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. 

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 (, [, ]) is a Lie algebra, then (, [, , ]) is a Lts, where [, , ] fl [[, ], ] (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 Homassociative 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][12][13][14][15][16][17][18][19][20][21][22][23]). Roughly speaking, Homalgebraic 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 selfmap is given by Yau in [24]. 
In [11,21], -ary Hom-algebra structures generalizing ary algebras of Lie type or associative type were considered. In particular, generalizations of -ary Nambu or Nambu-Lie algebras, called -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][26][27] as infinitesimal structures tangent to smooth Bol loops (some aspects of the theory of Bol algebras are discussed in [28][29][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. 

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 -ary Homalgebras 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 (, * , ) in which is a K-vector space, * : × → a bilinear map (the binary operation), and : → a linear map (the twisting map). 
The Hom-algebra is said to be multiplicative if ( * ) = () * () for all , ∈ . 
(ii) The Hom-Jacobian in (, * , ) is the trilinear map : × × → defined as (, , ) fl ↺ ,, ( * ) * (), where ↺ ,, denotes the sum over cyclic permutation of , , . 
(iii) The Hom-associator of a Hom-algebra (, * , ) is the trilinear map as : ⊗3 → defined as as(, , ) = ( * ) * () -() * ( * ). If as(, , ) = 0 for all , , ∈ , then (, * , ) is said to be Hom-associative. Remark 2. If = id (the identity map), then a Homalgebra (, * , ) reduces to an ordinary algebra (, * ), the Hom-Jacobian is the ordinary Jacobian , and the Homassociator is the usual associator for the algebra (, * ). One observes that, in general, the map 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 fl (, * , ) are attached two Hom-algebras: the commutator Hom-algebra A -fl (, [, ], ), where [, ] fl * - * (the commutator of and ), and the plus Hom-algebra A + fl (, ∘, ), where ∘ fl * + * (the Jordan product) for all , ∈ . 
For our purpose, we provide the following. 
Definition 3. The Hom-Jordan associator of a Hom-algebra A fl (, * , ) is the trilinear map as : ⊗3 → defined as as (, , ) = ( ∘ ) ∘ () -() ∘ ( ∘ ), where "∘" is the Jordan product on . 
If = id, the Hom-Jordan associator reduces to the usual Jordan associator. Definition 4. (i) A Hom-Lie algebra is a Hom-algebra (, * , ) such that the binary operation " * " is anticommutative and the Hom-Jacobi identity 
𝐽 𝛼 (𝑥, 𝑦, 𝑧) = 0 (1) 
holds for all , , and in ( [9]). 
(ii) A Hom-Maltsev algebra is a Hom-algebra (, * , ) such that the binary operation " * " is anticommutative and that the Hom-Maltsev identity 
𝐽 𝛼 (𝛼 (𝑥) , 𝛼 (𝑦) , 𝑥 * 𝑧) = 𝐽 𝛼 (𝑥, 𝑦, 𝑧) * 𝛼 2 (𝑥) (2) 
holds for all , , in ( [20]). 
(iii) A Hom-Jordan algebra is a Hom-algebra (, * , ) such that (, * ) is a commutative algebra and the Hom-Jordan identity as ( * , () , ()) = 0 
is satisfied for all , in ( [20]). 
(iv) A Hom-algebra (, * , ) is called a right Homalternative algebra if as (, , ) = 0 
for all , in . A Hom-algebra (, * , ) is called a left Homalternative algebra if as (, , ) = 0 
for all , in . A Hom-algebra (, * , ) is called a Homalternative algebra if it is both right and left Hom-alternative [18]. 
Remark 5. When = id, the Hom-Jacobi identity (1) is the usual Jacobi identity (, , ) = 0. Likewise, for = id, the Hom-Maltsev identity (2) reduces to the Maltsev identity (, , * ) = (, , ) * . Therefore a Lie (resp., Maltsev) algebra (, * ) 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 = 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 Homalternative 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 
𝐽 𝛼 (𝛼 (𝑥) , 𝛼 (𝑦) , 𝑤 * 𝑧) + 𝐽 𝛼 (𝛼 (𝑤) , 𝛼 (𝑦) , 𝑥 * 𝑧) = 𝐽 𝛼 (𝑥, 𝑦, 𝑧) * 𝛼 2 (𝑤) + 𝐽 𝛼 (𝑤, 𝑦, 𝑧) * 𝛼 2 (𝑥) (6)
