the methods used in the proof of results in [6, 29, 31] cannot be reported in the Hom-algebra setting at the present stage of the theory of Hom-algebras. In Section 5 we specify Theorem 23 to recover the construction of left Bol algebras from right alternative algebras (Theorem 26; one observes that, in our proof, we use essentially some fundamental properties of right alternative algebras). 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 23.

All vector spaces and algebras are meant over an algebraically closed ground field K of characteristic 0.

2. Some Basics on Hom-Algebras

We first recall some relevant definitions about binary and ternary Hom-algebras. In particular, we recall the notion of a Hom-Maltsev algebra as well as some of its equivalent defining identities. Although various types of n-ary Hom-algebras are introduced and discussed in [11, 21], for our purpose, we will consider ternary Hom-algebras (ternary Hom-Nambu algebras and Hom-Lts) and Hom-Bol algebras. For fundamentals on Hom-algebras, one may refer, for example, to [9-11, 13, 17, 24, 32]. Some aspects of the theory of binary Hom-algebras are considered in [33], while some classes of binary-ternary Hom-algebras are defined and discussed in [12, 14, 16].

Definition 1. (i) A Hom-algebra is a triple (A, *, [alpha]) in which A is a K-vector space, * : A x A [right arrow] A a bilinear map (the binary operation), and a : A [right arrow] A a linear map (the twisting map). The Hom-algebra A is said to be multiplicative if [alpha](x * y) = [alpha](x) * [alpha](y) for all x, y [member of] A.

(ii) The Hom-Jacobian in (A, *, [alpha]) is the trilinear map [J.sub.[alpha]] : A x A x A [right arrow] A defined as [J.sub.[alpha]] (x, y, z) = [[??].sub.x,y,z](x * y) * [alpha](z), where [[??].sub.x,y,z] denotes the sum over cyclic permutation of x, y, z.

(iii) The Hom-associator of a Hom-algebra (A, *, a) is the trilinear map as : [A.sup.[cross product]3] [right arrow] A defined as as(x, y, z) = (x * y) * [alpha](z) - [alpha](x) * (y * z). If as(x, y, z) = 0 for all x, y, z [member of] A, then (A, *, [alpha]) is said to be Hom-associative.

Remark 2. If [alpha] = id (the identity map), then a Hom-algebra (A, *, [alpha]) reduces to an ordinary algebra (A, *), the Hom-Jacobian [J.sub.[alpha]] is the ordinary Jacobian J, and the Hom-associator is the usual associator for the algebra (A, *). One observes that, in general, the map [alpha] is not always injective nor surjective (see [13, 15] for discussions on the subject). So, for example, a given algebra can be twisted into zero algebra and some properties of Hom-algebras may not be valid for corresponding ordinary algebras.

As for ordinary algebras, to each Hom-algebra A = (A, *, [alpha]) are attached two Hom-algebras: the commutator Hom-algebra [A.sup.-] = (A, [,], [alpha]), where [x, y] := x * y-y * x (the commutator of x and y), and the plus Hom-algebra [A.sup.+] := (A, [??],[alpha]), where x [??] y := x * y + y * x (the Jordan product) for all x, y [member of] A.

For our purpose, we provide the following.

Definition 3. The Hom-Jordan associator of a Hom-algebra A := (A, *, [alpha]) is the trilinear map [as.sup.J] : [A.sup.[cross product]3] [right arrow] A defined as [as.sup.J] (x, y, z) = (x [??] y) [??][alpha](z)- [alpha](x) [??] (y [??] z), where "[??]" is the Jordan product on A.

If [alpha] = id, the Hom-Jordan associator reduces to the usual Jordan associator.

Definition 4. (i) A Hom-Lie algebra is a Hom-algebra (A, *, [alpha]) such that the binary operation "*" is anti commutative and the Hom-Jacobi identity

[J.sub.[alpha]] (x, y, z) = 0 (1)

holds for all x, y, and z in A ([9]).

(ii) A Hom-Maltsev algebra is a Hom-algebra (A, *, [alpha]) such that the binary operation "*" is anticommutative and that the Hom-Maltsev identity

[J.sub.[alpha]] ([alpha](x), [alpha](y), x * z) = [J.sub.[alpha]] (x, y, z) * [[alpha].sup.2](x) (2)

holds for all x, y, z in A ([20]).

