On Killing Forms and Invariant Forms of Lie-Yamaguti Superalgebras

Abstract

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

1. Introduction

A Lie-Yamaguti algebra is a triple consisting of a vector space , a bilinear map , and a trilinear map such that(LY1),(LY2),(LY3),(LY4),(LY5),(LY6),for all , in , where denotes the sum over cyclic permutation of , , . The bilinear map sometimes will be denoted by juxtaposition. If , , one gets a Lie triple system , while in induces a Lie algebra .

Lie-Yamaguti algebras were introduced by Yamaguti [1] (who formerly called them “generalized Lie triple systems”) in an algebraic study of the characteristic properties of the torsion and curvature of a homogeneous space with canonical connection [2]. Later on, these algebraic objects were called “Lie triple algebras” [3] and the terminology of “Lie-Yamaguti algebras” is introduced in [4] for these algebras. For further development of the theory of Lie-Yamaguti algebras one may refer, for example, to [5–8]. From the standard enveloping Lie algebra of a given Lie-Yamaguti algebra, the notions of the Killing-Ricci form and the invariant form of a Lie-Yamaguti algebra are introduced and studied in [9]. Further properties of invariant forms of Lie-Yamaguti algebras were considered in [10].

Lie superalgebras as a -graded generalization of Lie algebras are considered in [11, 12] while a -graded generalization of Lie triple systems (called Lie supertriple systems) was first considered in [13]. For an application of Lie supertriple systems in physics, one may refer to [14]. Next, Lie-Yamaguti superalgebras as a -graded generalization of Lie-Yamaguti algebras were first considered in [15].

Definition 1 (see [16]). A Lie-Yamaguti superalgebra is a -graded vector space with a binary operation denoted by juxtaposition satisfying and a ternary operation satisfying () such that(LYS1),(LYS2),(LYS3),(LYS4),(LYS5),(LYS6),for all , in .

Observe that is a Lie-Yamaguti algebra.

As a part of the general theory of superalgebras, the notion of the Killing form of Lie algebras is extended to the one of Lie triple systems (see [17] and references therein), Lie superalgebras [12], and next Lie supertriple systems [18] (see also [19]).

In this paper we define and study the Killing form and invariant form of Lie-Yamaguti superalgebras as a generalization of the ones of both of Lie-Yamaguti algebras [9] and Lie supertriple systems [18, 19] including Lie superalgebras [12].

The paper is organized as follows. In Section 2 we record some useful results on Lie-Yamaguti superalgebras (see [16]). In Section 3 the Killing form of a Lie-Yamaguti superalgebra is defined (see Theorem 10 and Definition 11) and some of its properties are investigated (Proposition 13, Theorem 14, and Corollary 15). In Section 4 the invariant form of a Lie-Yamaguti superalgebra is defined (Definition 16) and, under some conditions, it is shown (Theorem 21) that the Killing form of a Lie-Yamaguti superalgebra is nondegenerate if and only if the standard enveloping Lie superalgebra of is semisimple.

All vector spaces and algebras are finite-dimensional over a fixed ground field of characteristic 0.

2. Some Basics on Lie-Yamaguti Superalgebras

We give here some definitions and results which can be found in [11, 12, 16].

A superalgebra over is a -graded algebra , where , (). The subspaces and are called the even and the odd parts of the superalgebra and so are called the elements from and from , respectively. Below, all the elements are assumed to be homogeneous, that is, either even or odd, and for a homogeneous element , , the notation is used and means the parity of .

Let be the Grassmann algebra over generated by the elements such that , for . The elements 1, , form a basis of . Denote by (resp., ) the span of the products of even length (resp., odd length) in the generators. The product of zero ’s is by convention equal to 1. Then is an associative and supercommutative superalgebra; that is, , where Let be a superalgebra. Consider the graded tensor product which becomes a superalgebra with the product given by , for homogeneous elements , and grading given by , The subalgebra is called the Grassmann envelope of the superalgebra .

Having in mind that if is a homogeneous variety of algebras [20], a superalgebra is called a -superalgebra, if its Grassmann envelope belongs to , we can state the following proposition.

