Hindawi International Journal of and Mathematical Sciences Volume 2017, Article ID 7842482, 9 pages https://doi.org/10.1155/2017/7842482

Research Article On Killing Forms and Forms of Lie-Yamaguti

Patricia L. Zoungrana1,2 and A. Nourou Issa1,2

1 Departement´ de Mathematiques´ de la Decision,´ Universite´ Ouaga 2, 12 BP 412, Ouagadougou 12, Burkina Faso 2Departement´ de Mathematiques,´ Universite´ d’Abomey-Calavi, 01 BP 4521, Cotonou, Benin

Correspondence should be addressed to Patricia L. Zoungrana; [email protected]

Received 15 October 2016; Accepted 20 December 2016; Published 12 January 2017

Academic Editor: Hernando Quevedo

Copyright © 2017 P. L. Zoungrana and A. N. Issa. 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.

The notions of the Killing form and invariant form in Lie are extended to the ones in Lie-Yamaguti superalgebras and some of their properties are investigated. These notions are also Z2-graded generalizations of the ones in Lie-Yamaguti algebras.

1. Introduction development of the theory of Lie-Yamaguti algebras one may refer, for example, to [5–8]. From the standard enveloping Lie (𝑇, ∗, [⋅, ⋅, ⋅]) A Lie-Yamaguti is a triple consisting of a algebra of a given Lie-Yamaguti algebra, the notions of the 𝑇 ∗:𝑇×𝑇→𝑇 vector space ,abilinearmap , and a trilinear Killing-Ricci form and the invariant form of a Lie-Yamaguti [⋅, ⋅, ⋅] : 𝑇 × 𝑇 × 𝑇 →𝑇 map such that algebra are introduced and studied in [9]. Further properties of invariant forms of Lie-Yamaguti algebras were considered (LY1) 𝑥∗𝑦=−𝑦∗𝑥, in [10]. (LY2) [𝑥, 𝑦, 𝑧] = −[𝑦,, 𝑥,𝑧] Lie superalgebras as a Z2-graded generalization of Lie algebras are considered in [11, 12] while a Z2-graded gener- (LY3) ↺𝑥,𝑦,𝑧{((𝑥∗𝑦)∗𝑧)+[𝑥,𝑦,𝑧]}=0, alization of Lie triple systems (called Lie supertriple systems) ↺ [𝑥∗𝑦,𝑧,𝑢]=0 (LY4) 𝑥,𝑦,𝑧 , was first considered in [13]. For an application of Lie super- (LY5) [𝑥,𝑦,𝑢∗V]=[𝑥,𝑦,𝑢]∗V +𝑢∗[𝑥,𝑦,V], triple systems in physics, one may refer to [14]. Next, Lie- Yamaguti superalgebras as a Z2-graded generalization of Lie- (LY6) [𝑢, V, [𝑥, 𝑦, 𝑧]] = [[𝑢, V, 𝑥], 𝑦, 𝑧] + [𝑥, [𝑢, V, 𝑦], 𝑧] + Yamaguti algebras were first considered in [15]. [𝑥,𝑦,[𝑢,V,𝑧]], Z 𝑢, V,𝑥,𝑦,𝑧 𝑇 ↺ Definition 1 (see [16]). A Lie-Yamaguti is a 2- for all ,in ,where 𝑥,𝑦,𝑧 denotes the sum over 𝑇=𝑇⊕𝑇 𝑥 𝑦 𝑧 ∗ graded vector space 0 1 with a binary operation cyclic of , , .Thebilinearmap sometimes 𝑇 𝑇 ⊆𝑇 𝑥∗𝑦 =0 ∀𝑥, 𝑦 ∈𝑇 denoted by juxtaposition satisfying 𝑖 𝑗 𝑖+𝑗 and a ternary will be denoted by juxtaposition. If , , [⋅,⋅,⋅] [𝑇 ,𝑇,𝑇 ]⊆𝑇 𝑖, 𝑗, 𝑘∈ Z one gets a Lietriplesystem(𝑇, [⋅, ⋅, ⋅]), while [𝑥,𝑦,𝑧] =0 in operation satisfying 𝑖 𝑗 𝑘 𝑖+𝑗+𝑘 ( 2) (𝑇, ∗, [⋅, ⋅, ⋅]) induces a Lie algebra (𝑇, ∗). such that 𝑥 𝑦 Lie-Yamaguti algebras were introduced by Yamaguti [1] (LYS1) 𝑥𝑦 = −(−1) 𝑦𝑥, 𝑥 𝑦 (who formerly called them “generalized Lie triple systems”) (LYS2) [𝑥,𝑦,𝑧]=−(−1) [𝑦, 𝑥, 𝑧], in an algebraic study of the characteristic properties of the 𝑥 𝑧 torsion and curvature of a homogeneous space with canonical (LYS3) ↺𝑥,𝑦,𝑧(−1) {((𝑥𝑦)𝑧) + [𝑥, 𝑦,, 𝑧]}=0 connection [2]. Later on, these algebraic objects were called 𝑥 𝑧 (LYS4) ↺𝑥,𝑦,𝑧(−1) [𝑥𝑦, 𝑧, 𝑢], =0 “Lie triple algebras” [3] and the terminology of “Lie-Yamaguti 𝑢(𝑥+𝑦) algebras” is introduced in [4] for these algebras. For further (LYS5) [𝑥,𝑦,𝑢V]=[𝑥,𝑦,𝑢]V +(−1) 𝑢[𝑥, 𝑦, V], 2 International Journal of Mathematics and Mathematical Sciences

