Research Article Introduction to -Triple Systems - Hindawi

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Feb 3, 2014 - ... of Mathematics, Southwestern Oklahoma State University, 100 Campus Drive, Weatherford, OK 73096, USA ...... International Journal of.
Hindawi Publishing Corporation ISRN Algebra Volume 2014, Article ID 738154, 5 pages http://dx.doi.org/10.1155/2014/738154

Research Article Introduction to 𝑔𝑏-Triple Systems Guy Roger Biyogmam Department of Mathematics, Southwestern Oklahoma State University, 100 Campus Drive, Weatherford, OK 73096, USA Correspondence should be addressed to Guy Roger Biyogmam; [email protected] Received 10 January 2014; Accepted 3 February 2014; Published 13 March 2014 Academic Editors: D. Herbera and H. Li Copyright Β© 2014 Guy Roger Biyogmam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper introduces the category of 𝑔𝑏-triple systems and studies some of their algebraic properties. Also provided is a functor from this category to the category of Leibniz algebras.

1. Introduction A Triple system is a vector space g over a field K together with a K-trilinear map 𝑇 : gβŠ—3 β†’ 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 𝑔𝑏triple systems, presented as another generalization of Leibniz algebras with the particularity that, for all π‘Ž, 𝑏 ∈ g, the map 𝑇 : g β†’ g, defined by π‘‡π‘Ž,𝑏 (π‘₯) = 𝑇(π‘Ž, π‘₯, 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 that K is a field of characteristic different to 2 and all tensor products are taken over K. Definition 1. A 𝑔𝑏-triple system is a K-vector space g equipped with a trilinear operation [βˆ’, βˆ’, βˆ’]g : gβŠ—3 󳨀→ g satisfying the identity

(1)

[π‘₯, 𝑦, [π‘Ž, 𝑏, 𝑐]g ]g = [π‘Ž, [π‘₯, 𝑦, 𝑏]g , 𝑐] βˆ’ [[π‘Ž, π‘₯, 𝑐]g , 𝑦, 𝑏]g g

βˆ’ [π‘₯, [π‘Ž, 𝑦, 𝑐]g , 𝑏] . g

(2)

Example 2. Let g be an 𝑛-dimensional vector space with basis {π‘Ž1 , π‘Ž2 , . . . , π‘Žπ‘› }. Define on gβŠ—3 the bracket [βˆ’, βˆ’, βˆ’] by if 𝑖 = 𝑝, 𝑗 = π‘ž, π‘˜ =ΜΈ 𝑝, π‘ž π‘Ž , { { 𝑝 [π‘Žπ‘– , π‘Žπ‘— , π‘Žπ‘˜ ]g = {βˆ’π‘Žπ‘ , if 𝑖 = π‘ž, 𝑗 = 𝑝, π‘˜ =ΜΈ 𝑝, π‘ž { else {0,

(3)

for fixed 𝑝 =ΜΈ π‘ž ∈ {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 [βˆ’, βˆ’, βˆ’]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 𝑔𝑏triple systems. The following example provides a Leibniz 3algebra that is not a 𝑔𝑏-triple system. Example 3. The two-dimensional complex Leibniz 3-algebra Ł (see [6, Theorem 2.14]) with basis {π‘Ž1 , π‘Ž2 }, dim([Ł, Ł, Ł]Ł ) = 1, and brackets π‘Ž, { { 1 [π‘Žπ‘– , π‘Žπ‘— , π‘Žπ‘˜ ]g = {π›Όπ‘Ž1 , { {0,

if 𝑖 = 1, 𝑗, π‘˜ = 2 if 𝑖, 𝑗, π‘˜ = 2 else,

(4)

with 𝛼 =ΜΈ 0, is not a 𝑔𝑏-triple system. It is easy to check that its bracket does not satisfy the identity (2).

2

ISRN Algebra

Definition 4. Let g, gσΈ€  be 𝑔𝑏-triple systems. A function 𝛼 : g β†’ gσΈ€  is said to be a homomorphism of 𝑔𝑏-triple systems if 𝛼 ([π‘₯, 𝑦, 𝑧]g ) = [𝛼 (π‘₯) , 𝛼 (𝑦) , 𝛼 (𝑧)]gσΈ€ 

βˆ€π‘₯, 𝑦, 𝑧 ∈ g.