is equivalent to (2) in any anticommutative Hom-algebra (, * , ) ([20], Proposition 2.7). In [34], it is proved that, in any anticommutative Hom-algebra (, * , ), the Hom-Maltsev identity (2) is equivalent to 
𝐽 𝛼 (𝛼 (𝑥) , 𝛼 (𝑦) , 𝑢 * V) = 𝛼 2 (𝑢) * 𝐽 𝛼 (𝑥, 𝑦, V) + 𝐽 𝛼 (𝑥, 𝑦, 𝑢) * 𝛼 2 (V) -2𝐽 𝛼 (𝛼 (𝑢) , 𝛼 (V) , 𝑥 * 𝑦) . (7) 
Therefore, apart from (2), identities ( 6) and ( 7) may be taken as defining identities of a Hom-Maltsev algebra. 
The Hom-algebras mentioned above are binary Homalgebras. 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 Homalgebras are generalized to -ary Hom-algebra structures in [11] (see also [21]). Definition 6 (see [11]). A ternary Hom-Nambu algebra is a triple (, [, , ], ) in which is a K-vector space, [, , ] : × × → is a trilinear map, and = ( 1 , 2 ) is a pair of linear maps (the twisting maps) such that the identity 
[𝛼 1 (𝑥) , 𝛼 2 (𝑦) , [𝑢, V, 𝑤]] = [[𝑥, 𝑦, 𝑢] , 𝛼 1 (V) , 𝛼 2 (𝑤)] + [𝛼 1 (𝑢) , [𝑥, 𝑦, V] , 𝛼 2 (𝑤)] + [𝛼 1 (𝑢) , 𝛼 2 (V) , [𝑥, 𝑦, 𝑤]] (8) 
holds for all , V, , , and in . Identity ( 8) is called the ternary Hom-Nambu identity. Remark 7. When 1 = id = 2 one recovers the usual ternary Nambu algebra. One may refer to [35] for the origins of Nambu algebras. In [11], examples of -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, [, , ], 𝛼 = (𝛼 1 , 𝛼 2 )) such that [𝑥, 𝑦, 𝑧] = -[𝑦, 𝑥, 𝑧] , (9) 
↺ ,, [, , ] = 0, (10) and the ternary Hom-Nambu identity (8) holds in (, [, , ], = ( 1 , 2 )). 
One notes that when the twisting maps 1 , 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 are all equal, 1 = 2 = and ([, , ]) = [(), (), ()] for all , , and in . The Hom-Lie triple system (, [, , ], ) is then said to be multiplicative [21]. In case of multiplicativity, the ternary Hom-Nambu identity (8) then reads 
[𝛼 (𝑥) , 𝛼 (𝑦) , [𝑢, V, 𝑤]] = [[𝑥, 𝑦, 𝑢] , 𝛼 (V) , 𝛼 (𝑤)] + [𝛼 (𝑢) , [𝑥, 𝑦, V] , 𝛼 (𝑤)] + [𝛼 (𝑢) , 𝛼 (V) , [𝑥, 𝑦, 𝑤]] . (11) 
In [14] a (multiplicative) Hom-triple system is defined as a (multiplicative) ternary Hom-algebra (, [, , ], ) 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 International Journal of Mathematics and Mathematical Sciences Hom-triple system is just the Hom-algebra (, [, , ], )). 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 Homtriple system structure if defining [, , ] fl [[, ], ()]as(, , ) + as(, , ). We note here that we get the same result if defining another ternary operation on a given Homalgebra. Specifically, we have the following result. 
for all , , and ∈ . Then (, (, , ), ) 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, * () means ( * ) * (). 
Various results and constructions related to Hom-Lts are given in [21]. In particular, it is shown that every Lts can be twisted along any self-morphism of into a multiplicative Hom-Lts. For our purpose we just mention the following result. 
Proposition 10 (see [21]). Let (, * ) be a Maltsev algebra and : → an algebra morphism. Then fl (, [, , ] , ) is a multiplicative Hom-Lts, where [, , ] = (2 * - * - * ), for all , , and in . 
One observes that the product [, , ] = 2 * - * - * is the one defined in [6] providing a Maltsev algebra (, * ) 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. 
and so (11) holds for [, , ] . Thus (, [, , ] , ) 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][26][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 (, [, ], (, , ), ) in which is a vector space, "[, ]" a binary operation, "(, , )" a ternary operation on , and : → a linear map such that (HB7) ( 2 (), 2 (), (, V, )) = ((, , ), 2 (V), 2 ()) + ( 2 (), (, , V), 2 ()) + ( 2 (), 2 (V), (, , )) 
for all , V, , , , and ∈ . 
Identities (HB1) and (HB2) mean the multiplicativity of (, [, ], (, , ), ). It is built into our definition for convenience. 
One observes that for = id identities (HB3)-(HB7) reduce to the defining identities of a (left) Bol algebra [25] (see also [29,30]). If [, ] = 0 for all , ∈ , then (, [, ], (, , ), ) becomes a (multiplicative) Hom-Lts (, (, , ), 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. 