𝑥(𝑢+V) (LYS6) [𝑢, V, [𝑥, 𝑦, 𝑧]] = [[𝑢, V, 𝑥], 𝑦, 𝑧] + (−1) [𝑥, [𝑢, V, Proposition 2. A superalgebra 𝑇=𝑇0 ⊕𝑇1 equipped with (𝑥+𝑦)(𝑢+V) 𝑦], 𝑧] + (−1) [𝑥,𝑦,[𝑢,V,𝑧]], bilinear and trilinear products verifying 𝑇𝑖𝑇𝑗 ⊆𝑇𝑖+𝑗 and [𝑇𝑖,𝑇𝑗,𝑇𝑘]⊆𝑇𝑖+𝑗+𝑘 is a Lie-Yamaguti superalgebra if its for all 𝑢, V,𝑥,𝑦,𝑧,in𝑇. Grassmann envelope 𝐺(𝑇)0 =𝐺 ⊗𝑇0⊕𝐺1⊗𝑇1 is a Lie-Yamaguti algebra under the following products: Observe that 𝑇0 is a Lie-Yamaguti algebra. As a part of the general theory of superalgebras, the (𝑥 ⊗ 𝑔 )(𝑦⊗𝑔 )=(−1)𝑥 𝑦 𝑥𝑦 ⊗𝑔 𝑔 ; notion of the Killing form of Lie algebras is extended to the 1 2 1 2 one of Lie triple systems (see [17] and references therein), Lie [𝑥 ⊗1 𝑔 ,𝑦⊗𝑔2,𝑧⊗𝑔3] (1) superalgebras [12], and next Lie supertriple systems [18] (see also [19]). 𝑥 𝑦+𝑦 𝑧+𝑥 𝑧 = (−1) [𝑥,𝑦,𝑧]⊗𝑔1𝑔2𝑔3. In this paper we define and study the Killing form and invariant form of Lie-Yamaguti superalgebras as a generaliza- Proof. The proof is straightforward by using the fact that, for tion of the ones of both of Lie-Yamaguti algebras [9] and Lie any element 𝑥⊗𝑔in 𝐺(𝑇),wehave𝑥=𝑔. supertriplesystems[18,19]includingLiesuperalgebras[12]. The paper is organized as follows. In Section 2 we record Example 3. (1) Lie superalgebras are Lie-Yamaguti superalge- some useful results on Lie-Yamaguti superalgebras (see [16]). bras with [𝑥,𝑦,𝑧]=0. In Section 3 the Killing form of a Lie-Yamaguti superalgebra (2) If 𝑥𝑦 =0 for any 𝑥, 𝑦0 ∈𝑇 ∪𝑇1 then (LYS2), (LYS3), is defined (see Theorem 10 and Definition 11) and some of and (LYS6) define a Lie supertriple system. its properties are investigated (Proposition 13, Theorem 14, (3) Let 𝑀=𝑀0 ⊕𝑀1 be a Malcev superalgebra; that is, andCorollary15).InSection4theinvariantformofaLie- for any 𝑥, 𝑦, 𝑧,𝑡 in 𝑇, Yamaguti superalgebra is defined (Definition 16) and, under some conditions, it is shown (Theorem 21) that the Killing 𝑥𝑦 =− (−1)𝑥 𝑦 𝑦𝑥; form of a Lie-Yamaguti superalgebra 𝑇 is nondegenerate if and only if the standard enveloping Lie superalgebra of 𝑇 is − (−1)𝑦 𝑧 (𝑥𝑧) (𝑦𝑡) = ((𝑥𝑦) 𝑧)𝑡 semisimple. 𝑥(𝑦+𝑧+𝑡) All vector spaces and algebras are finite-dimensional over + (−1) ((𝑦𝑧) 𝑡) 𝑥 (2) afixedgroundfieldK of characteristic 0. + (−1)(𝑥+𝑦)(𝑧+𝑡) ((𝑧𝑡) 𝑥) 𝑦 2. Some Basics on Lie-Yamaguti Superalgebras + (−1)𝑡(𝑥+𝑦+𝑧) ((𝑡𝑥) 𝑦) 𝑧. We give here some definitions and results which can be found in [11, 12, 16]. Itisshownin[16]that𝑀 becomes a Lie-Yamaguti super- AsuperalgebraoverK is a Z2-graded algebra 𝐴=𝐴0 ⊕ 𝑥 𝑦 algebra if we [𝑥,𝑦,𝑧] = 𝑥(𝑦𝑧)− (−1) 𝑦(𝑥𝑧) + (𝑥𝑦)𝑧. 𝐴1,where𝐴𝑖𝐴𝑗 ⊆𝐴𝑖+𝑗,(𝑖, 𝑗 ∈ Z2). The subspaces 𝐴0 and 𝐴1 Conversely, if on a Malcev superalgebra (𝑀, ⋅) we define a are called the even and the odd parts of the superalgebra and 𝑥 𝑦 𝐴 𝐴 trilinear operation by [𝑥,𝑦,𝑧]=𝑥⋅𝑦𝑧−(−1) 𝑦⋅𝑥𝑧+𝑥𝑦⋅𝑧 so are called the elements from 0 and from 1,respectively. (𝑀,⋅,[⋅,⋅,⋅]) Below, all the elements are assumed to be homogeneous, that then is a Lie-Yamaguti superalgebra. is, either even or odd, and for a homogeneous element 𝑥∈𝐴𝑖, Definition 4. Let 𝑇=𝑇0 ⊕𝑇1 be a Lie-Yamaguti superalgebra. 𝑖=0,1,thenotation𝑥=𝑖is used and means the parity of 𝑥. A graded subspace 𝐻=𝐻0 ⊕𝐻1 of 𝑇 is a graded Lie- Let 𝐺 be the Grassmann algebra over K generated by the 2 Yamaguti subalgebra of 𝑇 if 𝐻𝑖𝐻𝑗 ⊆𝐻𝑖+𝑗 and [𝐻𝑖,𝐻𝑗,𝐻𝑘]⊆ elements 1, 𝑒1,...,𝑒𝑛 such that 𝑒𝑖 =0, 𝑒𝑖𝑒𝑗 =−𝑒𝑗𝑒𝑖 for 𝑖 =𝑗̸ . 𝐻𝑖+𝑗+𝑘 for any 𝑖, 𝑗, 𝑘∈ Z2. The elements 1, 𝑒𝑖 𝑒𝑖 ⋅⋅⋅𝑒𝑖 , 𝑖1 <𝑖2 <⋅⋅⋅<𝑖𝑟 form a basis 𝐺 𝐺1 2 𝑟 𝐺 of .Denoteby 0 (resp., 1)thespanoftheproductsof Definition 5. A graded subalgebra of a Lie-Yamaguti superal- even length (resp., odd length) in the generators. The product gebra 𝑇 is an invariant graded subalgebra (resp., an ideal) of 𝑒 𝐺=𝐺 ⊕𝐺 of zero 𝑖’s is by convention equal to 1. Then 0 1 𝑇 if [𝑇,𝑇,𝐻]⊆𝐻(resp., 𝑇𝐻⊆ 𝐻 and [𝑇, 𝐻, 𝑇] ⊆𝐻). is an associative and supercommutative superalgebra; that is, 𝑔1𝑔2 𝑔1𝑔2 =(−1) 𝑔2𝑔1,where𝑔1,𝑔2 ∈𝐺0 ∪𝐺1. Let 𝐴=𝐴0 ⊕𝐴1 If 𝐻 is an ideal of 𝑇, it is an invariant graded subalgebra of be a superalgebra. Consider the graded product 𝐺⊗𝐴 𝑇.Obviouslythecenter𝑍(𝑇) of a Lie-Yamaguti superalgebra which becomes a superalgebra with the product given by (𝑥⊗ 𝑇 defined by 𝑍(𝑇) = {𝑥 ∈ 𝑇, 𝑥𝑦=0 and [𝑥,𝑦,𝑧] = 𝑥 𝑔 2 0, ∀𝑦, 𝑧 ∈ 𝑇} 𝑇 𝑔1)(𝑦 ⊗2 𝑔 )=(−1) 𝑥𝑦 ⊗𝑔1𝑔2, for homogeneous elements is an ideal of . 𝑔 ,𝑔 ∈𝐺𝑥, 𝑦 ∈𝐴 (𝐺 ⊗ 𝐴) =𝐺 ⊗ 1 2 , and grading given by 0 0 󸀠 󸀠 󸀠 𝐴 ⊕𝐺 ⊗𝐴 (𝐺 ⊗ 𝐴) =𝐺 ⊗𝐴 ⊕𝐺 ⊗𝐴 . Definition 6. Let 𝑇=𝑇0⊕𝑇1 and 𝑇 =𝑇0⊕𝑇1 be Lie-Yamaguti 0 1 1, 1 0 1 1 0 The subalgebra 󸀠 𝐺(𝐴) = (𝐺⊗𝐴) =𝐺 ⊗𝐴 ⊕𝐺 ⊗𝐴 superalgebras. A linear map 𝑓:𝑇→𝑇 is said to be of degree 0 0 0 1 1 is called the Grassmann 󸀠 envelope of the superalgebra 𝐴. 𝑟 if 𝑓(𝑇𝑖)⊆𝑇𝑟+𝑖 for all 𝑟, 𝑖 ∈ Z2. Having in mind that if 𝑉 is a homogeneous variety of 𝑇=𝑇⊕𝑇 𝑇󸀠 =𝑇󸀠 ⊕𝑇󸀠 algebras [20], a superalgebra 𝐴=𝐴0 ⊕𝐴1 is called a 𝑉- Definition 7. Let 0 1 and 0 1 be Lie- 󸀠 superalgebra, if its Grassmann envelope 𝐺(𝐴) belongs to 𝑉, Yamaguti superalgebras. A linear map 𝑓:𝑇→𝑇 is called a we can state the following proposition. homomorphism of Lie-Yamaguti superalgebras if International Journal of Mathematics and Mathematical Sciences 3

󸀠 (1) 𝑓 preserves the grading, that is, 𝑓(𝑇𝑖)⊆𝑇𝑖 , 𝑖∈Z2; the superderivation 𝐷𝑥,𝑦 of degree 𝑟. From (5) we also have that, for any 𝑥, 𝑦, 𝑧, V,𝑤∈𝑇0 ∪𝑇1, (2) 𝑓(𝑥𝑦) = 𝑓(𝑥)𝑓(𝑦);

(3) 𝑓([𝑥, 𝑦, 𝑧]) = [𝑓(𝑥), 𝑓(𝑦), 𝑓(𝑧)] for any 𝑥, 𝑦,0 𝑧∈𝑇 ∪ 𝑧(𝑥+𝑦) [𝐷𝑥,𝑦,𝐷𝑧,V]=𝐷[𝑥,𝑦,𝑧],V + (−1) 𝐷𝑧,[𝑥,𝑦,V]. (6) 𝑇1. 𝑉=𝑉⊕𝑉 Z Recall [11] that if 0 1 is a 2-graded vector 𝐷(𝑇, 𝑇) Z (𝑉) = {𝑓 ∈ (𝑉)/𝑓(𝑉)⊆𝑉 } It is clear from (6) that is a 2-graded Lie subalgebra space then, if we set End𝑟 End 𝑖 𝑟+𝑖 , of 𝐷(𝑇) called the Lie superalgebra of all inner superderiva- we obtain an associative superalgebra End(𝑉) = End0(𝑉) ⊕ (𝑉) (𝑉) 𝑉 tions of T. End1 ;End𝑟 consists of the linear mappings of into Now, let (𝑇, ⋅, [⋅, ⋅, ⋅]) be a Lie-Yamaguti superalgebra. itself which are homogeneous of degree 𝑟. The [𝑓,𝑔] = Set 𝐿𝑖(𝑇) =𝑖 𝑇 ⊕𝐷𝑖(𝑇, 𝑇), 𝑖=0,1, and define a new bracket 𝑓𝑔 − ( − 1) 𝑓𝑔𝑔𝑓 (𝑉) makes End into a Lie superalgebra which operation in 𝐿(𝑇) =𝐿0(𝑇) ⊕ 𝐿1(𝑇) = 𝑇 ⊕ 𝐷(𝑇, 𝑇) as follows: 𝑙(𝑉) 𝑙(𝑚, 𝑛) 𝑚= 𝑉 𝑛= we denote by or where dim 0 and for any 𝑥, 𝑦0 ∈𝑇 ∪𝑇1, 𝐷1,𝐷2 ∈𝐷0(𝑇, 𝑇)1 ∪𝐷 (𝑇, 𝑇), dim 𝑉1.Let𝑒1,...,𝑒𝑚,𝑒𝑚+1,...,𝑒𝑚+𝑛 be a basis of 𝑉.Inthis 𝛼𝛽 basis the matrix of 𝑎∈𝑙(𝑚,𝑛)is expressed as ( 𝛾𝛿), 𝛼 being [𝑥, 𝑦] = 𝑥𝑦𝑥,𝑦 +𝐷 ; an (𝑚 × 𝑚)-, 𝛿 an (𝑛 × 𝑛)-, 𝛽 an (𝑚 × 𝑛)-, and 𝛾 an (𝑛 × 𝑚)- 𝛼0 matrix.Thematricesofevenelementshavetheform( 0𝛿) 𝑥𝐷 0𝛽 𝛼𝛽 [𝐷, 𝑥] =−(−1) [𝑥,] 𝐷 =𝐷(𝑥) ; (7) and those of odd ones ( 𝛾0).For𝑎=(𝛾𝛿),thesupertraceof 𝑎 (𝑎) = 𝛼− 𝛿 𝐷1𝐷2 is defined by str tr tr and does not depend on the [𝐷1,𝐷2]=𝐷1𝐷2 − (−1) 𝐷2𝐷1. choice of a homogeneous basis. We have str([𝑎, 𝑏]) =0 that is 𝑎𝑏 −1 str(𝑎𝑏) = (−1) str(𝑏𝑎) and str(𝑎𝑏𝑎 )=str(𝑏). Theorem 9. Let 𝑇=𝑇0 ⊕𝑇1 be a Lie-Yamaguti superalgebra.