Proposition 2. A superalgebra equipped with bilinear and trilinear products verifying and is a Lie-Yamaguti superalgebra if its Grassmann envelope is a Lie-Yamaguti algebra under the following products:

Proof. The proof is straightforward by using the fact that, for any element in , we have .

Example 3. (1) Lie superalgebras are Lie-Yamaguti superalgebras with .
(2) If for any then (LYS2), (LYS3), and (LYS6) define a Lie supertriple system.
(3) Let be a Malcev superalgebra; that is, for any in , It is shown in [16] that becomes a Lie-Yamaguti superalgebra if we set . Conversely, if on a Malcev superalgebra we define a trilinear operation by then is a Lie-Yamaguti superalgebra.

Definition 4. Let be a Lie-Yamaguti superalgebra. A graded subspace of is a graded Lie-Yamaguti subalgebra of if and for any

Definition 5. A graded subalgebra of a Lie-Yamaguti superalgebra is an invariant graded subalgebra (resp., an ideal) of if (resp., and ).

If is an ideal of , it is an invariant graded subalgebra of . Obviously the center of a Lie-Yamaguti superalgebra defined by is an ideal of .

Definition 6. Let and be Lie-Yamaguti superalgebras. A linear map is said to be of degree if for all

Definition 7. Let and be Lie-Yamaguti superalgebras. A linear map is called a homomorphism of Lie-Yamaguti superalgebras if(1) preserves the grading, that is, , ;(2);(3) for any

Recall [11] that if is a -graded vector space then, if we set , we obtain an associative superalgebra ; consists of the linear mappings of into itself which are homogeneous of degree The bracket makes into a Lie superalgebra which we denote by or where and . Let be a basis of . In this basis the matrix of is expressed as , being an -, an -, an -, and an -matrix. The matrices of even elements have the form and those of odd ones . For , the supertrace of is defined by and does not depend on the choice of a homogeneous basis. We have that is and .

Definition 8. Let be a Lie-Yamaguti superalgebra; is a superderivation of if, for any ,

Let consist of all the superderivations of degree and It is easy to check that is a graded subalgebra of called the Lie superalgebra of superderivations of

Let be a Lie-Yamaguti superalgebra. For any , denote by the endomorphism of defined by for any We have, for any , , ; that is, is a linear map of degree Moreover, it comes from (LYS5) and (LYS6) that for any It follows that is a superderivation of called an inner superderivation of .

Let be the vector space spanned by all

We can define naturally a -gradation by setting , where consists of the superderivation of degree From (5) we also have that, for any , It is clear from (6) that is a -graded Lie subalgebra of called the Lie superalgebra of all inner superderivations of T.

Now, let be a Lie-Yamaguti superalgebra.

Set , , and define a new bracket operation in as follows: for any , ,

Theorem 9. Let be a Lie-Yamaguti superalgebra. Then (1) is a Lie superalgebra called the standard enveloping Lie superalgebra of and becomes a graded subalgebra of .(2)If is an ideal of then is an ideal of .

Proof. The bracket is bilinear by definition and for any by (LYS1) and (LYS2). Jacobi’s superidentity follows from (LTS3–6).
(2) is obvious.

3. Killing Forms of Lie-Yamaguti Superalgebras

The definition of the Killing form given here for Lie-Yamaguti superalgebras stems from [9] in the case of Lie-Yamaguti algebras and extends the one given in [18] for Lie supertriple systems. Let be an -dimensional Lie-Yamaguti superalgebra. Denote by the Killing form of the standard enveloping Lie superalgebra . Consider the bilinear form of obtained by restricting to . For any in , define the endomorphisms and of the vector space by and . It is clear that is of degree and .

Theorem 10. For , we have

Proof. Let , , , be bases of , , , , respectively. For these bases, we express the operations of and as follows: To prove the theorem, it suffices to show that , and . Since and , we have . Also, and give and then because of the consistency property of (). Hence, it remains to show that and . The operations in and the identities (7) imply the following:In a similar way, we getAlso,By interchanging and , we haveTherefore,It remains to show that ,Likewise, we haveTherefore,Now,By interchanging and , we haveHence the theorem is proved.

Definition 11. The bilinear form defined on the Lie-Yamaguti superalgebra by for is called the Killing form of .