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 (, * ) the following ternary operation: 
{𝑥, 𝑦, 𝑧} = 2𝑥𝑦 * 𝑧 -𝑦𝑧 * 𝑥 -𝑧𝑥 * 𝑦. (15) 
Then (, {, , }) turns out to be a Lts. This result, in the Homalgebra 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 (, * , ): 
{, , } = 2 * () - * () - * () . ( 16) 
From ( 16) it clearly follows that {, , } can also be written as {, , } = - (, , ) + 3 * () . 
One observes that when = id, we recover product (15). First, we prove the following. 
Lemma 13. Let (, * , ) be a Hom-Maltsev algebra. If one defines on (, * , ) a ternary operation "{, , } " by ( 16), then 
{𝛼 (𝑥) , 𝛼 (𝑦) , 𝑢 * V} 𝛼 = 𝛼 2 (𝑢) * {𝑥, 𝑦, V} 𝛼 + {𝑥, 𝑦, 𝑢} 𝛼 * 𝛼 2 (V) -𝐽 𝛼 (𝛼 (𝑢) , 𝛼 (V) , 𝑥 * 𝑦) ( 18 
)
for all , V, , and in . 
Proof. Let us write (7) as 
-𝐽 𝛼 (𝛼 (𝑥) , 𝛼 (𝑦) , 𝑢 * V) = -𝐽 𝛼 (𝑥, 𝑦, 𝑢) * 𝛼 2 (V) + 𝛼 2 (𝑢) * (-𝐽 𝛼 (𝑥, 𝑦, V)) + 3𝐽 𝛼 (𝛼 (𝑢) , 𝛼 (V) , 𝑥 * 𝑦) -𝐽 𝛼 (𝛼 (𝑢) , 𝛼 (V) , 𝑥 * 𝑦) . (19) 
That is, 
-𝐽 𝛼 (𝛼 (𝑥) , 𝛼 (𝑦) , 𝑢 * V) = -𝐽 𝛼 (𝑥, 𝑦, 𝑢) * 𝛼 2 (V) + 𝛼 2 (𝑢) * (-𝐽 𝛼 (𝑥, 𝑦, V)) + 3𝛼 (𝑢) 𝛼 (V) * 𝛼 (𝑥 * 𝑦) + 3 (𝛼 (V) * 𝑥𝑦) * 𝛼 2 (𝑢) + 3 (𝑥𝑦 * 𝛼 (𝑢)) * 𝛼 2 (V) -𝐽 𝛼 (𝛼 (𝑢) , 𝛼 (V) , 𝑥 * 𝑦) . (20) 
Therefore, by multiplicativity, we have 
-𝐽 𝛼 (𝛼 (𝑥) , 𝛼 (𝑦) , 𝑢 * V) + 3𝛼 (𝑥) 𝛼 (𝑦) * 𝛼 (𝑢 * V) = (-𝐽 𝛼 (𝑥, 𝑦, 𝑢) + 3𝑥𝑦 * 𝛼 (𝑢)) * 𝛼 2 (V) + 𝛼 2 (𝑢) * (-𝐽 𝛼 (𝑥, 𝑦, V) + 3𝑥𝑦 * 𝛼 (V)) -𝐽 𝛼 (𝛼 (𝑢) , 𝛼 (V) , 𝑥 * 𝑦) (21) 
and so, we get ( 18) by (17). 
We now prove the following. 
Theorem 14. Let (, * , ) be a multiplicative Hom-Maltsev algebra. If one defines on (, * , ) a ternary operation "{, , } " by (16), then (, {, , } , 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 (, * , ). First observe that the multiplicativity of (, * , ) implies that 2 ({, , } ) = { 2 (), 2 (), 2 ()} , with , , and in . 
From the skew-symmetry of " * " and (, , ), it clearly follows from (17) that {, , } = -{, , } which is (9) for "{, , } ." 
Next, using (17) and the skew-symmetry of (, , ) where applicable, we compute {, , } + {, , } + {, , } = - (, , ) + 3 * () - (, , ) + 3 * () - (, , ) + 3 * () = -3 (, , ) + 3 (, , ) = 0 (22) and thus ↺ ,, {, , } = 0, so we get (10) for "{, , } ." 
Consider now { 2 (), 2 (), {, V, } } in (, * , ). Then 