Definition 8. Let 𝑇=𝑇0 ⊕𝑇1 be a Lie-Yamaguti superalgebra; Then 𝐷∈End𝑟(𝑇) is a superderivation of 𝑇 if, for any 𝑥, 𝑦, 𝑧∈ 𝑇0 ∪𝑇1, (1) 𝐿(𝑇) is a Lie superalgebra called the standard envelop- ing Lie superalgebra of 𝑇 and 𝐷(𝑇, 𝑇) becomes a graded 𝐷(𝑥𝑦)=𝐷(𝑥) 𝑦+(−1)𝑟𝑥 𝑥𝐷 (𝑦) ; subalgebra of 𝐿(𝑇).

𝑟𝑥 𝐷 ([𝑥, 𝑦, 𝑧]) =[𝐷 (𝑥) ,𝑦,𝑧]+(−1) [𝑥, 𝐷 (𝑦) ,𝑧] (3) (2) If 𝐻 is an ideal of 𝑇 then 𝐻⊕ 𝐷(𝑇, 𝐻) is an ideal of 𝑇⊕𝐷(𝑇,𝑇). + (−1)𝑟(𝑥+𝑦) [𝑥, 𝑦, 𝐷 (𝑧)]. Proof. The bracket [⋅, ⋅] is bilinear by definition and 𝑋𝑌 = 𝑋 𝑌 Let 𝐷𝑟(𝑇) consist of all the superderivations of degree 𝑟 −(−1) 𝑌𝑋 for any 𝑋, 𝑌 ∈ 𝐿(𝑇) by (LYS1) and (LYS2). and 𝐷(𝑇)0 =𝐷 (𝑇) ⊕1 𝐷 (𝑇). It is easy to check that 𝐷(𝑇) is Jacobi’s superidentity follows from (LTS3–6). agradedsubalgebraofEnd(𝑇) called the Lie superalgebra of (2) is obvious. superderivations of 𝑇. Let 𝑇=𝑇0 ⊕𝑇1 be a Lie-Yamaguti superalgebra. For any 𝑥, 𝑦0 ∈𝑇 ∪𝑇1, denote by 𝐷𝑥,𝑦 the endomorphism of 𝑇 defined 3. Killing Forms of Lie-Yamaguti by 𝐷𝑥,𝑦(𝑧)= [𝑥,𝑦,𝑧]for any 𝑧∈𝑇.We have, for any 𝑥, 𝑦 ∈ Superalgebras 𝑇0 ∪𝑇1, 𝑟∈Z2, 𝐷𝑥,𝑦(𝑇𝑟)⊆𝑇𝑟+𝑥+𝑦;thatis,𝐷𝑥,𝑦 is a linear map of degree 𝑥+𝑦. Moreover, it comes from (LYS5) and (LYS6) The definition of the Killing form given here for Lie-Yamaguti that superalgebras stems from [9] in the case of Lie-Yamaguti algebras and extends the one given in [18] for Lie supertriple 𝑧(𝑥+𝑦) systems. Let 𝑇=𝑇0 ⊕𝑇1 be an 𝑛-dimensional Lie-Yamaguti 𝐷𝑥,𝑦 (𝑧𝑤) =𝐷𝑥,𝑦 (𝑧) 𝑤+(−1) 𝑧𝐷𝑥,𝑦 (𝑤) ; (4) superalgebra. Denote by 𝛼 the Killing form of the standard enveloping Lie superalgebra 𝐿(𝑇) =0 (𝑇 ⊕𝐷0(𝑇, 𝑇)) ⊕ 𝐷𝑥,𝑦 ([𝑧, V,𝑤]) =[𝐷𝑥,𝑦 (𝑧) , V,𝑤] (𝑇1 ⊕𝐷1(𝑇, 𝑇)). Consider the bilinear form 𝛽 of 𝑇 obtained 𝑧(𝑥+𝑦) by restricting 𝛼 to 𝑇×𝑇.Forany𝑥, 𝑦, 𝑧 in 𝑇, define the + (−1) [𝑧,𝑥,𝑦 𝐷 (V) ,𝑤] (5) endomorphisms 𝐿𝑥 and 𝑅𝑥,𝑦 of the vector space 𝑇 by 𝐿𝑥(𝑦) = (𝑧+V)(𝑥+𝑦) 𝑥𝑦 𝑅 (𝑧) = (−1)𝑧(𝑥+𝑦)[𝑧,𝑥,𝑦]= (−1)𝑧(𝑥+𝑦)𝐷 (𝑦) + (−1) [𝑧, V,𝐷𝑥,𝑦 (𝑤)] and 𝑥,𝑦 𝑧,𝑥 .Itis clear that 𝑅𝑥,𝑦 is of degree 𝑥+𝑦 and [𝐷𝑧,𝑡,𝑅𝑥,𝑦]=𝑅[𝑧,𝑡,𝑥],𝑦 + (𝑧+𝑡)𝑥 for any 𝑥, 𝑦, 𝑧, V,𝑤 ∈0 𝑇 ∪𝑇1. It follows that 𝐷𝑥,𝑦 is a (−1) 𝑅𝑥,[𝑧,𝑡,𝑦]. superderivation of 𝑇 called an inner superderivation of 𝑇. Let 𝐷(𝑇, 𝑇) bethevectorspacespannedbyall Theorem 10. For 𝑥, 𝑦,wehave ∈𝑇 𝐷𝑥,𝑦 (𝑥, 𝑦 ∈ 𝑇). We can define naturally a Z2-gradation by setting 𝑥 𝑦 𝛽(𝑥,𝑦)=str (𝐿𝑥𝐿𝑦)+str (𝑅𝑥,𝑦 + (−1) 𝑅𝑦,𝑥). (8) 𝐷(𝑇, 𝑇)0 =𝐷 (𝑇, 𝑇)1 ⊕𝐷 (𝑇, 𝑇),where𝐷𝑟(𝑇, 𝑇) consists of 4 International Journal of Mathematics and Mathematical Sciences

Proof. Let {𝑎𝑖}, {𝑏𝑖}, {𝑢𝑖}, {V𝑖} be bases of 𝑇0, 𝑇1, 𝐷0(𝑇, 𝑇), In a similar way, we get 𝐷1(𝑇, 𝑇), respectively. For these bases, we express the oper- ations of 𝑇 and 𝐷(𝑇, 𝑇) as follows: 𝑙 [𝑎 ,[𝑎 ,𝑏 ]] = ∑𝑇𝑚𝑇𝑙 𝑏 + ∑𝑇𝑚𝐶𝛼 V 𝑎𝑖𝑎𝑗 = ∑𝑆𝑖𝑗 𝑎𝑙;𝑎𝑖𝑎𝑗 ∈𝑇0, 𝑖 𝑗 𝑘 𝑗𝑘 𝑖𝑚 𝑙 𝑗𝑘 𝑖𝑚 𝛼 𝑚,𝛼 𝑙 𝑚,𝑙 𝑙 − ∑𝐶𝛼 𝐿𝑙 𝑏 , 𝑎𝑖𝑏𝑗 = ∑𝑇𝑖𝑗 𝑏𝑙;𝑎𝑖𝑏𝑗 ∈𝑇1, 𝑗𝑘 𝛼𝑖 𝑙 𝑙 𝛼,𝑙 𝑏 𝑏 = ∑𝑅𝑙 𝑎 ;𝑏𝑏 ∈𝑇, 𝑖 𝑗 𝑖𝑗 𝑙 𝑖 𝑗 0 [𝑎 ,[𝑎 ,𝑢 ]] = − [𝑎 ,𝑢 (𝑎 )] = − [𝑎 , ∑𝐾𝑚 𝑎 ] 𝑙 𝑖 𝑗 𝛼 𝑖 𝛼 𝑗 𝑖 𝛼𝑗 𝑚 𝑚 𝛼 (11) 𝐷 = ∑𝐷 𝑢 ;𝐷 ∈𝐷 (𝑇, 𝑇) , 𝑚 𝑙 𝑚 𝛽 𝑎𝑖,𝑎𝑗 𝑖𝑗 𝛼 𝑎𝑖,𝑎𝑗 0 =−∑𝐾 𝑆 𝑎 − ∑𝐾 𝐷 𝑢 , 𝛼 𝛼𝑗 𝑖𝑚 𝑙 𝛼𝑗 𝑖𝑚 𝛽 𝑚,𝑙 𝑚,𝛽 𝐷 = ∑𝐶𝛼V ;𝐷 ∈𝐷 (𝑇, 𝑇) , 𝑎𝑖,𝑏𝑗 𝑖𝑗 𝛼 𝑎𝑖,𝑏𝑗 1 𝛼 𝑚 [𝑎𝑖,[𝑎𝑗, V𝛼]] = − [𝑎𝑖, V𝛼 (𝑎𝑗)] = − [𝑎𝑖, ∑𝐿𝛼𝑗 𝑏𝑚] 𝐷 = ∑𝑋𝛼𝑢 ;𝐷 ∈𝐷 (𝑇, 𝑇) , (9) 𝑚 𝑏𝑖,𝑏𝑗 𝑖𝑗 𝛼 𝑏𝑖,𝑏𝑗 0 𝛼 𝑚 𝑙 𝑚 𝛽 =−∑𝐿𝛼𝑗 𝑇𝑖𝑚𝑏𝑙 − ∑𝐿𝛼𝑗 𝐶𝑖𝑚V𝛽. 𝑗 𝑚,𝑙 𝑚,𝛽 [𝑢𝛼,𝑎𝑖]=𝑢𝛼 (𝑎𝑖)=∑𝐾𝛼𝑖𝑎𝑗; 𝑗

