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

Authors: Sylvain Attan; A. Nourou Issa
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## Abstract

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.


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## Body

## 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 n-ary 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 fieldK 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 ofn-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) AHom-algebra is a triple (A,∗,α) in which A is a K-vector space, ∗:A×A→A a bilinear map (the binary operation), and α:A→A a linear map (the twisting map). The Hom-algebra A is said to bemultiplicative if α(x∗y)=α(x)∗α(y) for all x,y∈A.
(ii) TheHom-Jacobian in (A,∗,α) is the trilinear map Jα:A×A×A→A defined as Jα(x,y,z)≔↺x,y,z(x∗y)∗α(z), where ↺x,y,z denotes the sum over cyclic permutation of x,y,z.
(iii) TheHom-associator of a Hom-algebra (A,∗,α) is the trilinear map as:A⊗3→A defined as as(x,y,z)=(x∗y)∗α(z)-α(x)∗(y∗z). If as(x,y,z)=0 for all x,y,z∈A, then (A,∗,α) is said to beHom-associative.Remark 2.
Ifα=id (the identity map), then a Hom-algebra (A,∗,α) reduces to an ordinary algebra (A,∗), the Hom-Jacobian Jα is the ordinary Jacobian J, and the Hom-associator is the usual associator for the algebra (A,∗). One observes that, in general, the map α 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-algebraA≔(A,∗,α) are attached two Hom-algebras: thecommutator Hom-algebraA-≔(A,[,],α), where [x,y]≔x∗y-y∗x (the commutator of x and y), and theplus Hom-algebraA+≔(A,∘,α), where x∘y≔x∗y+y∗x (theJordan product) for all x,y∈A.For our purpose, we provide the following.Definition 3.
TheHom-Jordan associator of a Hom-algebra A≔(A,∗,α) is the trilinear map asJ:A⊗3→A defined as asJ(x,y,z)=(x∘y)∘α(z)-α(x)∘(y∘z), where “∘” is the Jordan product on A.Ifα=id, the Hom-Jordan associator reduces to the usual Jordan associator.Definition 4.
(i) AHom-Lie algebra is a Hom-algebra (A,∗,α) such that the binary operation “∗” is anticommutative and theHom-Jacobi identity(1)Jαx,y,z=0holds for all x,y, and z in A ([9]).
(ii) AHom-Maltsev algebra is a Hom-algebra (A,∗,α) such that the binary operation “∗” is anticommutative and that theHom-Maltsev identity(2)Jααx,αy,x∗z=Jαx,y,z∗α2xholds for all x,y,z in A ([20]).
(iii) AHom-Jordan algebra is a Hom-algebra (A,∗,α) such that (A,∗) is a commutative algebra and theHom-Jordan identity(3)asx∗x,αy,αx=0is satisfied for all x,y in A ([20]).
(iv) A Hom-algebra(A,∗,α) is called aright Hom-alternative algebra if (4)asx,y,y=0for all x,y in A. A Hom-algebra (A,∗,α) is called aleft Hom-alternative algebra if (5)asx,x,y=0for all x,y in A. A Hom-algebra (A,∗,α) is called aHom-alternative algebra if it is both right and left Hom-alternative [18].Remark 5.
Whenα=id, the Hom-Jacobi identity (1) is the usualJacobi identityJ(x,y,z)=0. Likewise, for α=id, the Hom-Maltsev identity (2) reduces to theMaltsev identityJ(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 α=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 isHom-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 isHom-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 (6)Jααx,αy,w∗z+Jααw,αy,x∗z=Jαx,y,z∗α2w+Jαw,y,z∗α2xis equivalent to (2) in any anticommutative Hom-algebra (A,∗,α) ([20], Proposition 2.7). In [34], it is proved that, in any anticommutative Hom-algebra (A,∗,α), the Hom-Maltsev identity (2) is equivalent to (7)Jααx,αy,u∗v=α2u∗Jαx,y,v+Jαx,y,u∗α2v-2Jααu,αv,x∗y.Therefore, apart from (2), identities (6) and (7) may be taken as defining identities of a Hom-Maltsev algebra.The Hom-algebras mentioned above arebinary 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]).