(5)

We may thus form the category 𝑔𝑏-TS of 𝑔𝑏-triple systems and 𝑔𝑏-triple system homomorphisms.

Lemma 10. For a 𝑔𝑏-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 [βˆ’, βˆ’, βˆ’]g .

Recall that if g is a vector space endowed with a trilinear operation 𝜎 : g Γ— g Γ— g β†’ g, then a map 𝐷 : g β†’ g is called a derivation with respect to 𝜎 if

The following theorem classifies a subfamily of twodimensional complex 𝑔𝑏-triple systems. This result was obtained by Camacho et al. in [6] for Leibniz 3-algebras.

𝐷 (𝜎 (π‘Ž, 𝑏, 𝑐)) = 𝜎 (𝐷 (π‘Ž) , 𝑏, 𝑐) + 𝜎 (π‘Ž, 𝐷 (𝑏) , 𝑐)

Theorem 11. Up to isomorphism, there are seven twodimensional complex 𝑔𝑏-triple systems with one-dimensional derived algebra.

(6)

+ 𝜎 (π‘Ž, 𝑏, 𝐷 (𝑐)) .

Lemma 5. Let g be a 𝑔𝑏-triple system and 𝑔, β„Ž ∈ g. Then the map π·π‘”β„Ž defined on g by π·π‘”β„Ž (π‘₯) = [𝑔, π‘₯, β„Ž]g

(7)

is a derivation with respect to the bracket [βˆ’, βˆ’, βˆ’]g of g.

Proof. The proof is similar to [6, Theorem 2.14]. Let g be a 𝑔𝑏-triple system with basis {π‘Ž1 , π‘Ž2 }, and assume that dim([g, g, g)]g ) = 1. Then write [π‘Žπ‘– , π‘Žπ‘— , π‘Žπ‘˜ ]g = π›Όπ‘–π‘—π‘˜ π‘Ž1 , 𝑖, 𝑗, π‘˜ = 1, 2. Then, using the identity (2), the only possible nonzero coefficients yield to the system of equations

Proof. By setting 𝜎 = [βˆ’, βˆ’, βˆ’]g and using the identity (2), we have 𝜎 (π‘₯, 𝑦, π·π‘”β„Ž (π‘Ž)) = [π‘₯, 𝑦, [𝑔, π‘Ž, β„Ž]g ]

𝛼221 (𝛼122 + 𝛼212 + 𝛼221 ) = 0, 𝛼122 (𝛼122 + 𝛼212 + 𝛼221 ) = 0,

g

= [𝑔, [π‘₯, 𝑦, π‘Ž]g , β„Ž] βˆ’ [[𝑔, π‘₯, β„Ž]g , 𝑦, π‘Ž] g

βˆ’ [π‘₯, [𝑔, 𝑦, β„Ž]g , π‘Ž]

𝛼222 (𝛼122 + 𝛼221 ) = 0, g

g

for which the solution provides the following 𝑔𝑏-triple systems with bracket operations: π›Όπ‘Ž , g1 : [π‘Žπ‘– , π‘Žπ‘— , π‘Žπ‘˜ ]g = { 1 0,

= π·π‘”β„Ž (𝜎 (π‘₯, 𝑦, π‘Ž)) βˆ’ 𝜎 (π·π‘”β„Ž (π‘₯) , 𝑦, π‘Ž) βˆ’ 𝜎 (π‘₯, π·π‘”β„Ž (𝑦) , π‘Ž) . (8)

Definition 6. A subspace 𝐼 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 subalgebra 𝐼 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 βŠ† 𝐼 (resp., [g, g, 𝐼]g βŠ† 𝐼, resp., [𝐼, g, g]g βŠ† 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 Example 2, the subspace I1 with basis {π‘Žπ‘ , π‘Žπ‘ž } is an ideal of g. However the subspace I2 with basis {π‘Žπ‘ž } is not an ideal of g, since, for π‘˜ =ΜΈ 𝑝, π‘ž, we have [π‘Žπ‘ , π‘Žπ‘ž , π‘Žπ‘˜ ] = π‘Žπ‘ βˆ‰ I2 . Definition 9. Given a 𝑔𝑏-triple system g, one defines the center of g and the derived algebra of g, respectively, by 𝑍 (g) = {π‘₯ ∈ g : [g, π‘₯, g]g = 0} , [g, g, g] = {[π‘Ž1 , π‘Ž2 , π‘Ž3 ]g , π‘Ž1 , π‘Ž2 , π‘Ž3 ∈ g} .