𝑗 [V𝛼,𝑎𝑖]=V𝛼 (𝑎𝑖)=∑𝐿𝛼𝑖𝑏𝑗; Also, 𝑗

𝑗 [𝑢𝛼,𝑏𝑖]=𝑢𝛼 (𝑏𝑖)=∑𝐻𝛼𝑖𝑏𝑗; 𝑝 𝑗 𝐿 𝐿 (𝑎 )=𝑎(𝑎 𝑎 )=𝑎(∑𝑆 𝑎 ) 𝑎𝑖 𝑎𝑗 𝑘 𝑖 𝑗 𝑘 𝑖 𝑗𝑘 𝑝 𝑝 𝑗 [V𝛼,𝑏𝑖]=V𝛼 (𝑏𝑖)=∑𝑄 𝑎𝑗. 𝛼𝑖 𝑝 𝑝 𝑙 𝑗 = ∑𝑆𝑗𝑘 (𝑎𝑖𝑎𝑝)=∑𝑆𝑗𝑘𝑆𝑖𝑝𝑎𝑙, 𝑝 𝑝,𝑙 To prove the theorem, it suffices to show that 𝛽(𝑎𝑖,𝑎𝑗)= 𝛼(𝑎𝑖,𝑎𝑗), 𝛽(𝑎𝑖,𝑏𝑗)=𝛼(𝑎𝑖,𝑏𝑗) and 𝛽(𝑏𝑖,𝑏𝑗)=𝛼(𝑏𝑖,𝑏𝑗). 𝑝 Since (𝐿𝑎 𝐿𝑏 )(𝑇0)⊆𝑇1 and (𝐿𝑎 𝐿𝑏 )(𝑇1)⊆𝑇0,wehave 𝐿 𝐿 (𝑏 )=𝑎(𝑎 𝑏 )=𝑎(∑𝑇 𝑏 ) 𝑖 𝑗 𝑖 𝑗 𝑎𝑖 𝑎𝑗 𝑘 𝑖 𝑗 𝑘 𝑖 𝑗𝑘 𝑝 (𝐿 𝐿 )=0 𝑅 (𝑇 )⊆𝑇 𝑅 (𝑇 )⊆𝑇 𝑝 str 𝑎𝑖 𝑏𝑗 .Also, 𝑎𝑖,𝑏𝑗 1 0 and 𝑎𝑖,𝑏𝑗 0 1 give (𝑅 +𝑅 )=0 𝛽(𝑎 ,𝑏)=0=𝛼(𝑎,𝑏) 𝑝 𝑝 𝑙 str 𝑎𝑖,𝑏𝑗 𝑏𝑗,𝑎𝑖 and then 𝑖 𝑗 𝑖 𝑗 because = ∑𝑇𝑗𝑘 (𝑎𝑖𝑏𝑝)=∑𝑇𝑗𝑘𝑇𝑖𝑝𝑏𝑙, of the property of 𝛼 (𝑎𝑖 ∈𝑇0 ⊕𝐷0(𝑇, 𝑇),𝑗 𝑏 ∈𝑇1 ⊕ 𝑝 𝑝,𝑙 𝐷1(𝑇, 𝑇)). Hence, it remains to show that 𝛽(𝑎𝑖,𝑎𝑗)=𝛼(𝑎𝑖,𝑎𝑗) (12) 𝛼 and 𝛽(𝑏𝑖,𝑏𝑗)=𝛼(𝑏𝑖,𝑏𝑗).Theoperationsin𝑇 and the identities 𝑅 (𝑎 )=[𝑎,𝑎,𝑎 ]=𝐷 (𝑎 )=∑𝐷 𝑢 (𝑎 ) 𝑎𝑖,𝑎𝑗 𝑘 𝑘 𝑖 𝑗 𝑎𝑘,𝑎𝑖 𝑗 𝑘𝑖 𝛼 𝑗 (7) imply the following: 𝛼 𝛼 𝑚 𝛼 𝑚 [𝑎𝑖,[𝑎𝑗,𝑎𝑘]] = [𝑎𝑖,𝑎𝑗𝑎𝑘 +𝐷𝑎 ,𝑎 ] = ∑𝐷𝑘𝑖∑𝐾𝛼𝑗 𝑎𝑚 = ∑𝐷𝑘𝑖𝐾𝛼𝑗 𝑎𝑚; 𝑗 𝑘 𝛼 𝑚 𝛼,𝑚

=[𝑎, ∑𝑆𝑚 𝑎 + ∑𝐷𝛼 𝑢 ] 𝑅 (𝑏 )=[𝑏,𝑎,𝑎 ]=𝐷 (𝑎 ) 𝑖 𝑗𝑘 𝑚 𝑗𝑘 𝛼 𝑎𝑖,𝑎𝑗 𝑘 𝑘 𝑖 𝑗 𝑏𝑘,𝑎𝑖 𝑗 𝑚 𝛼 =−∑𝐶𝛼 V (𝑎 )=−∑𝐶𝛼 ∑𝐿𝑚 𝑏 = ∑𝑆𝑚 (𝑎 𝑎 +𝐷 )−∑𝐷𝛼 ∑𝐾𝑙 𝑎 𝑖𝑘 𝛼 𝑗 𝑖𝑘 𝛼𝑗 𝑚 𝑗𝑘 𝑖 𝑚 𝑎𝑖,𝑎𝑚 𝑗𝑘 𝛼𝑖 𝑙 𝛼 𝛼 𝑚 𝑚 𝛼 𝑙 𝛼 𝑚 =−∑𝐶𝑖𝑘𝐿𝛼𝑗 𝑏𝑚. 𝛼,𝑚 = ∑𝑆𝑚 (∑𝑆𝑙 𝑎 + ∑𝐷𝛼 𝑢 ) 𝑗𝑘 𝑖𝑚 𝑙 𝑖𝑚 𝛼 (10) 𝑚 𝑙 𝛼 𝛼 𝑙 𝑖 𝑗 − ∑𝐷𝑗𝑘𝐾𝛼𝑖𝑎𝑙 By interchanging and ,wehave 𝛼,𝑙

𝑚 𝑙 𝑚 𝛼 𝛼 𝑚 = ∑𝑆𝑗𝑘𝑆𝑖𝑚𝑎𝑙 + ∑𝑆𝑗𝑘𝐷𝑖𝑚𝑢𝛼 𝑅 (𝑎 )=∑𝐷 𝐾 𝑎 ; 𝑚,𝛼 𝑎𝑗,𝑎𝑖 𝑘 𝑘𝑗 𝛼𝑖 𝑚 𝑚,𝑙 𝛼,𝑚 (13) 𝛼 𝑙 𝛼 𝑚 − ∑𝐷𝑗𝑘𝐾𝛼𝑖𝑎𝑙. 𝑅 (𝑏 )=−∑𝐶 𝐿 𝑏 . 𝑎𝑗,𝑎𝑖 𝑘 𝑗𝑘 𝛼𝑖 𝑚 𝛼,𝑙 𝛼,𝑚 International Journal of Mathematics and Mathematical Sciences 5

Therefore, Now, 𝛽(𝑎,𝑎 )=𝛼(𝑎,𝑎 ) 𝑖 𝑗 𝑖 𝑗 𝐿 𝐿 (𝑎 )=𝑏(𝑏 𝑎 )=−𝑏(∑𝑇𝑚𝑏 ) 𝑏𝑖 𝑏𝑗 𝑘 𝑖 𝑗 𝑘 𝑖 𝑗𝑘 𝑚 𝑚 = ∑𝑆𝑚 𝑆𝑘 − ∑𝐷𝛼 𝐾𝑘 𝑎 − ∑𝑇𝑚𝑇𝑘 + ∑𝐶𝛼 𝐿𝑘 𝑗𝑘 𝑖𝑚 𝑗𝑘 𝛼𝑖 𝑘 𝑗𝑘 𝑖𝑚 𝑗𝑘 𝛼𝑖 𝑚 𝑚 𝑙 𝑚,𝑘 𝛼,𝑘 𝑚,𝑘 𝛼,𝑘 =−∑𝑇𝑗𝑘 (𝑏𝑖𝑏𝑚)=−∑𝑇𝑗𝑘𝑅𝑖𝑚𝑎𝑙; 𝑚 𝑚,𝑙 𝑚 𝛼 𝑚 𝛼 − ∑𝐾𝛼𝑗 𝐷𝑖𝑚 + ∑𝐿𝛼𝑗 𝐶𝑖𝑚, 𝑚,𝛼 𝑚,𝛼 𝑚 𝐿𝑏 𝐿𝑏 (𝑏𝑘)=𝑏𝑖 (𝑏𝑗𝑏𝑘)=𝑏𝑖 (∑𝑅 𝑎𝑚) (14) 𝑖 𝑗 𝑗𝑘 (𝐿 𝐿 )+ (𝑅 +𝑅 ) 𝑚 str 𝑎𝑖 𝑎𝑗 str 𝑎𝑗,𝑎𝑖 𝑎𝑗,𝑎𝑖 =−∑𝑅𝑚 (𝑎 𝑏 )=−∑𝑅𝑚 𝑇𝑙 𝑏 ; 𝑝 𝑘 𝑝 𝑘 𝛼 𝑘 𝛼 𝑘 𝑗𝑘 𝑚 𝑖 𝑗𝑘 𝑚𝑖 𝑙 = ∑𝑆 𝑆 − ∑𝑇 𝑇 + ∑𝐷 𝐾 + ∑𝐷 𝐾 𝑚 𝑗𝑘 𝑖𝑝 𝑗𝑘 𝑖𝑝 𝑘𝑖 𝛼𝑗 𝑘𝑗 𝛼𝑖 𝑚,𝑙 (18) 𝑝,𝑘 𝑝,𝑘 𝛼,𝑘 𝛼,𝑘 𝑅 (𝑎 )=[𝑎,𝑏,𝑏]=𝐷 (𝑏 )=∑𝐶𝛼 V (𝑏 ) 𝑏𝑖,𝑏𝑗 𝑘 𝑘 𝑖 𝑗 𝑎𝑘,𝑏𝑖 𝑗 𝑘𝑖 𝛼 𝑗 𝛼 𝑘 𝛼 𝑘 𝛼 + ∑𝐶𝑖𝑘𝐿𝛼𝑗 + ∑𝐶𝑗𝑘𝐿𝛼𝑖 =𝛽(𝑎𝑖,𝑎𝑗). 𝛼,𝑘 𝛼,𝑘 𝛼 𝑙 𝛼 𝑙 = ∑𝐶𝑘𝑖∑𝑄𝛼𝑗 𝑎𝑙 = ∑𝐶𝑘𝑖𝑄𝛼𝑗 𝑎𝑙; 𝛽(𝑏 ,𝑏)= (𝐿 𝐿 )+ (𝑅 + 𝛼 𝑙 𝛼,𝑙 It remains to show that 𝑖 𝑗 str 𝑏𝑖 𝑏𝑗 str 𝑏𝑗,𝑏𝑖 𝑅𝑏 ,𝑏 ), 𝛼 𝑗 𝑖 𝑅 (𝑏 )=[𝑏,𝑏,𝑏]=𝐷 (𝑏 )=∑𝑋 𝑢 (𝑏 ) 𝑏𝑖,𝑏𝑗 𝑘 𝑘 𝑖 𝑗 𝑏𝑘,𝑏𝑖 𝑗 𝑘𝑖 𝛼 𝑗 𝛼 [𝑏 ,[𝑏,𝑏 ]] = [𝑏 ,𝑏𝑏 +𝐷 ] 𝑖 𝑗 𝑘 𝑖 𝑗 𝑘 𝑏𝑗,𝑏𝑘 𝛼 𝑙 𝛼 𝑙 = ∑𝑋𝑘𝑖∑𝐻𝛼𝑗 𝑏𝑙 = ∑𝑋𝑘𝑖𝐻𝛼𝑗 𝑏𝑙. 𝛼 𝑙 𝛼,𝑙 𝑚 𝛼 =[𝑏𝑖, ∑𝑅𝑗𝑘𝑎𝑚 + ∑𝑋𝑗𝑘𝑢𝛼] 𝑚 𝛼 By interchanging 𝑖 and 𝑗,wehave (15) =−∑𝑅𝑚 𝑇𝑙 𝑏 − ∑𝑅𝑚𝛼 V 𝑅 (𝑎 )=∑𝐶𝛼 𝑄𝑙 𝑎 ; 𝑗𝑘 𝑚𝑖 𝑙 𝑗𝑘𝑚𝑖 𝛼 𝑏𝑗,𝑏𝑖 𝑘 𝑘𝑗 𝛼𝑖 𝑙 𝑚,𝑙 𝑚,𝛼 𝛼,𝑙

