# Introduction togb-Triple Systems

Authors: Guy Roger Biyogmam
Journal: ISRN Algebra
 (2014)

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## Abstract

This paper introduces the category ofgb-triple systems and studies some of their algebraic properties. Also provided is a functor from this category to the category of Leibniz algebras.


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## Body

## 1. Introduction

A Triple system is a vector space𝔤 over a field 𝔎 together with a 𝔎-trilinear map T:𝔤⊗3→𝔤. Among the many examples known in the literature, one may mention 3-Lie algebras [1] and Lie triple systems [2] which are the generalizations of Lie algebras to ternary algebras, Jordan triple systems [2] which are the generalizations of Jordan algebras, and Leibniz 3-algebras [3] and Leibniz triple systems [4] which are generalizations of Leibniz algebras [5]. In this paper we enrich the family of triple systems by introducing the concept of gb-triple systems, presented as another generalization of Leibniz algebras with the particularity that, for all a,b∈𝔤, the map T:𝔤→𝔤, defined by Ta,b(x)=T(a,x,b), is a derivation of 𝔤, a property of great importance in Nambu Mechanics. We investigate some of their algebraic properties and provide a functorial connection with Leibniz algebras and Lie algebras.For the remaining of this paper, we assume that𝔎 is a field of characteristic different to 2 and all tensor products are taken over 𝔎.Definition 1.
Agb-triple system is a 𝔎-vector space 𝔤 equipped with a trilinear operation
(1)[-,-,-]𝔤:𝔤⊗3⟶𝔤satisfyingtheidentity(2)[x,y,[a,b,c]𝔤]𝔤=[a,[x,y,b]𝔤,c]𝔤-[[a,x,c]𝔤,y,b]𝔤-[x,[a,y,c]𝔤,b]𝔤.Example 2.
Let𝔤 be an n-dimensional vector space with basis {a1,a2,…,an}. Define on 𝔤⊗3 the bracket [-,-,-] by
(3)[ai,aj,ak]𝔤={ap,ifi=p,j=q,k≠p,q-ap,ifi=q,j=p,k≠p,q0,else
for fixed p≠q∈{1,2,…,n}. It is easy to check that the identity (2) is satisfied. So 𝔤 is a gb-triple system when endowed with the operation [-,-,-]𝔤.Because of the resemblance between the identity (2) and the generalized Leibniz identity [3], it is worth mentioning that, in general, Leibniz 3-algebras do not coincide with gb-triple systems. The following example provides a Leibniz 3-algebra that is not a gb-triple system.Example 3.
The two-dimensional complex Leibniz 3-algebraŁ (see [6, Theorem 2.14]) with basis {a1,a2}, dim⁡([Ł,Ł,Ł]Ł)=1, and brackets
(4)[ai,aj,ak]𝔤={a1,ifi=1,j,k=2αa1,ifi,j,k=20,else,
with α≠0, is not a gb-triple system. It is easy to check that its bracket does not satisfy the identity (2).Definition 4.
Let𝔤,𝔤′ be gb-triple systems. A function α:𝔤→𝔤′ is said to be a homomorphism of gb-triple systems if
(5)α([x,y,z]𝔤)=[α(x),α(y),α(z)]𝔤′∀x,y,z∈𝔤.
We may thus form the categorygb-TS of gb-triple systems and gb-triple system homomorphisms.Recall that if𝔤 is a vector space endowed with a trilinear operation σ:𝔤×𝔤×𝔤→𝔤, then a map D:𝔤→𝔤 is called a derivation with respect to σ if
(6)D(σ(a,b,c))=σ(D(a),b,c)+σ(a,D(b),c)+σ(a,b,D(c)).Lemma 5.
Let𝔤 be a gb-triple system and g,h∈𝔤. Then the map Dgh defined on 𝔤 by
(7)Dgh(x)=[g,x,h]𝔤
is a derivation with respect to the bracket [-,-,-]𝔤 of 𝔤.Proof.
By settingσ=[-,-,-]𝔤 and using the identity (2), we have
(8)σ(x,y,Dgh(a))=[x,y,[g,a,h]𝔤]𝔤=[g,[x,y,a]𝔤,h]𝔤-[[g,x,h]𝔤,y,a]𝔤-[x,[g,y,h]𝔤,a]𝔤=Dgh(σ(x,y,a))-σ(Dgh(x),y,a)-σ(x,Dgh(y),a).Definition 6.
A subspaceI of a gb-triple system 𝔤 is a subalgebra of 𝔤 if I is a gb-triple system when endowed with the trilinear operation of 𝔤.Definition 7.