(10)

(9)

if 𝑖, π‘˜ = 2, 𝑗 = 1 else,

π›Όπ‘Ž , if 𝑖, 𝑗, π‘˜ = 2 g2 : [π‘Žπ‘– , π‘Žπ‘— , π‘Žπ‘˜ ]g = { 1 0, else, π‘Ž, { { 1 g3 : [π‘Žπ‘– , π‘Žπ‘— , π‘Žπ‘˜ ]g = {βˆ’π‘Ž1 , { {0,

if 𝑖 = 1, 𝑗, π‘˜ = 2 if 𝑖, 𝑗 = 2, π‘˜ = 1 else,

π‘Ž, { { 1 g4 : [π‘Žπ‘– , π‘Žπ‘— , π‘Žπ‘˜ ]g = {βˆ’π‘Ž1 , { {0,

if 𝑖, π‘˜ = 2, 𝑗 = 1 if 𝑖, 𝑗 = 2, π‘˜ = 1 else,

if 𝑖, π‘˜ = 2, 𝑗 = 1 π‘Ž, { { 1 g5 : [π‘Žπ‘– , π‘Žπ‘— , π‘Žπ‘˜ ]g = {π›Όπ‘Ž1 , if 𝑖, 𝑗, π‘˜ = 2 { else, {0, π‘Ž, { { 1 g6 : [π‘Žπ‘– , π‘Žπ‘— , π‘Žπ‘˜ ]g = {βˆ’π‘Ž1 , { {π›Όπ‘Ž1 ,

if 𝑖 = 1, 𝑗, π‘˜ = 2 if 𝑖, 𝑗 = 2, π‘˜ = 1 if 𝑖, 𝑗, π‘˜ = 2,

βˆ’ (1 + 𝛼) π‘Ž1 , if 𝑖 = 1, 𝑗, π‘˜ = 2 { { g7 : [π‘Žπ‘– , π‘Žπ‘— , π‘Žπ‘˜ ]g = {π‘Ž1 , if 𝑖, π‘˜ = 2, 𝑗 = 1 { if 𝑖, 𝑗 = 2, π‘˜ = 1, {π›Όπ‘Ž1 , with 𝛼 =ΜΈ 0.

(11)

ISRN Algebra

3

Definition 12. Given a 𝑔𝑏-triple system g, one defines the left center and the right center of g, respectively, by

Lemma 17. Let 𝑆 be a subalgebra of a 𝑔𝑏-triple system g. Then N𝑙g (𝑆) and Nπ‘Ÿg (𝑆) are also subalgebras of g.

𝑍𝐿 (g) = {π‘₯ ∈ g : [π‘₯, g, g]g = 0} ,

Proof. To show that Nπ‘Ÿg (𝑆) is a subalgebra of g, let π‘₯, 𝑦, 𝑧 ∈ Nπ‘Ÿg (𝑆), 𝑒 ∈ 𝑆, and 𝑔 ∈ g. Then, by the identity (2), we have

(12)

𝑍𝑅 (g) = {π‘₯ ∈ g : [g, g, π‘₯]g = 0} .

Lemma 13. The left center 𝑍𝐿 (g) and the right center 𝑍𝑅 (g) are 3-sided ideals of g. σΈ€ 

Proof. To show that 𝑍𝐿 (g) is an ideal of g, let 𝑔, 𝑔 ∈ g and let π‘₯ ∈ 𝑍𝐿 (g). Then, for every 𝑒, V ∈ g, we have, by the identity (2), [[𝑔, π‘₯, 𝑔󸀠 ]g , 𝑒, V] = [𝑔, [π‘₯, 𝑒, V]g , 𝑔󸀠 ] βˆ’ [π‘₯, [𝑔, 𝑒, 𝑔󸀠 ] , V]g ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ g 0 [ ]g 0 βˆ’ [π‘₯, 𝑒, [𝑔, V, 𝑔󸀠 ]g ] = 0. ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟g 0

(13) So [g, 𝑍𝐿 (g), g]g βŠ† 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 a 𝑔𝑏-triple system g, we define left and right centralizers of a subalgebra 𝑆 in g by 𝐢g𝑙 (𝑆) = {π‘₯ ∈ g : [π‘₯, 𝑆, g]g = 0} ,