− ∑𝑋𝛼 𝐻𝑙 𝑏 . 𝑅 (𝑏 )=∑𝑋𝛼 𝐻𝑙 𝑏 , 𝑗𝑘 𝛼𝑖 𝑙 𝑏𝑗,𝑏𝑖 𝑘 𝑘𝑗 𝛼𝑖 𝑙 𝛼,𝑙 𝛼,𝑙

Likewise, we have (𝐿 𝐿 )+ (𝑅 +𝑅 ) str 𝑏𝑖 𝑏𝑗 str 𝑏𝑗,𝑏𝑖 𝑏𝑗,𝑏𝑖 (19) 𝑚 𝑙 𝑚 𝛼 [𝑏𝑖,[𝑏𝑗,𝑎𝑘]] = −∑𝑇𝑘𝑗 𝑅𝑖𝑚𝑎𝑙 − ∑𝑇𝑘𝑗 𝑋𝑖𝑚𝑢𝛼 𝑚 𝑘 𝑚 𝑘 𝛼 𝑘 𝛼 𝑘 𝑚,𝑙 𝑚,𝛼 =−∑𝑇𝑗𝑘𝑅𝑖𝑚 + ∑𝑅𝑗𝑘𝑇𝑚𝑖 + ∑𝐶𝑘𝑖𝑄𝛼𝑗 − ∑𝑋𝑘𝑖𝐻𝛼𝑗 𝑚,𝑘 𝑚,𝑘 𝛼,𝑘 𝛼,𝑘 𝛼 𝑙 − ∑𝐶𝑘𝑗 𝑄𝛼𝑖𝑎𝑙, 𝛼 𝑘 𝛼 𝑘 𝛼,𝑙 − ∑𝐶𝑘𝑗 𝑄𝛼𝑖 + ∑𝑋𝑘𝑗 𝐻𝛼𝑖𝑏𝑙 =𝛽(𝑏𝑖,𝑏𝑗). 𝛼,𝑘 𝛼,𝑘

𝑙 [𝑏𝑖,[𝑏𝑗,𝑢𝛼]] = − [𝑏𝑖,𝑢𝛼 (𝑏𝑗)] = − [𝑏𝑖, ∑𝐻𝛼𝑗 𝑏𝑙] Hence the theorem is proved. 𝑙 (16) Definition 11. The bilinear form 𝛽 defined on the Lie- 𝑙 𝑚 𝑙 𝛽 𝑇=𝑇 ⊕𝑇 =−∑𝐻𝛼𝑗 𝑅𝑖𝑙 𝑎𝑚 − ∑𝐻𝛼𝑗 𝑋𝑖𝑙𝑢𝛽, Yamaguti superalgebra 0 1 by 𝑙 𝑙,𝛽 𝑥 𝑦 𝛽(𝑥,𝑦)=str (𝐿𝑥𝐿𝑦)+str (𝑅𝑥,𝑦 + (−1) 𝑅𝑦,𝑥) (20) 𝑙 [𝑏𝑖,[𝑏𝑗, V𝛼]] = [𝑏𝑖, V𝛼 (𝑏𝑗)] = [𝑏𝑖, ∑𝑄𝛼𝑗 𝑎𝑙] for 𝑥, 𝑦 ∈𝑇 is called the Killing form of 𝑇. 𝑙 Remark 12. Recall that if 𝑇 is a Lie superalgebra, then the =−∑𝑄𝑙 𝑇𝑚𝑏 − ∑𝑄𝑙 𝐶𝛽V . 𝛼𝑗 𝑙𝑖 𝑚 𝛼𝑗 𝑙𝑖 𝛽 Killing form 𝛽 on 𝑇 is defined as 𝛽(𝑥, 𝑦) = str(𝐿𝑥𝐿𝑦), 𝑙,𝑚 𝑙,𝛽 𝑥, 𝑦.Likewise,if ∈𝑇 𝑇 is a Lie supertriple system (resp., a Lie-Yamaguti algebra), the Killing form on 𝑇 is defined Therefore, 𝑥 𝑦 as 𝛽(𝑥, 𝑦) = str(𝑅𝑥,𝑦 +(−1) 𝑅𝑦,𝑥) (resp., 𝛽(𝑥, 𝑦) = 𝛽(𝑎𝑖,𝑎𝑗)=𝛼(𝑎𝑖,𝑎𝑗) tr(𝐿𝑥𝐿𝑦)+tr(𝑅𝑥,𝑦 +𝑅𝑦,𝑥))with𝐿𝑢 and 𝑅𝑢,V defined according to the considered structure on 𝑇.SoifaLie-Yamaguti 𝑚 𝑘 𝛼 𝑘 𝑚 𝑘 𝑇 =−∑𝑇𝑘𝑗 𝑅𝑖𝑚 − ∑𝐶𝑘𝑗 𝑄𝛼𝑖 + ∑𝑅𝑗𝑘𝑇𝑚𝑖 superalgebra is reduced to a Lie superalgebra (resp., a Lie 𝑚,𝑘 𝛼,𝑘 𝑚,𝑘 (17) supertriple system, a Lie-Yamaguti algebra), then 𝛽 as defined 𝛼 𝑘 𝑙 𝛼 𝑙 𝛼 in Definition 11 is the Killing form of the Lie superalgebra + ∑𝑋𝑗𝑘𝐻𝛼𝑖 − ∑𝐻𝛼𝑗 𝑋𝑖𝑙 + ∑𝑄𝛼𝑗 𝐶𝑙𝑖. (resp., the Lie supertriple system, the Lie-Yamaguti algebra) 𝛼,𝑘 𝑙,𝛼 𝑙,𝛼 𝑇. 6 International Journal of Mathematics and Mathematical Sciences

Proposition 13. Let 𝑇=𝑇0 ⊕𝑇1 be a Lie-Yamaguti superal- Proof. The Killing form 𝛼 of 𝐿=(𝑇0 ⊕𝐷0(𝑇, 𝑇)) ⊕1 (𝑇 ⊕ 𝛽. 𝑥 𝑦 gebra with a Killing form denoted by Then, 𝐷1(𝑇, 𝑇)) satisfies 𝛼(𝑦, [𝑥, 𝑧]) +(−1) 𝛼([𝑥, 𝑦], 𝑧)=0;that 𝛼(𝑦, [𝑥, 𝑧]) = 𝛼([𝑦, 𝑥],𝑧) (1) 𝛽(𝑇0,𝑇1)=0(consistence), is, . But, using (7), we have 𝑥 𝑦 (2) 𝛽(𝑥, 𝑦) = (−1) 𝛽(𝑦, 𝑥) (), 𝛼 ([𝑥, 𝑦] , 𝑧) = 𝛼𝑥,𝑦 (𝑥𝑦+𝐷 ,𝑧) (3) 𝛽(𝐴(𝑥), 𝐴(𝑦)) = 𝛽(𝑥,𝑦), 𝐴∈Aut(𝑇). =𝛼(𝑥𝑦,𝑧)+𝛼(𝐷 ,𝑧) 𝐿 𝐿 (𝑇 )⊆𝑇 𝐿 𝐿 (𝑇 )⊆𝑇 𝑅 (𝑇 )⊆𝑇 𝑥,𝑦 Proof. As 𝑇0 𝑇1 1 0, 𝑇0 𝑇1 0 1, 𝑇0,𝑇1 0 1 and 𝑅𝑇 ,𝑇 (𝑇1)⊆𝑇0 we can state that 𝛽(𝑇0,𝑇1)=0. 0 1 = 𝛽 (𝑥𝑦, 𝑧)+ str (𝐷𝑥,𝑦𝐿𝑧) (2) comes from the definition of 𝛽. Now, for any 𝐴 in Aut(𝑇), 𝑥 in 𝑇, 𝐴(𝑥) = 𝑥,and = 𝛽 (𝑥𝑦, 𝑧) + 𝛾 (𝑥, 𝑦,𝑧), 𝛽(𝐴(𝑥) ,𝐴(𝑦)) (25) 𝛼(𝑦,[𝑥,] 𝑧 )=𝛼(𝑦,𝑥𝑧+𝐷𝑥,𝑧)