A subalgebraI of a gb-triple system 𝔤 is called ideal (resp., left ideal, resp., right ideal) of 𝔤 if it satisfies the condition [𝔤,I,𝔤]𝔤⊆I (resp., [𝔤,𝔤,I]𝔤⊆I, resp., [I,𝔤,𝔤]𝔤⊆I). If I satisfies the three conditions, then I is called a 3-sided ideal.Note that none of these three conditions implies the others as in the case of Lie triple systems.Example 8.
In Example2, the subspace ℑ1 with basis {ap,aq} is an ideal of 𝔤. However the subspace ℑ2 with basis {aq} is not an ideal of 𝔤, since, for k≠p,q, we have [ap,aq,ak]=ap∉ℑ2.Definition 9.
Given agb-triple system 𝔤, one defines the center of 𝔤 and the derived algebra of 𝔤, respectively, by
(9)Z(𝔤)={x∈𝔤:[𝔤,x,𝔤]𝔤=0},[𝔤,𝔤,𝔤]={[a1,a2,a3]𝔤,a1,a2,a3∈𝔤}.Lemma 10.
For agb-triple system 𝔤, Z(𝔤) and [𝔤,𝔤,𝔤] are ideals of 𝔤.Proof.
Clearly,[𝔤,Z(𝔤),𝔤]𝔤=0. So Z(𝔤) is an ideal of 𝔤. That [𝔤,𝔤,𝔤]𝔤 is an ideal follows from the fact that 𝔤 is closed under the operation [-,-,-]𝔤.The following theorem classifies a subfamily of two-dimensional complexgb-triple systems. This result was obtained by Camacho et al. in [6] for Leibniz 3-algebras.Theorem 11.
Up to isomorphism, there are seven two-dimensional complexgb-triple systems with one-dimensional derived algebra.Proof.
The proof is similar to [6, Theorem 2.14]. Let 𝔤 be a gb-triple system with basis {a1,a2}, and assume that dim⁡([𝔤,𝔤,𝔤)]𝔤)=1. Then write [ai,aj,ak]𝔤=αijka1, i,j,k=1,2. Then, using the identity (2), the only possible nonzero coefficients yield to the system of equations
(10)α221(α122+α212+α221)=0,α122(α122+α212+α221)=0,α222(α122+α221)=0,
for which the solution provides the following gb-triple systems with bracket operations:
(11)𝔤1:[ai,aj,ak]𝔤={αa1,ifi,k=2,j=10,else,𝔤2:[ai,aj,ak]𝔤={αa1,ifi,j,k=20,else,𝔤3:[ai,aj,ak]𝔤={a1,ifi=1,j,k=2-a1,ifi,j=2,k=10,else,𝔤4:[ai,aj,ak]𝔤={a1,ifi,k=2,j=1-a1,ifi,j=2,k=10,else,𝔤5:[ai,aj,ak]𝔤={a1,ifi,k=2,j=1αa1,ifi,j,k=20,else,𝔤6:[ai,aj,ak]𝔤={a1,ifi=1,j,k=2-a1,ifi,j=2,k=1αa1,ifi,j,k=2,𝔤7:[ai,aj,ak]𝔤={-(1+α)a1,ifi=1,j,k=2a1,ifi,k=2,j=1αa1,ifi,j=2,k=1,
with α≠0.Definition 12.
Given agb-triple system 𝔤, one defines the left center and the right center of 𝔤, respectively, by
(12)ZL(𝔤)={x∈𝔤:[x,𝔤,𝔤]𝔤=0},ZR(𝔤)={x∈𝔤:[𝔤,𝔤,x]𝔤=0}.Lemma 13.
The left centerZL(𝔤) and the right center ZR(𝔤) are 3-sided ideals of 𝔤.Proof.
To show thatZL(𝔤) is an ideal of 𝔤, let g,g′∈𝔤 and let x∈ZL(𝔤). Then, for every u,v∈𝔤, we have, by the identity (2),
(13)[[g,x,g′]𝔤,u,v]𝔤=[g,[x,u,v]𝔤︸0,g′]𝔤-[x,[g,u,g′],v]𝔤︸0-[x,u,[g,v,g′]𝔤]𝔤︸0=0.
So [𝔤,ZL(𝔤),𝔤]𝔤⊆ZL(𝔤). The proof that ZL(𝔤) is both left ideal and right ideal is similar, so is the case for ZR(𝔤).Definition 14.
Given agb-triple system 𝔤, we define left and right centralizers of a subalgebra S in 𝔤 by
(14)C𝔤l(S)={x∈𝔤:[x,S,𝔤]𝔤=0},(15)C𝔤r(S)={x∈𝔤:[𝔤,S,x]𝔤=0},
respectively.Lemma 15.
LetS be an ideal of a gb-triple system 𝔤. Then C𝔤r(S) and C𝔤l(S) are also ideals of 𝔤.Proof.
To show thatC𝔤r(S) is an ideal of 𝔤, let x∈C𝔤r(S), u∈S, and g,a,b∈𝔤. Then, by the identity (2),
(16)[g,u,[a,x,b]𝔤]𝔤=[a,[g,u,x]︸0,b]-[[a,g,b]𝔤,u,x]𝔤︸0-[g,[a,u,b]𝔤︸∈S,x]𝔤︸0=0.
So [𝔤,C𝔤r(S),𝔤]𝔤⊆C𝔤r(S). The proof for C𝔤l(S) is similar.Definition 16.
For agb-triple system 𝔤 and a subalgebra S of 𝔤, we define the left normalizer of S in 𝔤 by
(17)𝔑𝔤l(S):={x∈𝔤:[x,S,𝔤]⊆S},
and the right normalizer of S in 𝔤 by
(18)𝔑𝔤r(S):={x∈𝔤:[𝔤,S,x]⊆S}.Lemma 17.
LetS be a subalgebra of a gb-triple system 𝔤. Then 𝔑𝔤l(S) and 𝔑𝔤r(S) are also subalgebras of 𝔤.Proof.
To show that𝔑𝔤r(S) is a subalgebra of 𝔤, let x,y,z∈𝔑𝔤r(S), u∈S, and g∈𝔤. Then, by the identity (2), we have
(19)[g,u,[x,y,z]𝔤]𝔤=[x,[g,u,y]𝔤︸∈S,z]𝔤-[[x,g,z]𝔤,u,y]𝔤-[g,[x,u,z]𝔤︸∈S,y]𝔤∈S.
So [𝔑𝔤r(S),𝔑𝔤r(S),𝔑𝔤r(S)]𝔤⊆𝔑𝔤r(S). The proof for 𝔑𝔤l(S) is similar.Remark 18.
IfS is an ideal, then 𝔑𝔤l(S)=𝔤=𝔑𝔤r(S).