Aternary Hom-Nambu algebra is a triple (A,[,,],α) in which A is a K-vector space, [,,]:A×A×A→A is a trilinear map, and α=(α1,α2) is a pair of linear maps (the twisting maps) such that the identity (8)α1x,α2y,u,v,w=x,y,u,α1v,α2w+α1u,x,y,v,α2w+α1u,α2v,x,y,wholds for all u,v,w,x, and y in A. Identity (8) is called theternary Hom-Nambu identity.Remark 7.
Whenα1=id=α2 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]).
AHom-Lie triple system (Hom-Lts) is a ternary Hom-algebra (A,[,,],α=(α1,α2)) such that (9)x,y,z=-y,x,z,(10)↺x,y,zx,y,z=0,and the ternary Hom-Nambu identity (8) holds in (A,[,,],α=(α1,α2)).One notes that when the twisting mapsα1,α2 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 mapsαi are all equal, α1=α2=α and α([x,y,z])=[α(x),α(y),α(z)] for all x,y, and z in A. The Hom-Lie triple system (A,[,,],α) is then said to bemultiplicative [21]. In case of multiplicativity, the ternary Hom-Nambu identity (8) then reads (11)αx,αy,u,v,w=x,y,u,αv,αw+αu,x,y,v,αw+αu,αv,x,y,w.In [14] a (multiplicative)Hom-triple system is defined as a (multiplicative) ternary Hom-algebra (A,[,,],α) 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,[,,],α)). 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],α(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.
LetA=(A,∗,α) be a multiplicative Hom-algebra. Define on A the ternary operation(12)x,y,z≔asJy,z,xfor all x,y, and z∈A. Then (A,(,,),α) 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∗α(z) means (x∗y)∗α(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 α:A→A an algebra morphism. Then Aα≔(A,[,,]α,α) is a multiplicative Hom-Lts, where [x,y,z]α=α(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 α:A→A an algebra morphism. Let α0=id and, for any integer n≥1, αn=α∘αn-1. If one defines on A a trilinear operation [,,]αn by (13)x,y,zαn=αn2xy∗z-yz∗x-zx∗yfor all x,y, and z in A, then (A,[,,]αn,αn) is a multiplicative Hom-Lts.Proof.
Let[x,y,z]=2xy∗z-yz∗x-zx∗y and then [x,y,z]αn=αn([x,y,z]). We shall use the fact that (A,[,,]) is a Lts [6]. Identities (9) and (10) for [x,y,z]αn are quite obvious. Next, (14)αnx,αny,u,v,wαnαn=αnx,αny,αnu,v,wαn=α2nx,y,u,v,w=α2nx,y,u,v,w+α2nu,x,y,v,w+α2nu,v,x,y,w=αnx,y,u,αnv,αnwαn+αnu,αnx,y,v,αnwαn+αnu,αnv,αnx,y,wαn=x,y,uαn,αnv,αnwαn+αnu,x,y,vαn,αnwαn+αnu,αnv,x,y,wαnαnand so (11) holds for [,,]αn. Thus (A,[,,]αn,αn) 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]).