= str (𝐿𝐴(𝑥)𝐿𝐴(𝑦)) (21) =𝛼(𝑦,𝑥𝑧)+str (𝐿𝑦𝐷𝑥,𝑧) 𝑥 𝑦 + str (𝑅𝐴(𝑥),𝐴(𝑦) + (−1) 𝑅𝐴(𝑦),𝐴(𝑥)). 𝑦(𝑥+𝑧) = 𝛽 (𝑦, 𝑥𝑧) + (−1) str (𝐷𝑥,𝑧𝐿𝑦)

As 𝐴(𝑥𝑦) = 𝐴(𝑥)𝐴(𝑦) then 𝐴𝐿𝑥(𝑦) = 𝐿𝐴(𝑥)𝐴(𝑦);thatis, 𝑦(𝑥+𝑧) −1 = 𝛽 (𝑦, 𝑥𝑧) + (−1) 𝛾(𝑥,𝑧,𝑦). 𝐴𝐿𝑥 =𝐿𝐴(𝑥)𝐴 and 𝐴𝐿𝑥𝐴 =𝐿𝐴(𝑥). −1 −1 Hence, str(𝐿𝐴(𝑥)𝐿𝐴(𝑦))=str(𝐴𝐿𝑥𝐴 𝐴𝐿𝑦𝐴 )= 𝛼(𝑦, [𝑥, 𝑧])𝑥 +(−1) 𝑦𝛼([𝑥, 𝑦], 𝑧)=0 −1 Then the identity gives str(𝐴𝐿𝑥𝐿𝑦𝐴 )=str(𝐿𝑥𝐿𝑦).Also,𝐴[𝑥, 𝑦, 𝑧] = [𝐴(𝑥), 𝐴(𝑦), 𝛽(𝑥𝑦, 𝑧) + 𝛾(𝑥, 𝑦, 𝑧)+(−1)𝑥 𝑦𝛽(𝑦, 𝑥𝑧)𝑦 +(−1) 𝑧𝛾(𝑥, 𝑧, 𝑦)= 𝐴(𝑧)] 𝐴𝑅 (𝑥) = 𝑅 (𝐴(𝑥)) 𝐴𝑅 𝐴−1 = 𝑥 𝑦 gives 𝑦,𝑧 𝐴(𝑦),𝐴(𝑧) ;thatis, 𝑦,𝑧 0 that is 𝛽(𝑥𝑦, 𝑧) + (−1) 𝛽(𝑦, 𝑥𝑧) = −𝛾(𝑥, 𝑦,𝑧)− 𝑅 𝑦 𝑧 𝑥 𝑦 (𝑥+𝑦)𝑧 𝐴(𝑦),𝐴(𝑧).Then, (−1) 𝛾(𝑥, 𝑧, 𝑦) = (−1) 𝛾(𝑦, 𝑥, 𝑧) + (−1) 𝛾(𝑧, 𝑥, 𝑦) and 𝛽(𝐴(𝑥) , 𝐴 (𝑦)) (23) is obtained. From 𝛼([𝑦, 𝑥], 𝑧) = 𝛼(𝑦, [𝑥,𝑧]) we deduce 𝛼(𝑥, [𝑤, [𝑦, 𝑧]]) = 𝛼([𝑥, 𝑤], [𝑦, 𝑧]) = 𝛼([[𝑥, 𝑤],𝑦],𝑧) = str (𝐿𝐴(𝑥)𝐿𝐴(𝑦)) that is 𝑤(𝑦+𝑧) −(−1) 𝛼(𝑥, [[𝑦, 𝑧], 𝑤]) = 𝛼([[𝑥, 𝑤],𝑦],𝑧) and 𝑥 𝑦 + str (𝑅𝐴(𝑥),𝐴(𝑦) + (−1) 𝑅𝐴(𝑦),𝐴(𝑥)) 𝑤(𝑦+𝑧) 𝛼 (𝑥, [[𝑦,] 𝑧 ,𝑤]) + (−1) 𝛼 ([[𝑥,] 𝑤 ,𝑦] ,𝑧) =0. (26) = str (𝐿𝑥𝐿𝑦) (22) Then, using (7) again and developing (26), we have 𝛼(𝑥, [𝑦𝑧+ −1 𝑥 𝑦 −1 𝑤(𝑦+𝑧) + str (𝐴𝑅𝑥,𝑦𝐴 + (−1) 𝐴𝑅𝑦,𝑥𝐴 ) 𝐷𝑦,𝑧,𝑤])+(−1) 𝛼([𝑥𝑤𝑥,𝑤 +𝐷 , 𝑦], 𝑧) =0 and we get = (𝐿 𝐿 )+ (𝐴 (𝑅 + (−1)𝑥 𝑦 𝑅 )𝐴−1) str 𝑥 𝑦 str 𝑥,𝑦 𝑦,𝑥 𝛼 (𝑥, (𝑦𝑧)𝑦𝑧,𝑤 𝑤+𝐷 +[𝑦,𝑧,𝑤])

= (𝐿 𝐿 )+ (𝑅 + (−1)𝑥 𝑦 𝑅 ) 𝑤(𝑦+𝑧) str 𝑥 𝑦 str 𝑥,𝑦 𝑦,𝑥 + (−1) 𝛼((𝑥𝑤) 𝑦+𝐷𝑥𝑤,𝑦 + [𝑥, 𝑤, 𝑦] ,𝑧) (27) =𝛽(𝑥,𝑦). =0.

This gives Now, let 𝛾 be a trilinear form in 𝑇 given by 𝛾(𝑥, 𝑦, 𝑧)= (𝐷𝑥,𝑦𝐿𝑧) 𝑥, 𝑦, 𝑧∈𝑇 str for any .Wecaneasilyseethat,for 𝛽 (𝑥, (𝑦𝑧) 𝑤)𝑦𝑧,𝑤 +𝛼(𝑥,𝐷 ) + 𝛽 (𝑥, [𝑦, 𝑧, 𝑤]) 𝑥 𝑦 any 𝑥, 𝑦,, 𝑧∈𝑇 𝛾(𝑥, 𝑦, 𝑧) = −(−1) 𝛾(𝑦, 𝑥, 𝑧) and that 𝛾 vanishes identically if 𝑇 is reduced to Lie superalgebra or Lie + (−1)𝑤(𝑦+𝑧) 𝛽((𝑥𝑤) 𝑦, 𝑧) supertriple system. 𝑤(𝑦+𝑧) + (−1) 𝛼(𝐷𝑥𝑤,𝑦,𝑧) Theorem 14. Let 𝑇=𝑇0 ⊕𝑇1 be a Lie-Yamaguti superalgebra with a Killing form denoted by 𝛽. Then, 𝛽 satisfies the identities + (−1)𝑤(𝑦+𝑧) 𝛽 ([𝑥, 𝑤, 𝑦] , 𝑧) =0, 𝛽(𝑥𝑦,𝑧)+(−1)𝑥 𝑦 𝛽(𝑦,𝑥𝑧) (28) 𝛽 (𝑥, (𝑦𝑧) 𝑤)+ (𝐿𝑥𝐿𝐷 ) + 𝛽 (𝑥, [𝑦, 𝑧, 𝑤]) (23) str 𝑦𝑧,𝑤 = (−1)𝑥 𝑦 𝛾(𝑦,𝑥,𝑧)+(−1)(𝑥+𝑦)𝑧 𝛾 (𝑧, 𝑥, 𝑦); + (−1)𝑤(𝑦+𝑧) 𝛽((𝑥𝑤) 𝑦, 𝑧) 𝛽 (𝑥, [𝑦, 𝑧, 𝑤])+ (−1)𝑤(𝑦+𝑧) 𝛽 ([𝑥, 𝑤, 𝑦] ,𝑧) + (−1)𝑤(𝑦+𝑧) (𝐿 𝐿 ) (24) str 𝐷𝑥𝑤,𝑦 𝑧 = (−1)𝑤(𝑦+𝑧) 𝛾(𝑥,𝑤,𝑦𝑧)−(−1)𝑥(𝑦+𝑧) 𝛾(𝑦,𝑧,𝑥𝑤) + (−1)𝑤(𝑦+𝑧) 𝛽 ([𝑥, 𝑤, 𝑦] , 𝑧) =0. for all 𝑥, 𝑦, 𝑧 ∈𝑇. International Journal of Mathematics and Mathematical Sciences 7

Thus, Corollary 15. Let 𝑇=𝑇0 ⊕𝑇1 be a Lie-Yamaguti superalgebra with a Killing form denoted by 𝛽. Then, 𝛽 satisfies the following 𝛽 (𝑥, (𝑦𝑧) 𝑤)+ (−1)𝑥(𝑦+𝑧+𝑤) (𝐿 𝐿 ) for 𝑥, 𝑦,: 𝑧∈𝑇 str 𝐷𝑦𝑧,𝑤 𝑥 ([𝑦, 𝑧, 𝑥] ,𝑤)+ (−1)𝑥(𝑦+𝑧) 𝛽 (𝑥, [𝑦, 𝑧, 𝑤])=0; + 𝛽 (𝑥, [𝑦, 𝑧, 𝑤])+ (−1)𝑤(𝑦+𝑧) 𝛽((𝑥𝑤) 𝑦, 𝑧) (34) (29) 𝛽(𝑅 (𝑥) ,𝑧)−(−1)(𝑤+𝑦)𝑥+𝑤 𝑦 𝛽(𝑥,𝑅 (𝑧)) + (−1)𝑤(𝑦+𝑧) (𝐿 𝐿 ) 𝑤,𝑦 𝑦,𝑤 str 𝐷𝑥𝑤,𝑦 𝑧 = (−1)𝑥(𝑤+𝑦) 𝛾(𝑥,𝑤,𝑦𝑧) (35) + (−1)𝑤(𝑦+𝑧) 𝛽 ([𝑥, 𝑤, 𝑦] , 𝑧) =0; + (−1)𝑤(𝑦+𝑧)+𝑥 𝑧 𝛾(𝑦,𝑧,𝑥𝑤). that is, Proof. Using (24) we have 𝑥(𝑦+𝑧+𝑤) 𝛽 (𝑥, (𝑦𝑧) 𝑤)+ (−1) 𝛾(𝑦𝑧,𝑤,𝑥) 𝛽([𝑦,𝑧,𝑥],𝑤)