## 2. Fromgb-Triple Systems to Leibniz Algebras

Recall that a Leibniz algebra (sometimes called Loday algebra, named after Jean-Louis Loday) is a𝔎 vector space with a bilinear product [-,-] satisfying the Leibniz identity
(20)[x,[y,z]]=[[x,y]z]+[y,[x,z]].Proposition 19.
Let𝔤 be a gb-triple system. Define on 𝔤⊗2 the bracket operation [-,-] by
(21)[a1⊗a2,b1⊗b2]=[a1,b1,a2]𝔤⊗b2+b1⊗[a1,b2,a2]𝔤.
Then [-,-] satisfies the Leibniz identity.Proof.
On one hand, we have(22)[a1⊗a2,[b1⊗b2,c1⊗c2]]=[a1⊗a2,[b1,c1,b2]𝔤⊗c2]+[a1⊗a2,c1⊗[b1,c2,b2]𝔤]=[a1,[b1,c1,b2]𝔤,a2]𝔤⊗c2+[b1,c1,b2]𝔤⊗[a1,c2,a2]𝔤+[a1,c1,a2]𝔤⊗[b1,c2,b2]𝔤+c1⊗[a1,[b1,c2,b2]𝔤,a2]𝔤.
Also,
(23)[[a1⊗a2,b1⊗b2],c1⊗c2]=[[a1,b1,a2]𝔤⊗b2,c1⊗c2]+[b1⊗[a1,b2,a2]𝔤,c1⊗c2]=[[a1,b1,a2]𝔤,c1,b2]𝔤⊗c2+c1⊗[[a1,b1,a2]𝔤,c2,b2]𝔤+[b1,c1,[a1,b2,a2]𝔤]𝔤⊗c2+c1⊗[b1,c2,[a1,b2,a2]𝔤]𝔤.
On the other hand,
(24)[b1⊗b2,[a1⊗a2,c1⊗c2]]=[b1⊗b2,[a1,c1,a2]𝔤⊗c2]+[b1⊗b2,c1⊗[a1,c2,a2]𝔤]=[b1,[a1,c1,a2]𝔤,b2]𝔤⊗c2+[a1,c1,a2]𝔤⊗[b1,c2,b2]𝔤+[b1,c1,b2]𝔤⊗[a1,c2,a2]𝔤+c1⊗[b1,[a1,c2,a2]𝔤,b2]𝔤.
One checks using the identity (2) that the equality
(25)[a1⊗a2,[b1⊗b2,c1⊗c2]]=[[a1⊗a2,b1⊗b2],c1⊗c2]+[b1⊗b2,[a1⊗a2,c1⊗c2]]
holds.Corollary 20.
Let𝔤 be a gb-triple system; then 𝔤⊗2 endowed with the bilinear map [-,-] has a Leibniz algebra structure.Proof.
This is a consequence of Proposition19.Similarly, we have the following.Corollary 21.
Let𝔤 be a gb-triple system; then 𝔤∧2 has a Leibniz algebra structure, when endowed with the bilinear map defined by
(26)[a1∧a2,b1∧b2]=[a1,b1,a2]𝔤∧b2+b1∧[a1,b2,a2]𝔤.These determine two functors from the categorygb-TS of gb-triple systems to the category Lb of Leibniz algebras.Definition 22.
Let𝔤 be a gb-triple system and Ł a Leibniz algebra. The action of Ł on 𝔤 is a map A:𝔤⊗Ł→𝔤 satisfying