(iii) A Hom-Jordan algebra is a Hom-algebra (A, *, [alpha]) such that (A, *) is a commutative algebra and the Hom-Jordan identity

as (x * x, [alpha](y), [alpha](x)) = 0 (3)

is satisfied for all x, y in A ([20]).

(iv) A Hom-algebra (A, *, [alpha]) is called a right Hom-alternative algebra if

as (x, y, y) = 0 (4)

for all x, y in A.A Hom-algebra (A, *, [alpha]) is called a left Hom-alternative algebra if

as (x, x, y) = 0 (5)

for all x, y in A. A Hom-algebra (A, *, [alpha]) is called a Hom-alternative algebra if it is both right and left Hom-alternative [18].

Remark 5. When [alpha] = id, the Hom-Jacobi identity (1) is the usual Jacobi identity J(x, y, z) = 0. Likewise, for [alpha] = id, the Hom-Maltsev identity (2) reduces to the Maltsev identity J(x, y, x * z) = J(x, y, z) * x. Therefore a Lie (resp., Maltsev) algebra (A, *) may be seen as a Hom-Lie (resp., Hom-Maltsev) algebra with the identity map as the twisting map. Also Hom-Maltsev algebras generalize Hom-Lie algebras in the same way that Maltsev algebras generalize Lie algebras. For [alpha] = id in the Hom-Jordan identity, we recover the usual Jordan identity. Observe that the definition of the Hom-Jordan identity in [20] is slightly different from the one formerly given in [18].

Hom-Maltsev algebras are introduced in [20] in connection with a study of Hom-alternative algebras introduced in [18]. In fact it is proved ([20], Theorem 3.8) that every Hom-alternative algebra is Hom-Maltsev admissible; that is, the commutator Hom-algebra of any Hom-alternative algebra is a Hom-Maltsev algebra (this is the Hom-analogue of Maltsev's construction of Maltsev algebras as commutator algebras of alternative algebras [7]). This result is also mentioned in [16], Section 4, using an approach via Hom-Akivis algebras (this approach is close to the one of Maltsev in [7]). Also, every Hom-alternative algebra is Hom-Jordan admissible; that is, its plus Hom-algebra is a Hom-Jordan algebra ([20]). Examples of Hom-alternative algebras and Hom-Jordan algebras could be found in [18, 20]. An example of a right Hom-alternative algebra that is not left Hom-alternative is given in [23].

Equivalent to (2) defining identities of Hom-Maltsev algebras are found in [20] where, in particular, it is shown that the identity

[mathematical expression not reproducible] (6)

is equivalent to (2) in any anticommutative Hom-algebra (A, *, [alpha]) ([20], Proposition 2.7). In [34], it is proved that, in any anticommutative Hom-algebra (A, *, [alpha]), the Hom-Maltsev identity (2) is equivalent to

[mathematical expression not reproducible]. (7)

Therefore, apart from (2), identities (6) and (7) maybe taken as defining identities of a Hom-Maltsev algebra.

The Hom-algebras mentioned above are binary Hom-algebras. The first generalization of binary algebras was the ternary algebras introduced in [1]. Ternary algebraic structures also appeared in various domains of theoretical and mathematical physics (see, e.g., [35]). Likewise, binary Hom-algebras are generalized to n-ary Hom-algebra structures in [11] (see also [21]).

Definition 6 (see [11]). A ternary Hom-Nambu algebra is a triple (A, [,,],[alpha]) in which A is a K-vector space, [,,] : A x A x A [right arrow] A is a trilinear map, and [alpha] = ([[alpha].sub.1], [[alpha].sub.2]) is a pair of linear maps (the twisting maps) such that the identity

[mathematical expression not reproducible] (8)

holds for all u, v, w, x, and y in A. Identity (8) is called the ternary Hom-Nambu identity.

Remark 7. When [[alpha].sub.1] = id = [[alpha].sub.2] one recovers the usual ternary Nambu algebra. One may refer to [35] for the origins of Nambu algebras. In [11], examples of n-ary Hom-Nambu algebras that are not Nambu algebras are provided.

Definition 8 (see [21]). A Hom-Lie triple system (Hom-Lts) is a ternary Hom-algebra (A, [,,],[alpha] = ([[alpha].sub.1], [[alpha].sub.2])) such that

[x, y, z] = -[y, x, z], (9)

[[??].sub.[x,y,z]] [x, y, z] = 0, (10)

and the ternary Hom-Nambu identity (8) holds in (A, [,,],[alpha] = ([[alpha].sub.1], [[alpha].sub.2])).