{𝛼 2 (𝑥) , 𝛼 2 (𝑦) , {𝑢, V, 𝑤} 𝛼 } 𝛼 = {𝛼 2 (𝑥) , 𝛼 2 (𝑦) , 2𝑢V * 𝛼 (𝑤) -V𝑤 * 𝛼 (𝑢) -𝑤𝑢 * 𝛼 (V)} 𝛼 (by (16)) = {𝛼 2 (𝑥) , 𝛼 2 (𝑦) , 2𝑢V * 𝛼 (𝑤)} 𝛼 -{𝛼 2 (𝑥) , 𝛼 2 (𝑦) , V𝑤 * 𝛼 (𝑢)} 𝛼 -{𝛼 2 (𝑥) , 𝛼 2 (𝑦) , 𝑤𝑢 * 𝛼 (V)} 𝛼 = {𝛼 (𝑥) , 𝛼 (𝑦) , 2𝑢 * V} 𝛼 * 𝛼 3 (𝑤) + 𝛼 2 (2𝑢 * V) + 2 [𝐽 𝛼 (𝛼 (V * 𝑤) , 𝛼
Therefore, we obtain that (11) holds for "{, , } " and we conclude that (, {, , } , 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 () in a Maltsev algebra (, * ) 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 can be twisted into a Hom-Maltsev algebra along any linear self-map of . 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). 
Observe that though constructed in quite a different way, the operation "{, , } " in Proposition 16 above coincides with "[, , ] " in Proposition 11 for = 2. 
Combining Lemma 13 and Theorem 14, we get the following result. 
where "{, , } " is defined by (17), then (, * , (, , ) , ) is a Hom-Bol algebra. 
Proof. Definition (30) and Theorem 14 imply that (, (, , ) , 2 ) is a multiplicative Hom-Lts; that is, (HB4), (HB5), and (HB7) hold for (, * , (, , ) , ). Now, (HB1), (HB2), and (HB3) are, respectively, the multiplicativity and skew-symmetry of " * "; next, we are done if we prove (HB6) for (, * , (, , ) , ). From ( 17) and multiplicativity we have 
-𝐽 𝛼 (𝛼 (𝑢) , 𝛼 (V) , 𝑥 * 𝑦) = {𝛼 (𝑢) , 𝛼 (V) , 𝑥 * 𝑦} 𝛼 -3 (𝛼 (𝑢) 𝛼 (V)) * (𝛼 (𝑥) 𝛼 (𝑦)) (31) 
and then (18) takes the form 
{𝛼 (𝑥) , 𝛼 (𝑦) , 𝑢 * V} 𝛼 = {𝑥, 𝑦, 𝑢} 𝛼 * 𝛼 2 (V) + 𝛼 2 (𝑢) * {𝑥, 𝑦, V} 𝛼 + {𝛼 (𝑢) , 𝛼 (V) , 𝑥 * 𝑦} 𝛼 -3 (𝛼 (𝑢) 𝛼 (V)) * (𝛼 (𝑥) 𝛼 (𝑦)) . (32) 
Multiplying by 1/3 each member of this latter equality and using (30) 
and the multiplication table given by 
𝑒 1 * 𝑒 2 = -2𝑒 2 -2𝑒 4 (= -𝑒 2 * 𝑒 1 ) ; 𝑒 1 * 𝑒 3 = 𝑒 3 (= -𝑒 3 * 𝑒 1 ) ; 𝑒 1 * 𝑒 4 = -2𝑒 4 (= -𝑒 4 * 𝑒 1 ) ; 𝑒 2 * 𝑒 3 = -4𝑒 4 (= -𝑒 3 * 𝑒 2 ) (35) 
(only nonzero products are specified), then (, * , ) is a multiplicative Hom-Maltsev algebra. It is observed that (, * , ) is not a Hom-Lie algebra nor a Maltsev algebra. Now, by ( 17) and ( 30), one checks that the only nonzero ternary products (, , ) on with respect to the basis elements are 
(𝑒 1 , 𝑒 2 , 𝑒 ) 𝛼 = -4𝑒 2 (= -(𝑒 2 , 𝑒 1 , 𝑒 1 ) 𝛼 ) ; (𝑒 1 , 𝑒 3 , 𝑒 1 ) 𝛼 = -𝑒 3 (= -(𝑒 3 , 𝑒 1 , 𝑒 1 ) 𝛼 ) ; (𝑒 1 , 𝑒 4 , 𝑒 1 ) 𝛼 = -4𝑒 4 (= -(𝑒 4 , 𝑒 1 , 𝑒 1 ) 𝛼 ) . (36) 
By Theorem 17 we get that (, * , (, , ) , ) is a Hom-Bol algebra. 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. 