Remark 12. Recall that if is a Lie superalgebra, then the Killing form on is defined as , . Likewise, if is a Lie supertriple system (resp., a Lie-Yamaguti algebra), the Killing form on is defined as (resp., ) with and defined according to the considered structure on . So if a Lie-Yamaguti superalgebra is reduced to a Lie superalgebra (resp., a Lie supertriple system, a Lie-Yamaguti algebra), then as defined in Definition 11 is the Killing form of the Lie superalgebra (resp., the Lie supertriple system, the Lie-Yamaguti algebra) .

Proposition 13. Let be a Lie-Yamaguti superalgebra with a Killing form denoted by Then, (1) (consistence),(2) (supersymmetry),(3), .

Proof. As , , and we can state that .
(2) comes from the definition of .
Now, for any in , in , , and As then ; that is, and .
Hence, . Also, gives ; that is, . Then,

Now, let be a trilinear form in given by for any . We can easily see that, for any , and that vanishes identically if is reduced to Lie superalgebra or Lie supertriple system.

Theorem 14. Let be a Lie-Yamaguti superalgebra with a Killing form denoted by Then, satisfies the identities for all

Proof. The Killing form of satisfies ; that is, . But, using (7), we have Then the identity gives that is and (23) is obtained.
From we deduce that is and Then, using (7) again and developing (26), we have and we get This gives Thus,that is,This impliesBut (23) gives Then, Hence, and (24) is proved.

Corollary 15. Let be a Lie-Yamaguti superalgebra with a Killing form denoted by Then, satisfies the following for :

Proof. Using (24) we have By adding memberwise (36) and (37) we obtain the identity (34).
Also, the identity (37) is equivalent to Then, we obtain that is and the remaining assertion is proved.

4. Invariant Forms of Lie-Yamaguti Superalgebras

In this section we introduced the concept of invariant forms of Lie-Yamaguti superalgebras as generalizations of those of Lie superalgebras and Lie supertriple systems.

Definition 16. An invariant form of a Lie-Yamaguti superalgebra is a supersymmetric bilinear form on satisfying the identities for all in

Remark 17. (1) If , the Killing form of is an invariant form of .
(2) If is reduced to a Lie supertriple system (resp., a Lie superalgebra, a Lie-Yamaguti algebra), then is reduced to an invariant form of a Lie supertriple system [19] (resp., a Lie superalgebra [12], a Lie-Yamaguti algebra [10]).

Definition 18. Let be an invariant form of a Lie-Yamaguti superalgebra and a subset of . The orthogonal of with respect to is defined by . The invariant form is nondegenerate if .

Lemma 19. Let be an invariant form of a Lie-Yamaguti superalgebra . Then, for any , , , in , we have

Proof. By interchanging and in (39) we havethat is by supersymmetry. Also by switching and in (41), we obtain that is Thus adding (41) and (42) we get (40) whence the lemma.

Lemma 20. Let be an invariant form of a Lie-Yamaguti superalgebra . Then,(1) if is nondegenerate;(2)if is an ideal of then is an ideal of . In particular, is an ideal of .

Proof. Consider in . Then, for any , we have and . This implies, by (38) and (39), that and that is and . As is nondegenerate, we get and for any . This gives .
Conversely, if , we have, for any , and whence .
Now, suppose that is an ideal of that is and ; then for any , , and , we have and . Then and which proves (2).

We are now ready to prove the following theorem.

Theorem 21. Let be a Lie-Yamaguti superalgebra with . Then the Killing form is nondegenerate if and only if the standard enveloping Lie superalgebra is a semisimple Lie superalgebra.

Proof. Let be the Killing form of the Lie superalgebra . If , we have, for any , and Then, using the invariance of and (43), we have, for any ; that is, by (7),
and . This givesThus, if is nondegenerate, the restriction of on is nondegenerate and is nondegenerate.
Now, suppose that is degenerate. Then by the lemma above, is an ideal of so is a nonzero ideal of .
Using the identities (43) and (44) we get It comes that is degenerate and is not semisimple which proves the theorem.

The results of this paper could be used for a study of the structure of a pair consisting of a semisimple Lie superalgebra and its semisimple graded subalgebra.

Competing Interests

The authors declare that they have no competing interests.