One notes that when the twisting maps [[alpha].sub.1], [[alpha].sub.2] are both equal to the identity map id, then we recover the usual notion of a Lie triple system [2, 3]. Examples of Hom-Lts could be found in [21].

A particular situation, interesting for our setting, occurs when the twisting maps [[alpha].sub.i] are all equal, [[alpha].sub.1] = [[alpha].sub.2] = [alpha] and [alpha]([x, y, z]) = [[alpha](x), [alpha](y), [alpha](z)] for all x, y, and z in A. The Hom-Lie triple system (A, [,,],[alpha]) is then said to be multiplicative [21]. In case of multiplicativity, the ternary Hom-Nambu identity (8) then reads

[mathematical expression not reproducible]. (11)

In [14] a (multiplicative) Hom-triple system is defined as a (multiplicative) ternary Hom-algebra (A, [,,],[alpha]) such that (9) and (10) are satisfied (thus a multiplicative Hom-Lts is seen as a Hom-triple system in which identity (11) holds; observe that this definition of a Hom-triple system is different from the one formerly given in [21], where a Hom-triple system is just the Hom-algebra (A, [,,], [alpha])). With this vision of a Hom-triple system, it is shown [14] that every multiplicative non-Hom-associative algebra (i.e., not necessarily Hom-associative algebra) has a natural Hom-triple system structure if defining [x, y, z] := [[x, y], [alpha](z)]-as(x, y, z) + as(y, x, z). We note here that we get the same result if defining another ternary operation on a given Hom-algebra. Specifically, we have the following result.

Proposition 9. Let A = (A, *, [alpha]) be a multiplicative Hom-algebra. Define on A the ternary operation

(x, y, z) := [as.sup.J] (y, z, x) (12)

for all x, y, and z [member of] A. Then (A, (,,), [alpha]) is a multiplicative Hom-triple system.

Proof. A proof follows from the straightforward checking of identities (9) and (10) 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; that is, for example, xy * [alpha](z) means (x * y) * [alpha](z).

Various results and constructions related to Hom-Lts are given in [21]. 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.

Proposition 10 (see [21]). Let (A, *) be a Maltsev algebra and [alpha] : A [right arrow] A an algebra morphism. Then [A.sub.[alpha]] := (A, [[,,].sub.[alpha]], [alpha]) is a multiplicative Hom-Lts, where [[x, y, z].sub.[alpha]] = [alpha](2xy * z-yz * x-zx * y), for all x, y, and z in A.

One observes that the product [x, y, z] = 2xy * z-yz * x-zx * y is the one defined in [6] providing a Maltsev algebra (A, *) with a Lts structure. A construction describing another view of Proposition 10 above will be given in Section 3 (see Proposition 16) via Hom-Maltsev 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 Maltsev algebra.

Proposition 11. Let (A, *) be a Maltsev algebra and [alpha] : A [right arrow] A an algebra morphism. Let [[alpha].sup.0] = id and, for any integer n [greater than or equal to] 1, [[alpha].sup.n] = [alpha] [??] [[alpha].sup.n-1]. If one defines on A a trilinear operation [mathematical expression not reproducible] by

[mathematical expression not reproducible] (13)

for all x, y, and z in A, then [mathematical expression not reproducible] is a multiplicative Hom-Lts.

Proof. Let [mathematical expression not reproducible]. We shall use the fact that (A, [,,]) is a Lts [6]. Identities (9) and (10) for [mathematical expression not reproducible] are quite obvious. Next,

[mathematical expression not reproducible] (14)

and so (11) holds for [mathematical expression not reproducible]. Thus [mathematical expression not reproducible] is a multiplicative Hom-Lts.

In [12] we defined a Hom-Bol algebra as a twisted generalization of a (left) Bol algebra. For the introduction and original studies of Bol algebras, we refer to [25-27] (the defining identities of left Bol algebras are recalled in Section 5 of the present paper). Bol algebras are further considered in, for example, [29, 30].

Definition 12 (see [12]). A Hom-Bol algebra is a quadruple (A, [,], (,,), [alpha]) in which A is a vector space, "[,]" a binary operation, "(,,)" a ternary operation on A, and [alpha] : A [right arrow] A a linear map such that

(HB1) [alpha]([x, y]) = [[alpha](x), [alpha](y)].

(HB2) [alpha]((x, y, z)) = ([alpha](x), [alpha](y), [alpha](z)).

(HB3) [x, y] = -[y, x].