(14)

𝐢gπ‘Ÿ (𝑆) = {π‘₯ ∈ g : [g, 𝑆, π‘₯]g = 0} ,

(15)

Lemma 15. Let 𝑆 be an ideal of a 𝑔𝑏-triple system g. Then 𝐢gπ‘Ÿ (𝑆) and 𝐢g𝑙 (𝑆) are also ideals of g. Proof. To show that 𝐢gπ‘Ÿ (𝑆) is an ideal of g, let π‘₯ ∈ 𝐢gπ‘Ÿ (𝑆), 𝑒 ∈ 𝑆, and 𝑔, π‘Ž, 𝑏 ∈ g. Then, by the identity (2), [𝑔, 𝑒, π‘₯], 𝑏] βˆ’ [[π‘Ž, 𝑔, 𝑏]g , 𝑒, π‘₯] [𝑔, 𝑒, [π‘Ž, π‘₯, 𝑏]g ]g = [π‘Ž, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟g 0

0

(16)

Definition 16. For a 𝑔𝑏-triple system g and a subalgebra 𝑆 of g, we define the left normalizer of 𝑆 in g by (17)

and the right normalizer of 𝑆 in g by Nπ‘Ÿg (𝑆) := {π‘₯ ∈ g : [g, 𝑆, π‘₯] βŠ† 𝑆} .

So [Nπ‘Ÿg (𝑆), Nπ‘Ÿg (𝑆), Nπ‘Ÿg (𝑆)]g βŠ† Nπ‘Ÿg (𝑆). The proof for N𝑙g (𝑆) is similar. Remark 18. If 𝑆 is an ideal, then N𝑙g (𝑆) = g = Nπ‘Ÿg (𝑆).

2. From 𝑔𝑏-Triple Systems to Leibniz Algebras Recall that a Leibniz algebra (sometimes called Loday algebra, named after Jean-Louis Loday) is a K vector space with a bilinear product [βˆ’, βˆ’] satisfying the Leibniz identity (20)

Proposition 19. Let g be a 𝑔𝑏-triple system. Define on gβŠ—2 the bracket operation [βˆ’, βˆ’] by [π‘Ž1 βŠ— π‘Ž2 , 𝑏1 βŠ— 𝑏2 ] = [π‘Ž1 , 𝑏1 , π‘Ž2 ]g βŠ— 𝑏2 + 𝑏1 βŠ— [π‘Ž1 , 𝑏2 , π‘Ž2 ]g . (21) Then [βˆ’, βˆ’] satisfies the Leibniz identity.

[π‘Ž1 βŠ— π‘Ž2 , [𝑏1 βŠ— 𝑏2 , 𝑐1 βŠ— 𝑐2 ]] = [π‘Ž1 βŠ— π‘Ž2 , [𝑏1 , 𝑐1 , 𝑏2 ]g βŠ— 𝑐2 ] + [π‘Ž1 βŠ— π‘Ž2 , 𝑐1 βŠ— [𝑏1 , 𝑐2 , 𝑏2 ]g ]

So [g, 𝐢gπ‘Ÿ (𝑆), g]g βŠ† 𝐢gπ‘Ÿ (𝑆). The proof for 𝐢g𝑙 (𝑆) is similar.

(𝑆) := {π‘₯ ∈ g : [π‘₯, 𝑆, g] βŠ† 𝑆} ,

(19)

Proof. On one hand, we have

𝑒, 𝑏]g , π‘₯] = 0. βˆ’ [𝑔, [π‘Ž, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ βˆˆπ‘† [ ]g ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

N𝑙g

βˆ’ [𝑔, [π‘₯, 𝑒, 𝑧]g , 𝑦] ∈ 𝑆. ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ βˆˆπ‘† [ ]g

[π‘₯, [𝑦, 𝑧]] = [[π‘₯, 𝑦] 𝑧] + [𝑦, [π‘₯, 𝑧]] .

respectively.