Since any


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 Homalternative algebras that could be found in [18,23]. 
The linearized form of the right Hom-alternative identity as(, , ) = 0 is given by the following result. 
Lemma 20 (see [18]). If (, * , ) is a Hom-algebra, then the following statements are equivalent. holds for all , , and ∈ . 

Proposition 9 .

 9 Let A = (, * , ) be a multiplicative Homalgebra. Define on A the ternary operation (, , ) fl (, , ) 


Proposition 11 .

 11 Let (, * ) be a Maltsev algebra and : → an algebra morphism. Let 0 = and, for any integer ≥ 1, = ∘ -1 . If one defines on a trilinear operation [, , ] by [, , ] = (2 * - * - * ) 


(

  HB1) ([, ]) = [(), ()]. (HB2) ((, , )) = ((), (), ()). (HB3) [, ] = -[, ]. (HB4) (, , ) = -(, , ). (HB5) ↺ ,, (, , ) = 0. (HB6) ((), (), [, V]) = [(, , ), 2 (V)] + [ 2 (), (, , V)] + ((), (V), [, ]) -[[(), (V)], [(), ()]]. 




  2 () , ( * )) + ( (V) , () , * ) * 3 ()] = 2 [- ( 2 () , ( * ) , () * (V)) - ( 2 () , ( * ) , () * (V)) - ( () , (V) , * ) * 3 () + ( (V) , () , * ) * 3 ()] = 2 [- ( () , * , (V)) * 3 () - ( () , * , (V)) * 3 () - ( () , (V) , * ) * 3 () + ( (V) , () , * ) * 3 ()] (applying (6) to - ( 2 () , ( * ) , () * (V)) - ( 2 () , ( * ) , () * (V))) = 0 (by the skew-symmetry of (, , )) . 


Proposition 16 .

 16 Let (, * ) be a Maltsev algebra and any selfmorphism of (, * ). If one defines on (, * ) a ternary operation "{, , } " by {, , } = 2 (2 * - * - * ) , (27) then (, {, , } , 2 ) is a multiplicative Hom-Lts. Proof. One knows ([20], Theorem 2.12) that, from (, * ) and any self-morphism of (, * ), we get a (multiplicative) Hom-Maltsev algebra (, * , ), where * = ( * ) for all , in . Next, if one defines on (, * , ) a ternary operation {, , } fl 2 ( * ) * () -( * ) * () -( * ) * () , (28) then, by Theorem 14, (, {, , } , 2 ) is a Hom-Lts and "{, , } " is expressed through " * " as {, , } = 2 ( ( * ) * ()) - ( ( * ) * ()) - ( ( * ) * ()) = 2 2 ( * ) - 2 ( * ) - 2 ( * ) = 2 (2 * - * - * ) . 


Theorem 17 .

 17 Let (, * , ) be a multiplicative Hom-Maltsev algebra. If one defines on (, * , ) a ternary operation (, , ) by (, , ) fl 1 3 {, , } , 


Corollary 19 .