(HB4) (x, y, z) = -(y, x, z).

(HB5) [[??].sub.x,y,z] (x, y, z) = 0.

(HB6) [mathematical expression not reproducible].

(HB7) [mathematical expression not reproducible] for all u, v, w, x, y, and z [member of] A.

Identities (HB1) and (HB2) mean the multiplicativity of (A, [,], (,,), [alpha]). It is built into our definition for convenience.

One observes that for [alpha] = id identities (HB3)-(HB7) reduce to the defining identities of a (left) Bol algebra [25] (see also [29, 30]). If [x, y] = 0 for all x, y [member of] A, then (A, [,,], (,,), [alpha]) becomes a (multiplicative) Hom-Lts (A, (,,), [[alpha].sup.2]).

Construction results and some examples of Hom-Bol algebras are given in [12]. In particular, Hom-Bol algebras can be constructed from Maltsev algebras. The Hom-analogues of the construction of Bol algebras from Maltsev algebras [25] or from right alternative algebras [31] (see also [29]) are considered in this paper.

3. Hom-Lts and Hom-Bol Algebras from Hom-Maltsev Algebras

In this section, we prove that every multiplicative Hom-Maltsev algebra has a natural multiplicative Hom-Lts structure (Theorem 14) and, moreover, a natural Hom-Bol algebra structure (Theorem 17). Theorem 14 could be seen as the Hom-analogue of Loos' result ([6], Satz 1) although his proof cannot be reproduced here. Besides identities (6) and (7), Lemma 13 below is a tool in the proof of this result. Theorem 17 could be seen as the Hom-analogue of a construction by Mikheev [25] of Bol algebras from Maltsev algebras. Proposition 16 is another view of a result in [21] (see Proposition 10 above).

In his work [6], Loos considered in a Maltsev algebra (A, *) the following ternary operation:

{x, y, z} = 2xy * z - yz * x - zx * y. (15)

Then (A, {,,}) turns out to be a Lts. This result, in the Hom-algebra setting, looks as in Theorem 14 below. Similarly as in the Loos construction, our investigations are based on the following ternary operation in a Hom-Maltsev algebra (A, *, [alpha]):

[{x, y, z}.sub.[alpha]] = 2xy * [alpha](z) - yz * [alpha](x) - zx * [alpha](y). (16)

From (16) it clearly follows that [{,,}.sub.[alpha]] can also be written as

[{x, y, z}.sub.[alpha]] = -[J.sub.[alpha]] (x, y, z) + 3xy * [alpha](z). (17)

One observes that when [alpha] = id, were cover product (15). First, we prove the following.

Lemma 13. Let (A, *, [alpha]) be a Hom-Maltsev algebra. If one defines on (A, *, [alpha]) a ternary operation "[{,,}.sub.[alpha]]" by (16), then

[mathematical expression not reproducible] (18)

for all u, v, x, and y in A.

Proof. Let us write (7) as

[mathematical expression not reproducible]. (19)

That is,

[mathematical expression not reproducible]. (20)

Therefore, by multiplicativity, we have

[mathematical expression not reproducible] (21)

and so, we get (18) by (17).

We now prove the following.

Theorem 14. Let (A, *, [alpha]) be a multiplicative Hom-Maltsev algebra. If one defines on (A, *, [alpha]) a ternary operation "[{,,}.sub.[alpha]]" by (16), then (A, [{,,}.sub.[alpha]], [[alpha].sup.2]) is a multiplicative Hom-Lts.

Proof. We must prove the validity of (9), (10), and (11) for operation (16) in the Hom-Maltsev algebra (A, *, [alpha]).

First observe that the multiplicativity of (A, *, [alpha]) implies that [[alpha].sup.2] ([{x, y, z}.sub.[alpha]]) = [[alpha].sup.2](x), [[alpha].sup.2](y), [[alpha].sup.2](z)}.sub.[alpha]], with x, y, and z in A.

From the skew-symmetry of "*" and [J.sub.[alpha]](x, y, z), it clearly follows from (17) that [{x, y, z}.sub.[alpha]] = -[{y, x, z}.sub.[alpha]] which is (9) for "[{,,}.sub.[alpha]]."

Next, using (17) and the skew-symmetry of [J.sub.[alpha]](x, y, z) where applicable, we compute

[mathematical expression not reproducible] (22)

and thus [[??].sub.x,y,z] [{x, y, z}.sub.[alpha]] = 0, so we get (10) for "[{,,}.sub.[alpha]]."