0

[𝑔, 𝑒, [π‘₯, 𝑦, 𝑧]g ] = [π‘₯, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ [𝑔, 𝑒, 𝑦]g , 𝑧] βˆ’ [[π‘₯, 𝑔, 𝑧]g , 𝑒, 𝑦] g g βˆˆπ‘† [ ]g

= [π‘Ž1 , [𝑏1 , 𝑐1 , 𝑏2 ]g , π‘Ž2 ] βŠ— 𝑐2 g

+ [𝑏1 , 𝑐1 , 𝑏2 ]g βŠ— [π‘Ž1 , 𝑐2 , π‘Ž2 ]g + [π‘Ž1 , 𝑐1 , π‘Ž2 ]g βŠ— [𝑏1 , 𝑐2 , 𝑏2 ]g + 𝑐1 βŠ— [π‘Ž1 , [𝑏1 , 𝑐2 , 𝑏2 ]g , π‘Ž2 ] . g

(18)

(22)

4

ISRN Algebra Definition 22. Let g be a 𝑔𝑏-triple system and Ł a Leibniz algebra. The action of Ł on g is a map 𝐴 : gβŠ—Ε β†’ g satisfying

Also, [[π‘Ž1 βŠ— π‘Ž2 , 𝑏1 βŠ— 𝑏2 ] , 𝑐1 βŠ— 𝑐2 ] = [[π‘Ž1 , 𝑏1 , π‘Ž2 ]g βŠ— 𝑏2 , 𝑐1 βŠ— 𝑐2 ] + [𝑏1 βŠ— [π‘Ž1 , 𝑏2 , π‘Ž2 ]g , 𝑐1 βŠ— 𝑐2 ]

(1) 𝐴 ([𝑔1 , 𝑔2 , 𝑔3 ]g βŠ— π‘₯) = [𝐴 (𝑔1 βŠ— π‘₯) , 𝑔2 , 𝑔3 ]g + [𝑔1 , 𝐴 (𝑔2 βŠ— π‘₯) , 𝑔3 ]g

= [[π‘Ž1 , 𝑏1 , π‘Ž2 ]g , 𝑐1 , 𝑏2 ] βŠ— 𝑐2 g

(27)

+ [𝑔1 , 𝑔2 , 𝐴 (𝑔3 βŠ— π‘₯)]g ,

+ 𝑐1 βŠ— [[π‘Ž1 , 𝑏1 , π‘Ž2 ]g , 𝑐2 , 𝑏2 ]

g

+ [𝑏1 , 𝑐1 , [π‘Ž1 , 𝑏2 , π‘Ž2 ]g ] βŠ— 𝑐2

(2) 𝐴 (𝑔 βŠ— [π‘₯, 𝑦]) = 𝐴 (𝐴 (𝑔 βŠ— 𝑦) βŠ— π‘₯) βˆ’ 𝐴 (𝐴 (𝑔 βŠ— π‘₯) βŠ— 𝑦) (28)

g

+ 𝑐1 βŠ— [𝑏1 , 𝑐2 , [π‘Ž1 , 𝑏2 , π‘Ž2 ]g ] . g

(23) On the other hand,

for all 𝑔1 , 𝑔2 , 𝑔3 ∈ g and π‘₯, 𝑦 ∈ Ł. Proposition 23. Let g be a 𝑔𝑏-triple system; then the Leibniz algebra gβŠ—2 acts on g via the map 𝐴 : g βŠ— gβŠ—2 β†’ g defined by 𝐴(𝑧 βŠ— 𝑔1 βŠ— 𝑔2 ) = [𝑔1 , 𝑧, 𝑔2 ]g . Proof. The first condition of Definition 22 follows by (2). To show (28), we have

[𝑏1 βŠ— 𝑏2 , [π‘Ž1 βŠ— π‘Ž2 , 𝑐1 βŠ— 𝑐2 ]] = [𝑏1 βŠ— 𝑏2 , [π‘Ž1 , 𝑐1 , π‘Ž2 ]g βŠ— 𝑐2 ] + [𝑏1 βŠ— 𝑏2 , 𝑐1 βŠ— [π‘Ž1 , 𝑐2 , π‘Ž2 ]g ] = [𝑏1 , [π‘Ž1 , 𝑐1 , π‘Ž2 ]g , 𝑏2 ] βŠ— 𝑐2