𝑤(𝑦+𝑧) +𝛽(𝑥,[𝑦,𝑧,𝑤])+(−1) 𝛽((𝑥𝑤) 𝑦, 𝑧) = (−1)(𝑤+𝑥)(𝑦+𝑧) 𝛽 ([𝑥, 𝑤, 𝑦] ,𝑧) (30) 𝑤(𝑦+𝑧) (36) + (−1) 𝛾(𝑥𝑤,𝑦,𝑧) − (−1)(𝑤+𝑥)(𝑦+𝑧) 𝛾(𝑥,𝑤,𝑦𝑧)

𝑤(𝑦+𝑧) + (−1) 𝛽 ([𝑥, 𝑤, 𝑦] , 𝑧) =0. − (−1)𝑤 𝑥 𝛾(𝑦,𝑧,𝑤𝑥);

𝑥(𝑦+𝑧) This implies (−1) 𝛽 (𝑥, [𝑦, 𝑧, 𝑤])

(𝑤+𝑥)(𝑦+𝑧) 𝛽 (𝑥, [𝑦, 𝑧, 𝑤])+ (−1)𝑤(𝑦+𝑧) 𝛽 ([𝑥, 𝑤, 𝑦] ,𝑧) =−(−1) 𝛽 ([𝑥, 𝑤, 𝑦] ,𝑧) (37) (𝑤+𝑥)(𝑦+𝑧) = −𝛽 (𝑥, (𝑦𝑧) 𝑤)− (−1)𝑤(𝑦+𝑧) 𝛽((𝑥𝑤) 𝑦, 𝑧) + (−1) 𝛾(𝑥,𝑤,𝑦𝑧) (31) 𝑤 𝑥 − (−1)𝑥(𝑦+𝑧+𝑤) 𝛾(𝑦𝑧,𝑤,𝑥) + (−1) 𝛾(𝑦,𝑧,𝑤𝑥).

− (−1)𝑤(𝑦+𝑧) 𝛾(𝑥𝑤,𝑦,𝑧). By adding memberwise (36) and (37) we obtain the identity (34). 𝑦 𝑧 Also, the identity (37) is equivalent to −(−1) 𝛽(𝑥, [𝑧, 𝑦, But (23) gives 𝑤]) + (−1)𝑤(𝑦+𝑧)𝛽([𝑥, 𝑤, 𝑦], 𝑧)𝑤( =(−1) 𝑦+𝑧)𝛾(𝑥, 𝑤, 𝑦𝑧) − (−1)𝑥(𝑦+𝑧)𝛾(𝑦, 𝑧,𝑥𝑤). (−1)𝑤 𝑦+𝑥(𝑤+𝑦)𝛽(𝑅 (𝑥), 𝑤 𝑥 Then, we obtain 𝑤,𝑦 (−1) 𝛽 (𝑥, 𝑤 (𝑦𝑧)) + 𝛽 (𝑤𝑥,𝑦𝑧) 𝑤 𝑦 𝑥(𝑦+𝑧)+𝑤 𝑧 𝑧, 𝑧) − 𝛽(𝑥,𝑦,𝑤 𝑅 (𝑧)) = (−1) 𝛾(𝑤,𝑥, 𝑦𝑧) −(−1) 𝛾(𝑦, 𝑤 𝑥 (𝑤+𝑦)𝑥+𝑤 𝑦 = (−1) 𝛾(𝑥,𝑤,𝑦𝑧) 𝑧, 𝑥𝑤) that is 𝛽(𝑅𝑤,𝑦(𝑥), 𝑧) − (−1) 𝛽(𝑥,𝑦,𝑤 𝑅 (𝑧)) = (−1)𝑥(𝑤+𝑦)𝛾(𝑥, 𝑤, 𝑦𝑧) +(−1)𝑤(𝑦+𝑧)+𝑥 𝑧𝛾(𝑦, 𝑧, 𝑥𝑤) (𝑤+𝑥)(𝑦+𝑧) and the + (−1) 𝛾(𝑦𝑧,𝑤,𝑥), (32) remaining assertion is proved. (−1)𝑤 𝑥 𝛽((𝑥𝑤) 𝑦, 𝑧) + 𝛽 (𝑤𝑥, 𝑦𝑧) 4. Invariant Forms of Lie-Yamaguti =𝛾(𝑤𝑥,𝑦,𝑧)+(−1)(𝑤+𝑥)(𝑦+𝑧)+𝑦 𝑧 𝛾(𝑧,𝑦,𝑤𝑥). Superalgebras In this section we introduced the concept of invariant forms Then, of Lie-Yamaguti superalgebras as generalizations of those of LiesuperalgebrasandLiesupertriplesystems. − 𝛽 (𝑥, 𝑤 (𝑦𝑧)) − (−1)𝑤(𝑦+𝑧) 𝛽((𝑥𝑤) 𝑦, 𝑧) Definition 16. An invariant form 𝑏 of a Lie-Yamaguti super- 𝑤(𝑦+𝑧) = (−1) 𝛾(𝑥,𝑤,𝑦𝑧) algebra 𝑇=𝑇0 ⊕𝑇1 is a supersymmetric bilinear form on 𝑇 satisfying the identities 𝑥(𝑤+𝑦+𝑧) + (−1) 𝛾(𝑦𝑧,𝑤,𝑥) (33) 𝑥 𝑦 𝑏(𝑥𝑦,𝑧)+(−1) 𝑏(𝑦,𝑥𝑧)=0; (38) − (−1)𝑤(𝑥+𝑦+𝑧) 𝛾(𝑤𝑥,𝑦,𝑧) 𝑤(𝑦+𝑧) 𝑏 ([𝑥, 𝑤,] 𝑦 ,𝑧) + (−1) 𝑏 (𝑥, [𝑦, 𝑧,]) 𝑤 =0 (39) − (−1)𝑥(𝑤+𝑦+𝑧)+𝑦 𝑧 𝛾 (𝑧, 𝑦, 𝑤𝑥). for all 𝑥, 𝑦, 𝑧,𝑤 in 𝑇. 𝑤(𝑦+𝑧) 𝑤(𝑦+𝑧) Hence, 𝛽(𝑥,[𝑦,𝑧,𝑤])+(−1) 𝛽([𝑥,𝑤,𝑦], 𝑧)= (−1) 𝛾(𝑥, Remark 17. (1) If 𝛾=0,theKillingformof𝑇 is an invariant 𝑥(𝑦+𝑧) 𝑤, 𝑦𝑧) − (−1) 𝛾(𝑧, 𝑦, 𝑤𝑥) and (24) is proved. form of 𝑇. 8 International Journal of Mathematics and Mathematical Sciences