Consider now [{[[alpha].sup.2](x), [[alpha].sup.2](y), [{u, v, w}.sub.[alpha]]}.sub.[alpha]] in (A, *, [alpha]).

Then

[mathematical expression not reproducible]. (23)

In this latest expression, denote by N(u, v, w, x, y) the expression in "[***]"; to conclude, we proceed to show that N(u, v, w, x, y) = 0.

Observe first that, by (6), we have

[mathematical expression not reproducible]. (24)

That is,

[mathematical expression not reproducible]. (25)

With this observation, the expression N(u, v, w, x, y) is transformed as follows:

[mathematical expression not reproducible]. (26)

Therefore, we obtain that (11) holds for "[{,,}.sub.[alpha]]" and we conclude that (A, [{,,}.sub.[alpha]], [[alpha].sup.2]) is a Hom-Lts.

Remark 15. In the proof of his result, Loos ([6], Satz 1) used essentially the fact that the left translations L(x) in a Maltsev algebra (A, *) are derivations with respect to the ternary operation "{,,}" defined by (15). Unfortunately, for Hom-Maltsev algebras such a tool is still not available at hand.

From [20] (Theorem 2.12) we know that any Maltsev algebra A can be twisted into a Hom-Maltsev algebra along any linear self-map of A. Consistent with this result, we recall the following method for constructing Hom-Lts which, in fact, is a result in [21] (see also Propositions 10 and 11 above) but using a Hom-Maltsev algebra construction in our proof (as a consequence of Theorem 14).

Proposition 16. Let (A, *) be a Maltsev algebra and a any self-morphism of (A, *). If one defines on (A, *) a ternary operation "[{,,}.sub.[alpha]]" by

[{x, y, z}.sub.[alpha]] = [[alpha].sup.2] (2xy * z - yz * x - zx * y), (27)

then (A, [{,,}.sub.[alpha]], [[alpha].sup.2]) is a multiplicative Hom-Lts.

Proof. One knows ([20], Theorem 2.12) that, from (A, *) and any self-morphism a of (A, *),we get a (multiplicative) Hom-Maltsev algebra (A, [??],[alpha]), where x [??] y = [alpha](x * y) for all x, y in A. Next, if one defines on (A, [??],[alpha]) a ternary operation

[mathematical expression not reproducible], (28)

then, by Theorem 14, (A, [{,,}.sub.[alpha]], [[alpha].sup.2]) is a Hom-Lts and "[{,,}.sub.[alpha]]" is expressed through "*" as

[mathematical expression not reproducible]. (29)

Observe that though constructed in quite a different way, the operation "[{,,}.sub.[alpha]]" in Proposition 16 above coincides with "[mathematical expression not reproducible]" in Proposition 11 for n = 2.

Combining Lemma 13 and Theorem 14, we get the following result.

Theorem 17. Let (A, *, [alpha]) be a multiplicative Hom-Maltsev algebra. If one defines on (A, *, [alpha]) a ternary operation [(,,).sub.[alpha]] by

[(x, y, z).sub.[alpha]] := 1/3 [{x, y, z}.sub.[alpha]], (30)

where "[{,,}.sub.[alpha]]" is defined by (17), then (A, *, [(,,).sub.[alpha]], [alpha]) is a Hom-Bol algebra.

Proof. Definition (30) and Theorem 14 imply that (A, [(,,).sub.[alpha]], [[alpha].sup.2]) is a multiplicative Hom-Lts; that is, (HB4), (HB5), and (HB7) hold for (A, *, [(,,).sub.[alpha]], [alpha]). Now, (HB1), (HB2), and (HB3) are, respectively, the multiplicativity and skew-symmetry of "*"; next, we are done if we prove (HB6) for (A, *, [(,,).sub.[alpha]], [alpha]).

From (17) and multiplicativity we have

[mathematical expression not reproducible] (31)

and then (18) takes the form

[mathematical expression not reproducible]. (32)

Multiplying by 1/3 each member of this latter equality and using (30), we get

[mathematical expression not reproducible] (33)

which is (HB6) for (A, *, [(,,).sub.[alpha]], [alpha]). So (A, *, [(,,).sub.[alpha]], [alpha]) is a Hom-Bol algebra.