𝐴 (𝐴 (𝑧 βŠ— 𝑏1 βŠ— 𝑏2 ) βŠ— π‘Ž1 βŠ— π‘Ž2 ) βˆ’ 𝐴 (𝐴 (𝑧 βŠ— π‘Ž1 βŠ— π‘Ž2 ) βŠ— 𝑏1 βŠ— 𝑏2 ) = [π‘Ž1 , [𝑏1 , 𝑧, 𝑏2 ]g , π‘Ž2 ] βˆ’ [𝑏1 , [π‘Ž1 , 𝑧, π‘Ž2 ]g , 𝑏2 ]

g

g

+ [π‘Ž1 , 𝑐1 , π‘Ž2 ]g βŠ— [𝑏1 , 𝑐2 , 𝑏2 ]g + [𝑏1 , 𝑐1 , 𝑏2 ]g βŠ— [π‘Ž1 , 𝑐2 , π‘Ž2 ]g + 𝑐1 βŠ— [𝑏1 , [π‘Ž1 , 𝑐2 , π‘Ž2 ]g , 𝑏2 ] . g

(24) One checks using the identity (2) that the equality

g

= [[π‘Ž1 , 𝑏1 , π‘Ž2 ]g , 𝑧, 𝑏2 ] + [𝑏1 , 𝑧, [π‘Ž1 , 𝑏2 , π‘Ž2 ]]g g

by (2)

= 𝐴 (𝑧 βŠ— [π‘Ž1 , 𝑏1 , π‘Ž2 ]g βŠ— 𝑏2 ) + 𝐴 (𝑧 βŠ— 𝑏1 βŠ— [π‘Ž1 , 𝑏2 , π‘Ž2 ]g ) = 𝐴 (𝑧 βŠ— ([π‘Ž1 , 𝑏1 , π‘Ž2 ]g βŠ— 𝑏2 + 𝑏1 βŠ— [π‘Ž1 , 𝑏2 , π‘Ž2 ]g )) = 𝐴 (𝑧 βŠ— [π‘Ž1 βŠ— π‘Ž2 , 𝑏1 βŠ— 𝑏2 ])

by (21) . (29)

[π‘Ž1 βŠ— π‘Ž2 , [𝑏1 βŠ— 𝑏2 , 𝑐1 βŠ— 𝑐2 ]] = [[π‘Ž1 βŠ— π‘Ž2 , 𝑏1 βŠ— 𝑏2 ] , 𝑐1 βŠ— 𝑐2 ] + [𝑏1 βŠ— 𝑏2 , [π‘Ž1 βŠ— π‘Ž2 , 𝑐1 βŠ— 𝑐2 ]] (25) holds.

Now let 𝑔1 , 𝑔2 ∈ g and consider the map 𝐴 𝑔1 βŠ—π‘”2 : g β†’ g defined by 𝐴 𝑔1 βŠ—π‘”2 (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.

Corollary 20. Let g be a 𝑔𝑏-triple system; then gβŠ—2 endowed with the bilinear map [βˆ’, βˆ’] has a Leibniz algebra structure.

Proposition 24. For a 𝑔𝑏-triple system g, the subspace A(g) = {𝐴 𝑔1 βŠ—π‘”2 | 𝑔1 , 𝑔2 ∈ g} is a Lie algebra with respect to the product

Proof. This is a consequence of Proposition 19. Similarly, we have the following. Corollary 21. Let g be a 𝑔𝑏-triple system; then g∧2 has a Leibniz algebra structure, when endowed with the bilinear map defined by [π‘Ž1 ∧ π‘Ž2 , 𝑏1 ∧ 𝑏2 ] = [π‘Ž1 , 𝑏1 , π‘Ž2 ]g ∧ 𝑏2 + 𝑏1 ∧ [π‘Ž1 , 𝑏2 , π‘Ž2 ]g . (26) These determine two functors from the category 𝑔𝑏-TS of 𝑔𝑏-triple systems to the category Lb of Leibniz algebras.

[𝐴 π‘Ž1 βŠ—π‘Ž2 , 𝐴 𝑏1 βŠ—π‘2 ] = 𝐴 π‘Ž1 βŠ—π‘Ž2 ∘ 𝐴 𝑏1 βŠ—π‘2 βˆ’ 𝐴 𝑏1 βŠ—π‘2 ∘ 𝐴 π‘Ž1 βŠ—π‘Ž2 .