 19 Hom-alternative algebra is Hom-Maltsev admissible ([20], Theorem 3.8), from Theorem 17 we have the following. Let (, * , ) be a multiplicative Hom-alternative algebra. Then (, [, ], (, , ) , ) is a Hom-Bol algebra, where (, , ) fl -(1/3)(2[[, ], ()] -[[, ], ()] -[[, ], ()]), for all , , and ∈ . The aim of Section 4 is a generalization of Corollary 19 to multiplicative right (or left) Hom-alternative algebras. 


Lemma 21 .

 21 (i) (, * , ) is right Hom-alternative. (ii) (, * , ) satisfies (, , ) = - (, , ) (37) for all , , and ∈ . (iii) (, * , ) satisfies () * ( + ) = * () + * () (38) for all , , and ∈ . Observe that if (, * , ) is a right Hom-alternative algebra, then (, * op , ) is a left Hom-alternative algebra, where * op fl * . So the mirrors of (37) and (38) hold for (, * op , ): as (, , ) = -as (, , ) , (39) (( * op ) + ( * op )) * op () = () * op ( * op ) + () * op ( * op ) . In any multiplicative right Hom-alternative algebra (, * , ), the identity ([, V] , () , ()) = [ (, , ) , 2 (V)] + [ 2 () , (V, , )] + ( (V) , () , [, ]) - ( () , (V) , [, ]) (41) 




  (10)6].Identities(9)and(10)for [, , ] are quite obvious.Next,[ () , () , [, V, ] ] = [ () , () , ([, V, ])] ([, , ]) , (V) , ()] + [ () , ([, , V]) , ()] + [ () , (V) , ([, , ])]  


	= 𝛼 2𝑛 ([𝑥, 𝑦, [𝑢, V, 𝑤]])
	= 𝛼 2𝑛 ([[𝑥, 𝑦, 𝑢] , V, 𝑤]) + 𝛼 2𝑛 ([𝑢, [𝑥, 𝑦, V] , 𝑤])
	+ 𝛼 2𝑛 ([𝑢, V, [𝑥, 𝑦, 𝑤]])
	= [𝛼

) for all , , and in , then (, [, , ] , ) is a multiplicative Hom-Lts.Proof.Let [, , ] = 2 * - * - * and then [, , ] = ([, , ]).We shall use the fact that (, [, , ])is a = [[, , ] , (V) , ()] + [ () , [, , V] , ()] + [ () , (V) , [, , ] ]  


  [20]So (, * , (, , ) , ) is a Hom-Bol algebra.Example 18.Let be a vector space with basis { 1 , 2 , 3 , 4 }.From[20](Example 2.13) we know that if one considers the linear map : → given by 


	International Journal of Mathematics and Mathematical Sciences
	which is (HB6) for (𝐴, * , (, , ) 𝛼 (𝑒 1 ) = 𝑒 1 + 𝑒 3 ;
	𝛼 (𝑒 2 ) = 2𝑒 2 + 2𝑒 4 ;
	𝛼 (𝑒 3 ) = -𝑒 3 ;
	𝛼 (𝑒 4 ) = -2𝑒 4
	(33)