(2) If 𝑇 is reduced to a Lie supertriple system (resp., a Lie Then, using the invariance of 𝛼 and (43), we have, for any superalgebra, a Lie-Yamaguti algebra), then 𝑏 is reduced to 𝑥, 𝑦, 𝑧,: 𝑤∈𝑇 𝛼([𝑥, 𝑦],𝑧,𝑤 𝐷 ) = 𝛼(𝑥, [𝑦,𝑧,𝑤 𝐷 ]);thatis,by an invariant form of a Lie supertriple system [19] (resp., a Lie (7), 𝑦(𝑧+𝑤) superalgebra [12], a Lie-Yamaguti algebra [10]). 𝛼(𝑥𝑦𝑥,𝑦 +𝐷 ,𝐷𝑧,𝑤) = −(−1) 𝛼(𝑥, [𝑧, 𝑤, 𝑦]) and 𝑦(𝑧+𝑤) 𝛼(𝐷𝑥,𝑦,𝐷𝑧,𝑤) = −(−1) 𝛼(𝑥, [𝑧, .Thisgives𝑤, 𝑦]) Definition 18. Let 𝑏 be an invariant form of a Lie-Yamaguti ⊥ superalgebra 𝑇 and 𝑆 asubsetof𝑇.Theorthogonal𝑆 of 𝑆 𝑦(𝑧+𝑤) ⊥ 𝛼(𝐷 ,𝐷 )=−(−1) 𝛽 (𝑥, [𝑧, 𝑤, 𝑦]). with respect to 𝑏 is defined by 𝑆 ={𝑥∈𝑇,𝑏(𝑥,𝑦)=0,∀𝑦∈ 𝑥,𝑦 𝑧,𝑤 (44) ⊥ 𝑆}.Theinvariantform𝑏 is nondegenerate if 𝑇 ={0}. Thus, if 𝛽 is nondegenerate, the restriction of 𝛼 on 𝐷(𝑇, 𝑇) × 𝐷(𝑇, 𝑇) 𝛼 Lemma 19. Let 𝑏 be an invariant form of a Lie-Yamaguti is nondegenerate and is nondegenerate. Now, suppose that 𝛽 is degenerate. Then by the lemma superalgebra 𝑇.Then,forany𝑥, 𝑦, 𝑧, 𝑤 in 𝑇,wehave ⊥ ⊥ ⊥ above, 𝑇 is an ideal of 𝑇 so 𝑇 ⊕𝐷(𝑇,𝑇 ) is a nonzero ideal (𝑥+𝑦)𝑧 𝑏 ([𝑥, 𝑦,] 𝑤 ,𝑧) + (−1) 𝑏 (𝑤, [𝑥, 𝑦,]) 𝑧 =0. (40) of 𝑇. Using the identities (43) and (44) we get Proof. By interchanging 𝑦 and 𝑤 in (39) we have ⊥ ⊥ 𝑦(𝑤+𝑧) 𝛼(𝑇 ⊕𝐷(𝑇,𝑇 ),𝑇⊕𝐷(𝑇, 𝑇)) 𝑏 ([𝑥, 𝑦, 𝑤] ,𝑧)+ (−1) 𝑏 (𝑥, [𝑤, 𝑧, 𝑦])=0 (41) =𝛼(𝑇⊥,𝑇)+𝛼(𝑇⊥,𝐷(𝑇, 𝑇))+𝛼(𝐷(𝑇,𝑇⊥),𝑇) (−1)(𝑥+𝑦)𝑧𝑏(𝑧,[𝑥,𝑦,𝑤])+(−1)𝑦(𝑤+𝑧)+𝑤 𝑧𝑏(𝑥, [𝑤, 𝑧, 𝑦])= that is (45) 0 by supersymmetry. Also by switching 𝑧 and 𝑤 in (41), we +𝛼(𝐷(𝑇,𝑇⊥),𝐷(𝑇, 𝑇)) 𝑦(𝑤+𝑧) obtain 𝑏([𝑥,𝑦,𝑧],𝑤)̃ + (−1) 𝑏(𝑥, [𝑧, 𝑤, 𝑦])=0 that is =𝛼(𝑇⊥,𝑇)+𝛽(𝑇,[𝑇,𝑇⊥,𝑇])=0. 𝑦(𝑤+𝑧)+𝑤 𝑧 𝑏 ([𝑥, 𝑦,] 𝑧 ,𝑤) − (−1) 𝑏 (𝑥, [𝑤, 𝑧, 𝑦]) =0. (42) It comes that 𝛼 is degenerate and 𝑇⊕𝐷(𝑇, 𝑇) is not semisimple Thus adding (41) and (42) we get (40) whence the lemma. which proves the theorem. Lemma 20. 𝑏 Let be an invariant form of a Lie-Yamaguti Theresultsofthispapercouldbeusedforastudyofthe 𝑇 superalgebra .Then, structure of a pair consisting of a semisimple Lie superalgebra ⊥ (1) (𝑇 + [𝑇, 𝑇, 𝑇]) = 𝑍(𝑇) if 𝑏 is nondegenerate; and its semisimple graded subalgebra. ⊥ (2) if 𝐻 is an ideal of 𝑇 then 𝐻 is an ideal of 𝑇.In ⊥ particular, 𝑇 is an ideal of 𝑇. Competing Interests

⊥ Proof. Consider 𝑥 in (𝑇 + [𝑇, 𝑇, 𝑇]) .Then,forany𝑢, V,𝑤 ∈ The authors declare that they have no competing interests. 𝑇,wehave𝑏(𝑥, 𝑢V)=0and 𝑏(𝑥, [𝑢, V,𝑤]) =.This 0 𝑢 𝑥 implies, by (38) and (39), that (−1) 𝑏(𝑢𝑥, V)=0and 𝑤(𝑢+V) References (−1) 𝑏([𝑥, 𝑤, 𝑢], V)=0that is 𝑏(𝑢𝑥, V)=0and 𝑏([𝑥, 𝑤, 𝑢], V)=0.As𝑏 is nondegenerate, we get 𝑢𝑥 =0 and [1] K. Yamaguti, “On the Lie triple system and its generalization,” [𝑥,𝑤,𝑢]=0for any 𝑢, V,𝑤∈𝑇.Thisgives𝑥 ∈ 𝑍(𝑇). Journal of Science of the Hiroshima University, Series A,vol.21, 󸀠 󸀠 󸀠 Conversely, if 𝑥 ∈ 𝑍(𝑇),wehave,forany𝑢, V,𝑢 , V ,𝑤 ∈ pp.107–113,1957/1958. 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 𝑇, 𝑏(𝑥, 𝑢V +[𝑢, V ,𝑤 ]) = 𝑏(𝑥, 𝑢V)+𝑏(𝑥,[𝑢, V ,𝑤 ]) = 0 and [2] K. Nomizu, “Invariant affine connections on homogeneous ⊥ ⊥ 𝑥 ∈ (𝑇 + [𝑇, 𝑇, 𝑇]) whence (𝑇 + [𝑇, 𝑇, 𝑇]) = 𝑍(𝑇). spaces,” American Journal of Mathematics,vol.76,pp.33–65, Now, suppose that 𝐻 is an ideal of 𝑇 that is 𝑇𝐻⊆ 𝐻 and 1954. ⊥ [𝑇,𝐻,𝑇]⊆𝐻;thenforany𝑥, 𝑦, ∈𝑇 𝑢∈𝐻 ,andℎ∈𝐻,we [3] M. Kikkawa, “ of homogeneous Lie loops,” Hiroshima 𝑥 𝑢 Mathematical Journal,vol.5,no.2,pp.141–179,1975. have 𝑏(𝑥𝑢, ℎ) = −(−1) 𝑏(𝑢, 𝑥ℎ) =0 and 𝑏([𝑥, 𝑢, 𝑦], ℎ)= 𝑥 𝑢 𝑥(𝑦+ℎ)+𝑥 𝑢 [4] M. K. Kinyon and A. Weinstein, “Leibniz algebras, Courant −(−1) 𝑏([𝑢, 𝑥, 𝑦], ℎ) = (−1) 𝑏(𝑢, [𝑦, ℎ,. 𝑥])=0 ⊥ ⊥ ⊥ ⊥ algebroids, and multiplications on reductive homogeneous Then 𝑇𝐻 ⊆𝐻 and [𝑇, 𝐻 ,𝑇]⊆𝐻 which proves (2). spaces,” American Journal of Mathematics,vol.123,no.3,pp. 525–550, 2001. We are now ready to prove the following theorem. [5] P. Benito, A. Elduque, and F. Mart´ın-Herce, “Irreducible Lie- Theorem 21. 𝑇=𝑇 ⊕𝑇 Yamaguti algebras,” Journal of Pure and Applied Algebra,vol.213, Let 0 1 be a Lie-Yamaguti superalgebra no.5,pp.795–808,2009. with 𝛾=0.ThentheKillingform𝛽 is nondegenerate if and only 𝐿(𝑇) = 𝑇⊕𝐷(𝑇, 𝑇) [6] P. Benito, A. Elduque, and F. Mart´ın-Herce, “Irreducible Lie- if the standard enveloping Lie superalgebra Yamaguti algebras of generic type,” Journal of Pure and Applied is a semisimple Lie superalgebra. Algebra,vol.215,no.2,pp.108–130,2011. [7] J. Lin, L. Y. Chen, and Y. Ma, “On the deformation of Lie- Proof. Let 𝛼 be the Killing form of the Lie superalgebra 𝐿(𝑇) 𝛾=0 𝑥, 𝑦, 𝑧∈𝑇 𝛾(𝑥, 𝑦, 𝑧)= Yamaguti algebras,” Acta Mathematica Sinica (English Series), .If ,wehave,forany , vol. 31, no. 6, pp. 938–946, 2015. str(𝐷𝑥,𝑦𝐿𝑧)=0and [8] T. Zhang and J. Li, “Deformations and extensions of Lie- 𝛼(𝐷 ,𝑧)=0. Yamaguti algebras,” Linear and ,vol.63,no. 𝑥,𝑦 (43) 11,pp.2212–2231,2015. International Journal of Mathematics and Mathematical Sciences 9

[9] M. Kikkawa, “On Killing-Ricci forms of Lie triple algebras,” Pacific Journal of Mathematics,vol.96,no.1,pp.153–161,1981. [10] M. Kikkawa, “Remarks on invariant forms of Lie triple algebras,” Memoirs of the Faculty of Science and Engineering, Shimane University,vol.16,pp.23–27,1982. [11] V.G. Kac, “Lie superalgebras,” Advances in Mathematics,vol.26, no.1,pp.8–96,1977. [12] M. Scheunert, The Theory of Lie Superalgebras, Springer, Berlin, Germany, 1979. [13] H. Tilgner, “A graded generalization of Lie triples,” Journal of Algebra, vol. 47, no. 1, pp. 190–196, 1977. [14] S. Okubo, “Parastatistics as Lie-supertriple systems,” Journal of ,vol.35,no.6,pp.2785–2803,1994. [15]M.F.Ouedraogo,´ Sur les Superalgebres` triples de Lie, These` de Doctorat 3ecycleMath` ematiques´ [Ph.D. thesis],Universitede´ Ouagadougou, Ouagadougou, Burkina Faso, 1999. [16] P. L. Zoungrana, “A note on Lie-Yamaguti superalgebras,” Far East Journal of Mathematical Sciences,vol.100,no.1,pp.1–18, 2016. [17] T. S. Ravisankar, “Some remarks on Lie triple systems,” Kumamoto Journal of Science (Mathematics), vol. 11, pp. 1–8, 1974. [18] S. Okubo and N. Kamiya, “Quasi-classical Lie superalgebras and Lie supertriple systems,” Communications in Algebra,vol.30,no. 8, pp. 3825–3850, 2002. [19] Z. Zhixue and J. Peipei, “The Killing forms and decomposition theorems of Lie supertriple systems,” Acta Mathematica Scien- tia. Series B,vol.29,no.2,pp.360–370,2009. [20] I. P.Shestakov, “Prime Mal’tsev superalgebras,” Matematicheskii Sbornik,vol.182,no.9,pp.1357–1366,1991(Russian). Advances in Advances in Journal of Journal of Decision Sciences Algebra Probability and Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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