(27)(1)A([g1,g2,g3]𝔤⊗x)=[A(g1⊗x),g2,g3]𝔤+[g1,A(g2⊗x),g3]𝔤+[g1,g2,A(g3⊗x)]𝔤,(28)(2)A(g⊗[x,y])=A(A(g⊗y)⊗x)-A(A(g⊗x)⊗y)
for all g1,g2,g3∈𝔤 and x,y∈Ł.Proposition 23.
Let𝔤 be a gb-triple system; then the Leibniz algebra 𝔤⊗2 acts on 𝔤 via the map A:𝔤⊗𝔤⊗2→𝔤 defined by A(z⊗g1⊗g2)=[g1,z,g2]𝔤.Proof.
The first condition of Definition22 follows by (2). To show (28), we have
(29)A(A(z⊗b1⊗b2)⊗a1⊗a2)-A(A(z⊗a1⊗a2)⊗b1⊗b2)=[a1,[b1,z,b2]𝔤,a2]𝔤-[b1,[a1,z,a2]𝔤,b2]𝔤=[[a1,b1,a2]𝔤,z,b2]𝔤+[b1,z,[a1,b2,a2]]𝔤by(2)=A(z⊗[a1,b1,a2]𝔤⊗b2)+A(z⊗b1⊗[a1,b2,a2]𝔤)=A(z⊗([a1,b1,a2]𝔤⊗b2+b1⊗[a1,b2,a2]𝔤))=A(z⊗[a1⊗a2,b1⊗b2])by(21).Now letg1,g2∈𝔤 and consider the map Ag1⊗g2:𝔤→𝔤 defined by Ag1⊗g2(z)=[g1,z,g2]𝔤, z∈𝔤. Clearly, this map is a derivation of 𝔤 as it is induced by the action (Proposition 23) defined above.Proposition 24.
For agb-triple system 𝔤, the subspace 𝔄(𝔤)={Ag1⊗g2∣g1,g2∈𝔤} is a Lie algebra with respect to the product
(30)[Aa1⊗a2,Ab1⊗b2]=Aa1⊗a2∘Ab1⊗b2-Ab1⊗b2∘Aa1⊗a2.
More precisely, it is an ideal of the Lie algebra Der(𝔤) of derivations of 𝔤.Proof.
To show that𝔄(𝔤) is a Lie subalgebra of Der(𝔤), let a1,a2,b1,b2∈𝔤. Then, for all z∈𝔤,
(31)[Aa1⊗a2,Ab1⊗b2]Der(𝔤)(z)=Aa1⊗a2∘Ab1⊗b2(z)-Ab1⊗b2∘Aa1⊗a2(z)=[a1,[b1,z,b2]𝔤,a2]𝔤-[b1,[a1,z,a2]𝔤,b2]𝔤=A(A(z⊗b1⊗b2)⊗a1⊗a2)-A(A(z⊗a1⊗a2)⊗b1⊗b2)=A(z⊗[a1⊗a2,b1⊗b2])byProposition23=A[a1⊗a2,b1⊗b2](z).
So 𝔄(𝔤) is closed under the bracket of Der(𝔤). Also, for any derivation ∈Der(𝔤), we have, for all z∈𝔤,
(32)[d,Aa1⊗a2]Der(𝔤)(z)=(d∘Aa1⊗a2)(z)-(Aa1⊗a2∘d)(z)=d(Aa1⊗a2(z))-Aa1⊗a2(d(z))=d([a1,z,a2]𝔤)-[a1,d(z),a2]𝔤=[d(a1),z,a2]𝔤+[a1,z,d(a2)]𝔤by(6)=Ad(a1)⊗a2(z)+Aa1⊗d(a2)(z).
Hence [d,Aa1⊗a2]Der(𝔤)=Ad(a1)⊗a2+Aa1⊗d(a2)∈𝔄(𝔤).



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