Example 18. Let A be a vector space with basis {[e.sub.1], [e.sub.2], [e.sub.3], [e.sub.4]}. From [20] (Example 2.13) we know that if one considers the linear map [alpha]: A [right arrow] A given by

[mathematical expression not reproducible] (34)

and the multiplication table given by

[mathematical expression not reproducible] (35)

(only nonzero products are specified), then (A, *, [alpha]) is a multiplicative Hom-Maltsev algebra. It is observed that (A, *, [alpha]) is not a Hom-Lie algebra nor a Maltsev algebra.

Now, by (17) and (30), one checks that the only nonzero ternary products [(x, y, z).sub.[alpha]] on A with respect to the basis elements are

[mathematical expression not reproducible]. (36)

By Theorem 17 we get that (A, *, [(,,).sub.[alpha]], [alpha]) is a Hom-Bol algebra.

Since any Hom-alternative algebra is Hom-Maltsev admissible ([20], Theorem 3.8), from Theorem 17 we have the following.

Corollary 19. Let (A, *, [alpha]) be a multiplicative Hom-alternative algebra. Then (A, [,], [(,,).sub.[alpha]], [alpha]) is a Hom-Bol algebra, where [(x,y,z).sub.[alpha]] := -(1/3)(2[[x, y], [alpha] (z)]-[[y, z], [alpha](x)]-[[z,x], [alpha](y)]), for all x, y, and z [member of] A.

The aim of Section 4 is a generalization of Corollary 19 to multiplicative right (or left) Hom-alternative algebras.

Various constructions of Hom-Lts are offered in [21] starting from either Hom-associative algebras, Hom-Lie algebras, Hom-Jordan triple systems, ternary totally Hom-associative algebras, Maltsev algebras, or alternative algebras. In practice, it is easier to construct Hom-Lts or Hom-Bol algebras from well-known (binary) algebras such as alternative algebras or Maltsev algebras. From this point of view, our construction results (Theorem 14, Proposition 16, and Theorem 17) have rather a theoretical feature (the extension to Hom-algebra setting of Loos' result [6] and a result by Mikheev [25]) 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-Maltsev algebra without resorting to the corresponding Maltsev algebra.

4. 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 [31] and by Hentzel and Peresi [29] although with a different scheme of proof.

First we recall some few basic properties of right Hom-alternative algebras that could be found in [18, 23].

The linearized form of the right Hom-alternative identity as (x, y, y) = 0 is given by the following result.

Lemma 20 (see [18]). If (A, *, [alpha]) is a Hom-algebra, then the following statements are equivalent.

(i) (A, *, [alpha]) is right Hom-alternative.

(ii) (A, *, [alpha]) satisfies

as (x, y, z) = -as (x, z, y) (37)

for all x, y, and z [member of] A.

(iii) (A, *, [alpha]) satisfies

[alpha](x) * (yz + zy) = xy * [alpha](z) + xz * [alpha](y) (38)

for all x, y, and z [member of] A.

Observe that if (A, *, [alpha]) is a right Hom-alternative algebra, then (A, [*.sup.op], [alpha]) is a left Hom-alternative algebra, where x [*.sup.op] y := y * x. So the mirrors of (37) and (38) hold for (A, [*.sup.op], [alpha]):

as (x, y, z) = -as (y, x, z), (39)

[mathematical expression not reproducible]. (40)

Now we have the following.

Lemma 21. In any multiplicative right Hom-alternative algebra (A, *, [alpha]), the identity

[mathematical expression not reproducible] (41)

holds for all x, y, and z [member of] A.

Proof. The identity

[mathematical expression not reproducible] (42)

holds in any right Hom-alternative algebra (see [23], Theorem 7.1 (7.1.1c)). Next, in this identity, switching u and v, we have

[mathematical expression not reproducible]. (43)

Then, subtracting memberwise this latter equality from the one above and using the linearity of as, we get (41).

Note that in the case when (A,*,a) is a left Hom-alternative algebra, identity (41) reads as

[mathematical expression not reproducible]. (44)

In any multiplicative right (or left) Hom-alternative algebra (A, *, [alpha]) we consider the ternary operation defined by (12); that is,

(x, y, z) := [as.sup.J] (y, z, x), (45)

where [as.sup.J] is the Hom-Jordan associator defined in Section 2. Observe that for [alpha] = id the ternary operation "(,,)" is precisely the one defined in [29] (see also [31], Remark 2) and that makes any right (or left) alternative algebra into a left Bol algebra. In [29], Hentzel and Peresi used the approach of Mikheev [31] who formerly proved that the commutator algebra of any right alternative algebra has a left Bol algebra structure.