, we get ( () , () , * V) = (, , ) * 2 (V) + 2 () * (, , V) + ( () , (V) , * ) -( () (V)) * ( () ()) 
 2 ()} + { 2 () , 2 (V) , {, , } } + [-2 ( ( () , (V) , * ) * 3 () + ( ( * V) , 2 () , ( * ))) + ( (V) , () , * ) * 3 () + ( (V * ) , 2 () , ( * )) + ( () , () , * ) * 3 (V) + ( ( * ) , 2 (V) , ( * ))] . (23) In this latest expression, denote by (, V, , , ) the expression in "[⋅ ⋅ ⋅ ]"; to conclude, we proceed to show that (, V, , , ) = 0. 
Observe first that, by (6), we have ( () , * , ()) * 2 ( (V)) 
+ ( (V) , * , ()) * 2 ( ()) 
= ( 2 () , ( * ) , (V) * ()) 
+ ( 2 (V) , ( * ) , () * ()) . 
( 
That is, ( () , () , * ) * 3 (V) 
+ ( ( * ) , 2 (V) , ( * )) 
= ( (V * ) , 2 () , ( * )) 
+ ( (V) , () , * ) * 3 () . 
( 
With this observation, the expression (, V, , , ) is transformed as follows: 
 (, V, , , ) = 2 [- ( () , (V) , * ) * 3 () - ( ( * V) , 2 () , ( * ))] 
Proof. The identity as (V, () , ()) = as (, , ) 2 (V) 
holds in any right Hom-alternative algebra (see [23], Theorem 7.1 (7.1.1c)). Next, in this identity, switching and V, we have as (V, () , ()) = as (V, , ) 2 () 
Then, subtracting memberwise this latter equality from the one above and using the linearity of as, we get (41). 
Note that in the when (, * , ) is a left Homalternative algebra, identity (41) reads as 
In any multiplicative right (or left) Hom-alternative algebra (, * , ) we consider the ternary operation defined by (12); that is, (, , ) fl as (, , ) , 
where as is the Hom-Jordan associator defined in Section 2. 
Observe that for = 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. 
for all , , and ∈ . 
Proof. (i) From (12) we have 
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 (, * , ) be a multiplicative right (resp., left) Hom-alternative algebra. If one defines on a ternary operation "(, , )" by (46) (resp., (47)), then (, (, , ), 2 ) is a Hom-Lts and (, [, ], (, , ), ) is a Hom-Bol algebra. 
Proof. We prove the theorem for a multiplicative right Homalternative algebra (, * , ) (the proof of the left case is the mirror of the right one). 
Identities (HB1) and (HB2) follow from the multiplicativity of (, * , ). 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 (, ∘, ), define a ternary operation by 
then (, [, , ], 2 ) is a multiplicative Hom-Lts (see [22] 
Multiplying each member of (41) by -2 and next using (50), we get 
That is, 
Therefore, (52) now reads 
and so (HB6) holds for (, [, ], (, , ), ). Thus we conclude that (, [, ], (, , ), ) is a Hom-Bol algebra. One gets the same result in the case when (, * , ) is a multiplicative left Homalternative algebra and essentially using (47) and (44). This finishes the proof. 
(again, only nonzero products are specified). Then (, * , ) is a multiplicative right Hom-alternative algebra (see [23], Example 2.9). Then, using [, ] = * - * and (46), one could find (although the computation is somewhat lengthy) all the nonzero products "[, ]" and "(, , )" with respect to the basis elements , , V, , and of . We just point out that they are nonzero products; for example, [, ] = -V, [, ] = - + , (, , ) = - -V, and (, , ) = - -. Therefore, Theorem 23 implies that (, [, ], (, , ), ) is a Hom-Bol algebra. 

The Construction of Bol Algebras from Right Alternative Algebras Revisited

As already mentioned in Section 2, for = 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 (, [, ], (, , )) in which is a vector space, "[, ]" a binary operation, and "(, , )" a ternary operation on such that 
(B5) (, , (, V, )) = ((, , ), V, ) + (, (, , V), ) + (, V, (, , )), for all , V, , , , ∈ . 
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 = id in Theorem 23 and specifying the right alternative case, we get the following. Proof. Identities (B1) and (B2) are obvious. For = id, the Hom-Jordan associator (see Definition 3) reduces to the usual Jordan associator as (, V, ) fl ( ∘ V) ∘ - ∘ (V ∘ ) in (, * ). The fact that (B3) and (B5) hold in (, * ) follows from the equality (, , ) = as (, , ) that holds in right alternative algebras (the untwisted form of ( 46)) and from that right alternative algebras are Jordan admissible [36]. Therefore (, (, , )) is a Lts since any Jordan algebra is a Lts with respect to the operation (, , ) = as (, , ) (see [1]). So we get the untwisted version of (HB5) and (HB7). 
In order to show that (B4) holds in (, * ), we proceed as follows. First, recall that the identity as (V, , ) = as (V, , ) + as (, V, ) + as (, V, )as (, V, ) 
holds in any algebra. Also, in a right alternative algebra (, * ) (over a ground field of characteristic different from 2), the following identity holds [37]: 
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 (, [, ], (, , )). Thus we get that (, [, ], (, , )) 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 (, * ) is a left alternative algebra, it is also possible to get a natural left Bol algebra structure on (, * ). Indeed, one needs to consider the counterparts of (, , ) and (63) that looks, respectively, as ( 
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). 

Conflicts of Interest

The authors declare that they have no conflicts of interests.
Sylvain Attan, A Nourou Issa
Hom-Lie Triple System and Hom-Bol Algebra Structures on Hom-Maltsev and Right Hom-Alternative Algebras