AHom-Bol algebra is a quadruple (A,[,],(,,),α) in which A is a vector space, “[,]” a binary operation, “(,,)” a ternary operation on A, and α:A→A a linear map such that(HB1)
α([x,y])=[α(x),α(y)].(HB2)
α((x,y,z))=(α(x),α(y),α(z)).(HB3)
[x,y]=-[y,x].(HB4)
(x,y,z)=-(y,x,z).(HB5)
↺x,y,z(x,y,z)=0.(HB6)
αx,αy,u,v=x,y,u,α2v+α2u,x,y,v+αu,αv,x,y-[αu,αv,[αx, αy]].(HB7)
(α2(x),α2(y),(u,v,w))=((x,y,u),α2(v),α2(w))+(α2(u),(x,y,v),α2(w))+(α2(u),α2(v),(x,y,w))for allu,v,w,x,y, and z∈A.Identities (HB1) and (HB2) mean the multiplicativity of(A,[,],(,,),α). It is built into our definition for convenience.One observes that forα=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∈A, then (A,[,],(,,),α) becomes a (multiplicative) Hom-Lts (A,(,,),α2).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 (Theorem14) 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: (15)x,y,z=2xy∗z-yz∗x-zx∗y.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,∗,α): (16)x,y,zα=2xy∗αz-yz∗αx-zx∗αy.From (16) it clearly follows that {,,}α can also be written as (17)x,y,zα=-Jαx,y,z+3xy∗αz.One observes that when α=id, we recover product (15). First, we prove the following.Lemma 13.
Let(A,∗,α) be a Hom-Maltsev algebra. If one defines on (A,∗,α) a ternary operation “{,,}α” by (16), then (18)αx,αy,u∗vα=α2u∗x,y,vα+x,y,uα∗α2v-Jααu,αv,x∗yfor all u,v,x, and y in A.Proof.
Let us write (7) as (19)-Jααx,αy,u∗v=-Jαx,y,u∗α2v+α2u∗-Jαx,y,v+3Jααu,αv,x∗y-Jααu,αv,x∗y.That is,(20)-Jααx,αy,u∗v=-Jαx,y,u∗α2v+α2u∗-Jαx,y,v+3αuαv∗αx∗y+3αv∗xy∗α2u+3xy∗αu∗α2v-Jααu,αv,x∗y.Therefore, by multiplicativity, we have (21)-Jααx,αy,u∗v+3αxαy∗αu∗v=-Jαx,y,u+3xy∗αu∗α2v+α2u∗-Jαx,y,v+3xy∗αv-Jααu,αv,x∗yand so, we get (18) by (17).We now prove the following.Theorem 14.
Let(A,∗,α) be a multiplicative Hom-Maltsev algebra. If one defines on (A,∗,α) a ternary operation “{,,}α” by (16), then (A,{,,}α,α2) 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,∗,α).
First observe that the multiplicativity of(A,∗,α) implies that α2({x,y,z}α)={α2(x),α2(y),α2(z)}α, with x,y, and z in A.
From the skew-symmetry of “∗” and Jα(x,y,z), it clearly follows from (17) that {x,y,z}α=-{y,x,z}α which is (9) for “,,α.”
Next, using (17) and the skew-symmetry of Jα(x,y,z) where applicable, we compute(22)x,y,zα+y,z,xα+z,x,yα=-Jαx,y,z+3xy∗αz-Jαy,z,x+3yz∗αx-Jαz,x,y+3zx∗αy=-3Jαx,y,z+3Jαx,y,z=0and thus ↺x,y,z{x,y,z}α=0, so we get (10) for “{,,}α.”
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.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(24)Jααu,x∗y,αw∗α2αv+Jααv,x∗y,αw∗α2αu=Jαα2u,αx∗y,αv∗αw+Jαα2v,αx∗y,αu∗αw.That is,(25)Jααw,αu,x∗y∗α3v+Jααw∗u,α2v,αx∗y=Jααv∗w,α2u,αx∗y+Jααv,αw,x∗y∗α3u. With this observation, the expression N(u,v,w,x,y) is transformed as follows:(26)Nu,v,w,x,y=2-Jααu,αv,x∗y∗α3w-Jααu∗v,α2w,αx∗y+2Jααv∗w,α2u,αx∗y+Jααv,αw,x∗y∗α3u=2-Jαα2w,αx∗y,αu∗αv-Jαα2u,αx∗y,αw∗αv-Jααu,αv,x∗y∗α3w+Jααv,αw,x∗y∗α3u=2-Jααw,x∗y,αv∗α3u-Jααu,x∗y,αv∗α3w-Jααu,αv,x∗y∗α3w+Jααv,αw,x∗y∗α3uapplying6to-Jαα2w,αx∗y,αu∗αv-Jαα2u,αx∗y,αw∗αv=0bytheskew-symmetryofJαx,y,z. Therefore, we obtain that (11) holds for “{,,}α” and we conclude that (A,{,,}α,α2) 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 α any self-morphism of (A,∗). If one defines on (A,∗) a ternary operation “{,,}α” by(27)x,y,zα=α22xy∗z-yz∗x-zx∗y,then (A,{,,}α,α2) is a multiplicative Hom-Lts.Proof.