Proposition 22. (i) If (A, *, [alpha]) is a multiplicative right Hom-alternative algebra, then

(x, y, z) = [[x, y], [alpha](z)] - 2as (z, x, y) (46)

for all x, y, and z [member of] A.

(ii) If (A, *, [alpha]) is a multiplicative left Hom-alternative algebra, then

(x, y, z) = [[x, y], [alpha](z)] - 2as (x, y, z) (47)

for all x, y, and z [member of] A.

Proof. (i) From (12) we have

[mathematical expression not reproducible] (48)

and so we get (46).

(ii) Proceeding as above, but using (40) and then (39), one gets (47).

We are now in a position to prove the main result of this section.

Theorem 23. Let (A, *, [alpha]) be a multiplicative right (resp., left) Hom-alternative algebra. If one defines on A a ternary operation "(,,)" by (46) (resp., (47)), then (A, (,,), [[alpha].sup.2]) is a Hom-Lts and (A, [,], (,,), [alpha]) is a Hom-Bol algebra.

Proof. We prove the theorem for a multiplicative right Hom-alternative algebra (A, *, [alpha]) (the proof of the left case is the mirror of the right one).

Identities (HB1) and (HB2) follow from the multiplicativity of (A, *, [alpha]). Identities (HB3) and (HB4) are obvious from the definition of "[,]" and "(,,)"; identity (HB5) follows from Proposition 9.

In [22] Yau showed that if, on a multiplicative Hom-Jordan algebra (A, [??],[alpha]), define a ternary operation by

[x, y, z] := 2 ([alpha](x) [??] (y [??] z) - [alpha](y) [??] (x [??] z)), (49)

then (A, [,,], [[alpha].sup.2]) is a multiplicative Hom-Lts (see [22], Corollary 4.1). Now, observe that [x, y, z] = 2[as.sup.J] (y, z, x); that is, [x, y, z] = 2(x, y, z). Therefore, since every multiplicative right Hom-alternative algebra is Hom-Jordan admissible (see [23], Theorem 4.3), we conclude that (A,(,,), [[alpha].sup.2]) is a multiplicative Hom-Lts and so identity (HB7) holds for (A, [,], (,,), [alpha]).

Next, (A, [,], (,,),[alpha]) is a Hom-Bol algebra if we prove that (HB6) additionally holds.

Write (46) as

-2as (z, x, y) = (x, y, z) - [[x, y], [alpha](z)]. (50)

Multiplying each member of (41) by -2 and next using (50), we get

[mathematical expression not reproducible]. (51)

That is,

[mathematical expression not reproducible]. (52)

Observe that

[mathematical expression not reproducible]. (53)

[mathematical expression not reproducible] (54)

and so (HB6) holds for (A, [,], (,,), [alpha]). Thus we conclude that (A, [,], (,,), [alpha]) is a Hom-Bol algebra. One gets the same result in the case when (A, *, [alpha]) is a multiplicative left Hom-alternative algebra and essentially using (47) and (44). This finishes the proof.

Example 24. Let A be a five-dimensional vector space with basis {e, u, v, w, z} and let [alpha]: A [right arrow] A be a linear map given by

[mathematical expression not reproducible]. (55)

Define on A a binary operation "*" by

[mathematical expression not reproducible] (56)

(again, only nonzero products are specified). Then (A, *, [alpha]) is a multiplicative right Hom-alternative algebra (see [23], Example 2.9). Then, using [x, y] = x * y-y * x and (46), one could find (although the computation is somewhat lengthy) all the nonzero products "[,]" and "(,,)" with respect to the basis elements e, u, v, w, and z of A. We just point out that they are nonzero products; for example, [e, u] = u-v, [e, w] = -w + z, (e, u, e) = -u-v, and (e, w, e) = -w-z. Therefore, Theorem 23 implies that (A, [,], (,,), [alpha]) is a Hom-Bol algebra.

5. The Construction of Bol Algebras from Right Alternative Algebras Revisited

As already mentioned in Section 2, for [alpha] = id in Definition 12 we get the definition of a left Bol algebra.

Definition 25 (see [25, 27]). A left Bol algebra is a triple (A, [,], (,,)) in which A is a vector space, "[,]" a binary operation, and "(,,)" a ternary operation on A such that

(B1) [x, y] = -[y, x],

(B2) (x, y, z) = -(y, x, z),

(B3) [[??].sub.x,y,z] (x, y, z) = 0,

(B4) (x, y, [u, v]) = [(x, y, u), v] + [u, (x, y, v)] + (u, v, [x, y]) - [[u, v], [x, y]],

(B5) (x, y, (u, v, w)) = ((x, y, u), v, w) + (u, (x, y, v), w) + (u, v, (x, y, w)), for all u, v, w, x, y, z [member of] A.