(30)

More precisely, it is an ideal of the Lie algebra Der(g) of derivations of g. Proof. To show that A(g) is a Lie subalgebra of Der(g), let π‘Ž1 , π‘Ž2 , 𝑏1 , 𝑏2 ∈ g. Then, for all 𝑧 ∈ g, [𝐴 π‘Ž1 βŠ—π‘Ž2 , 𝐴 𝑏1 βŠ—π‘2 ]Der(g) (𝑧) = 𝐴 π‘Ž1 βŠ—π‘Ž2 ∘ 𝐴 𝑏1 βŠ—π‘2 (𝑧) βˆ’ 𝐴 𝑏1 βŠ—π‘2 ∘ 𝐴 π‘Ž1 βŠ—π‘Ž2 (𝑧) = [π‘Ž1 , [𝑏1 , 𝑧, 𝑏2 ]g , π‘Ž2 ] βˆ’ [𝑏1 , [π‘Ž1 , 𝑧, π‘Ž2 ]g , 𝑏2 ] g

g

ISRN Algebra

5

= 𝐴 (𝐴 (𝑧 βŠ— 𝑏1 βŠ— 𝑏2 ) βŠ— π‘Ž1 βŠ— π‘Ž2 ) βˆ’ 𝐴 (𝐴 (𝑧 βŠ— π‘Ž1 βŠ— π‘Ž2 ) βŠ— 𝑏1 βŠ— 𝑏2 ) = 𝐴 (𝑧 βŠ— [π‘Ž1 βŠ— π‘Ž2 , 𝑏1 βŠ— 𝑏2 ])

by Proposition 23

= 𝐴 [π‘Ž1 βŠ—π‘Ž2 ,𝑏1 βŠ—π‘2 ] (𝑧) . (31) So A(g) is closed under the bracket of Der(g). Also, for any derivation ∈ Der(g), we have, for all 𝑧 ∈ g, [𝑑, 𝐴 π‘Ž1 βŠ—π‘Ž2 ]Der(g) (𝑧) = (𝑑 ∘ 𝐴 π‘Ž1 βŠ—π‘Ž2 ) (𝑧) βˆ’ (𝐴 π‘Ž1 βŠ—π‘Ž2 ∘ 𝑑) (𝑧) = 𝑑 (𝐴 π‘Ž1 βŠ—π‘Ž2 (𝑧)) βˆ’ 𝐴 π‘Ž1 βŠ—π‘Ž2 (𝑑 (𝑧)) = 𝑑 ([π‘Ž1 , 𝑧, π‘Ž2 ]g ) βˆ’ [π‘Ž1 , 𝑑 (𝑧) , π‘Ž2 ]g = [𝑑 (π‘Ž1 ) , 𝑧, π‘Ž2 ]g + [π‘Ž1 , 𝑧, 𝑑 (π‘Ž2 )]g

by (6)

= 𝐴 𝑑(π‘Ž1 )βŠ—π‘Ž2 (𝑧) + 𝐴 π‘Ž1 βŠ—π‘‘(π‘Ž2 ) (𝑧) .

(32)

Hence [𝑑, 𝐴 π‘Ž1 βŠ—π‘Ž2 ]Der(g) = 𝐴 𝑑(π‘Ž1 )βŠ—π‘Ž2 + 𝐴 π‘Ž1 βŠ—π‘‘(π‘Ž2 ) ∈ A(g).

Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.

References [1] V. T. Filippov, β€œπ‘›-Lie algebras,” Sibirski˘ı Matematicheski˘ı Zhurnal, vol. 26, no. 6, pp. 126–140, 1985. [2] N. Jacobson, β€œLie and Jordan triple systems,” American Journal of Mathematics, vol. 71, pp. 149–170, 1949. [3] J. M. Casas, J.-L. Loday, and T. Pirashvili, β€œLeibniz 𝑛-algebras,” Forum Mathematicum, vol. 14, no. 2, pp. 189–207, 2002. [4] M. R. Bremner and J. SΒ΄anchez-Ortega, β€œLeibniz triple systems,” Communications in Contemporary Mathematics, vol. 14, pp. 189–207, 2013. [5] J.-L. Loday, β€œUne version non commutative des alg`ebres de Lie: les alg`ebres de Leibniz,” L’Enseignement MathΒ΄ematique, vol. 39, no. 3-4, pp. 269–293, 1993. [6] L. M. Camacho, J. M. Casas, J. R. GΒ΄omez, M. Ladra, and B. A. Omirov, β€œOn nilpotent Leibniz 𝑛-algebras,” Journal of Algebra and its Applications, vol. 11, no. 3, Article ID 1250062, 17 pages, 2012.

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