One knows ([20], Theorem 2.12) that, from (A,∗) and any self-morphism α of (A,∗), we get a (multiplicative) Hom-Maltsev algebra (A,∗~,α), where x∗~y=α(x∗y) for all x,y in A. Next, if one defines on (A,∗~,α) a ternary operation(28)x,y,zα≔2x∗~y∗~αz-y∗~z∗~αx-z∗~x∗~αy,then, by Theorem 14, (A,{,,}α,α2) is a Hom-Lts and “{,,}α” is expressed through “∗” as(29)x,y,zα=2ααx∗y∗αz-ααy∗z∗αx-ααz∗x∗αy=2α2xy∗z-α2yz∗x-α2zx∗y=α22xy∗z-yz∗x-zx∗y.Observe that though constructed in quite a different way, the operation “{,,}α” in Proposition 16 above coincides with “[,,]αn” in Proposition 11 for n=2.Combining Lemma13 and Theorem 14, we get the following result.Theorem 17.
Let(A,∗,α) be a multiplicative Hom-Maltsev algebra. If one defines on (A,∗,α) a ternary operation (,,)α by (30)x,y,zα≔13x,y,zα,where “{,,}α” is defined by (17), then (A,∗,(,,)α,α) is a Hom-Bol algebra.Proof.
Definition (30) and Theorem 14 imply that (A,(,,)α,α2) is a multiplicative Hom-Lts; that is, (HB4), (HB5), and (HB7) hold for (A,∗,(,,)α,α). Now, (HB1), (HB2), and (HB3) are, respectively, the multiplicativity and skew-symmetry of “∗”; next, we are done if we prove (HB6) for (A,∗,(,,)α,α).
From (17) and multiplicativity we have (31)-Jααu,αv,x∗y=αu,αv,x∗yα-3αuαv∗αxαyand then (18) takes the form (32)αx,αy,u∗vα=x,y,uα∗α2v+α2u∗x,y,vα+αu,αv,x∗yα-3αuαv∗αxαy.Multiplying by 1/3 each member of this latter equality and using (30), we get (33)αx,αy,u∗vα=x,y,uα∗α2v+α2u∗x,y,vα+αu,αv,x∗yα-αuαv∗αxαywhich is (HB6) for (A,∗,(,,)α,α). So (A,∗,(,,)α,α) is a Hom-Bol algebra.Example 18.
LetA be a vector space with basis {e1,e2,e3,e4}. From [20] (Example 2.13) we know that if one considers the linear map α:A→A given by (34)αe1=e1+e3;αe2=2e2+2e4;αe3=-e3;αe4=-2e4and the multiplication table given by (35)e1∗e2=-2e2-2e4=-e2∗e1;e1∗e3=e3=-e3∗e1;e1∗e4=-2e4=-e4∗e1;e2∗e3=-4e4=-e3∗e2(only nonzero products are specified), then (A,∗,α) is a multiplicative Hom-Maltsev algebra. It is observed that (A,∗,α) 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)α on A with respect to the basis elements are (36)e1,e2,e1α=-4e2=-e2,e1,e1α;e1,e3,e1α=-e3=-e3,e1,e1α;e1,e4,e1α=-4e4=-e4,e1,e1α.By Theorem 17 we get that (A,∗,(,,)α,α) 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,∗,α) be a multiplicative Hom-alternative algebra. Then (A,[,],(,,)α,α) is a Hom-Bol algebra, where (x,y,z)α≔-1/32[[x,y],α(z)]-[[y,z],α(x)]-[[z,x],α(y)], for all x,y, and z∈A.The aim of Section4 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 identityas(x,y,y)=0 is given by the following result.Lemma 20 (see [18]).