In this section we show how the construction of Hom-Bol algebras from right or left Hom-alternative algebras described in Section 4 can be specified to the ordinary untwisted case of construction of (left) Bol algebras from right or left alternative algebras ([29, 31]). In fact, for [alpha] = id in Theorem 23 and specifying the right alternative case, we get the following.

Theorem 26. Let (A, *) be a right alternative algebra. If one defines on A a ternary operation "(,,)" by

(x, y, z) = [[x, y], z] - 2 as (z, x, y), (57)

where as (u, v, w) = uv * w-u * vw, then (A, (,,)) is a Lts and (A, [,], (,,)) is a left Bol algebra.

Proof. Identities (B1) and (B2) are obvious. For [alpha] = id, the Hom-Jordan associator (see Definition 3) reduces to the usual Jordan associator [as.sup.J] (u, v, w) := (u [??] v) [??] w-u [??] (v [??] w) in (A, *). The fact that (B3) and (B5) hold in (A, *) follows from the equality (x, y, z) = [as.sup.J] (y, z, x) that holds in right alternative algebras (the untwisted form of (46)) and from that right alternative algebras are Jordan admissible [36]. Therefore (A, (,,)) is a Lts since any Jordan algebra is a Lts with respect to the operation (x, y, z) = [as.sup.J] (y, z, x) (see [1]). So we get the untwisted version of (HB5) and (HB7).

In order to show that (B4) holds in (A, *), we proceed as follows. First, recall that the identity

as (uv, y, x) = u as (v, y, x) + as (u, v, y) x + as (u, vy, x) - as (u, v, yx) (58)

holds in any algebra. Also, in a right alternative algebra (A, *) (over a ground field of characteristic different from 2), the following identity holds [37]:

as (u, v, v * y) = as (u, v, y) * v; (59)

that is, by linearization and right alternativity,

as (u, v * y, x) = as (u, v, x * y) - as (u, v, y) * x - as (u, x, y) * v. (60)

Putting (60) in (58), we get

as (u * v, y, x) = u * as (v, y, x) - as (u, x, y) * v + as (u, V, [x, y]); (61)

that is, by right alternativity,

as (u * v, x, y) = as (u, x, y) * v + u * as (v, x, y) - as (u, v, [x, y]). (62)

Now, in (62) switching u and v and then subtracting the obtained equality from (62), one gets

as ([u, v], x, y) = [as (u, x, y), v] + [u, as (v, x, y)] + as (v, u, [x, y]) - as (u, v, [x, y])

(observe that (63) is the untwisted form of (41)). Next, write (57) as

-2as (z, x, y) = (x, y, z) - [[x, y], z]. (64)

Then multiplying (63) by -2 and using the equality above, and next proceeding as in the proof of Theorem 23, one proves the validity of (B4) for (A, [,], (,,)). Thus we get that (A, [,], (,,)) is a left Bol algebra.

Remark 27. (i) The process of constructing left Bol algebras from right alternative algebras described in Theorem 26 above is different from the ones given in [29, 31]. In our approach here, we rely essentially on fundamental properties of right alternative algebras (see, e.g., [36, 37]) without subsidiary constructions.

(ii) If (A, *) is a left alternative algebra, it is also possible to get a natural left Bol algebra structure on (A, *). Indeed, one needs to consider the counterparts of (x, y, z) and (63) that looks, respectively, as

(x, y, z) = [[x, y], z] - as (x, y, z) (65)

(the untwisted version of (47)) and

[mathematical expression not reproducible]. (66)

Next one proceeds as in Theorem 26 observing that a left alternative algebra is also Jordan-admissible (see [36], Theorem 2, for right alternative algebras).

https://doi.org/10.n55/2018/4528685

Conflicts of Interest

The authors declare that they have no conflicts of interests.

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Sylvain Attan and A. Nourou Issa (iD)

Departement de Mathematiques, Universite d'Abomey-Calavi, 01 BP4521 Cotonou, Benin

Correspondence should be addressed to A. Nourou Issa; woraniss@yahoo.fr

Received 18 December 2017; Accepted 6 March 2018; Published 2 May 2018