If(A,∗,α) is a Hom-algebra, then the following statements are equivalent.
(i)(A,∗,α) is right Hom-alternative.
(ii)(A,∗,α) satisfies (37)asx,y,z=-asx,z,yfor all x,y, and z∈A.
(iii)(A,∗,α) satisfies (38)αx∗yz+zy=xy∗αz+xz∗αyfor all x,y, and z∈A.Observe that if(A,∗,α) is a right Hom-alternative algebra, then (A,∗op,α) is a left Hom-alternative algebra, where x∗opy≔y∗x. So the mirrors of (37) and (38) hold for (A,∗op,α): (39)asx,y,z=-asy,x,z,(40)x∗opy+y∗opx∗opαz=αx∗opy∗opz+αy∗opx∗opz.Now we have the following.Lemma 21.
In any multiplicative right Hom-alternative algebra(A,∗,α), the identity(41)asu,v,αx,αy=asu,x,y,α2v+α2u,asv,x,y+asαv,αu,x,y-asαu,αv,x,yholds for all x,y, and z∈A.Proof.
The identity(42)asuv,αx,αy=asu,x,yα2v+α2uasv,x,y-asαu,αv,x,yholds in any right Hom-alternative algebra (see [23], Theorem 7.1 (7.1.1c)). Next, in this identity, switching u and v, we have (43)asvu,αx,αy=asv,x,yα2u+α2vasu,x,y-asαv,αu,x,y.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,∗,α) is a left Hom-alternative algebra, identity (41) reads as (44)asαx,αy,u,v=asx,y,u,α2v+α2u,asx,y,v+asx,y,αv,αu-asx,y,αu,αv.In any multiplicative right (or left) Hom-alternative algebra(A,∗,α) we consider the ternary operation defined by (12); that is, (45)x,y,z≔asJy,z,x,where asJ is the Hom-Jordan associator defined in Section 2. Observe that for α=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,∗,α) is a multiplicative right Hom-alternative algebra, then(46)x,y,z=x,y,αz-2asz,x,yfor all x,y, and z∈A.
(ii) If(A,∗,α) is a multiplicative left Hom-alternative algebra, then(47)x,y,z=x,y,αz-2asx,y,zfor all x,y, and z∈A.Proof.
(i) From (12) we have(48)x,y,z=y∘z∘αx-αy∘z∘x=y∗z+z∗y∗αx+αx∗y∗z+z∗y-αy∗z∗x+x∗z-z∗x+x∗z∗αy=y∗z+z∗y∗αx+x∗y∗αz+x∗z∗αy-y∗z∗αx+y∗x∗αz-z∗x+x∗z∗αyby38=z∗y∗αx+x∗y∗αz-y∗x∗αz-z∗x∗αy=z∗y∗αx-z∗x∗αy+x,y∗αz=z∗y∗αx-z∗x∗αy+x,y,αz+αz∗x,y=x,y,αz+z∗y∗αx-αz∗y∗x-z∗x∗αy+αz∗x∗y=x,y,αz+asz,y,x-asz,x,y=x,y,αz-2asz,x,yby37 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,∗,α) be a multiplicative right (resp., left) Hom-alternative algebra. If one defines on A a ternary operation “(,,)” by (46) (resp., (47)), then (A,(,,),α2) is a Hom-Lts and (A,[,],(,,),α) is a Hom-Bol algebra.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).
Identities (HB1) and (HB2) follow from the multiplicativity of(A,∗,α). 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,∘,α), define a ternary operation by (49)x,y,z≔2αx∘y∘z-αy∘x∘z,then (A,[,,],α2) is a multiplicative Hom-Lts (see [22], Corollary 4.1). Now, observe that [x,y,z]=2asJ(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,(,,),α2) is a multiplicative Hom-Lts and so identity (HB7) holds for (A,[,],(,,),α).
Next,(A,[,],(,,),α) is a Hom-Bol algebra if we prove that (HB6) additionally holds.
Write (46) as(50)-2asz,x,y=x,y,z-x,y,αz.Multiplying each member of (41) by -2 and next using (50), we get (51)αx,αy,u,v-αx,αy,αu,v=x,y,u-x,y,αu,α2v+α2u,x,y,v-x,y,αv+αu,x,y,αv-αu,x,y,α2v-αv,x,y,αu+αv,x,y,α2u.That is,(52)αx,αy,u,v=x,y,u,α2v+α2u,x,y,v-x,y,αu,αv+x,y,αv,αu+αx,y,u,v.Observe that(53)-x,y,αu,αv+x,y,αv,αu=αu,x,y,αv+x,y,αv,αu=-αv,αu,x,ysince↺a,b,ca,b,c=0byHB5=αu,αv,x,y.Therefore, (52) now reads (54)αx,αy,u,v=x,y,u,α2v+α2u,x,y,v+αu,αv,x,y-αu,v,x,yand so (HB6) holds for (A,[,],(,,),α). Thus we conclude that (A,[,],(,,),α) is a Hom-Bol algebra. One gets the same result in the case when (A,∗,α) is a multiplicative left Hom-alternative algebra and essentially using (47) and (44). This finishes the proof.Example 24.
LetA be a five-dimensional vector space with basis {e,u,v,w,z} and let α:A→A be a linear map given by (55)αe=e+u+v;αu=-u;αv=-v;αw=-w;αz=-z.Define on A a binary operation “∗” by (56)e∗e=e+u+v;e∗u=-v;e∗w=-w+z;e∗z=-z;u∗e=-u;z∗e=-z(again, only nonzero products are specified). Then (A,∗,α) 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,[,],(,,),α) is a Hom-Bol algebra.


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

As already mentioned in Section2, for α=id in Definition 12 we get the definition of a left Bol algebra.Definition 25 (see [25, 27]).
Aleft 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)
↺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 allu,v,w,x,y,z∈A.In this section we show how the construction of Hom-Bol algebras from right or left Hom-alternative algebras described in Section4 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 α=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 (57)x,y,z=x,y,z-2asz,x,y,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. Forα=id, the Hom-Jordan associator (see Definition 3) reduces to the usual Jordan associator asJ(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)=asJ(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)=asJ(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(58)asuv,y,x=uasv,y,x+asu,v,yx+asu,vy,x-asu,v,yxholds in any algebra. Also, in a right alternative algebra (A,∗) (over a ground field of characteristic different from 2), the following identity holds [37]:(59)asu,v,v∗y=asu,v,y∗v;that is, by linearization and right alternativity, (60)asu,v∗y,x=asu,v,x∗y-asu,v,y∗x-asu,x,y∗v.Putting (60) in (58), we get (61)asu∗v,y,x=u∗asv,y,x-asu,x,y∗v+asu,v,x,y;that is, by right alternativity, (62)asu∗v,x,y=asu,x,y∗v+u∗asv,x,y-asu,v,x,y.Now, in (62) switching u and v and then subtracting the obtained equality from (62), one gets (63)asu,v,x,y=asu,x,y,v+u,asv,x,y+asv,u,x,y-asu,v,x,y(observe that (63) is the untwisted form of (41)). Next, write (57) as(64)-2asz,x,y=x,y,z-x,y,z.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 Theorem26 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(65)x,y,z=x,y,z-asx,y,z(the untwisted version of (47)) and (66)asx,y,u,v=asx,y,u,v+u,asx,y,v+asx,y,v,u-asx,y,u,v.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).



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