On the Cohomology and Extensions of -ary Multiplicative Hom-Nambu-Lie Superalgebras

Abstract

In this paper, we discuss the representations of -ary multiplicative Hom-Nambu-Lie superalgebras as a generalization of the notion of representations for -ary multiplicative Hom-Nambu-Lie algebras. We also give the cohomology of an -ary multiplicative Hom-Nambu-Lie superalgebra and obtain a relation between extensions of an -ary multiplicative Hom-Nambu-Lie superalgebra by an abelian one and . We also introduce the notion of -extensions of -ary multiplicative Hom-Nambu-Lie superalgebras and prove that every finite-dimensional nilpotent metric -ary multiplicative Hom-Nambu-Lie superalgebra over an algebraically closed field of characteristic not 2 in the case is a surjection is isometric to a suitable -extension.

1. Introduction

In 1996, the concept of -Lie superalgebras was firstly introduced by Daletskii and Kushnirevich in [1]. Moreover, Cantarini and Kac gave a more general concept of -Lie superalgebras again in 2010 in [2]. -Lie superalgebras are more general structures including -Lie algebras (-ary Nambu-Lie algebras), -ary Nambu-Lie superalgebras, and Lie superalgebras [3].

The general Hom-algebra structures arose first in connection with quasideformation and discretizations of Lie algebras of vector fields. These quasideformations lead to quasi-Lie algebras, a generalized Lie algebra structure in which the skew symmetry and Jacobi conditions are twisted. Hom-Lie algebras, Hom-Lie superalgebras, Hom-Lie bialgebras, Hom-Lie -algebras, and quasi-Hom-Lie algebras are discussed in [4–14]. The -ary Hom-Nambu-Lie algebras have been introduced in [15]. It is the generalization of -ary algebras of Lie type by twisting the identities using linear maps. It includes -ary Hom-algebra structures generalizing the -ary algebras of Lie type such as -ary Nambu algebras, -ary Nambu-Lie algebras, and -ary Lie algebras [16].

Cohomologies are powerful tools in mathematics, which can be applied to algebras and topologies as well as the theory of smooth manifolds or of holomorphic functions. The cohomology of Lie algebras was defined by Chevalley and Eilenberg in order to give an algebraic construction of the cohomology of the underlying topological spaces of compact Lie groups in [17]. The cohomology of Lie superalgebras was introduced by Scheunert and Zhang in [18] and was used in mathematics and theoretical physics: the theory of cobordisms, invariant differential operators, central extensions, and deformations. The theory of cohomology for -ary Hom-Nambu-Lie algebras and -Lie superalgebras can be found in [19, 20]. This paper generalizes it to -ary multiplicative Hom-Nambu-Lie superalgebras.

The extension is an important way to find a larger algebra and there are many extensions such as general extensions, abelian extensions, nonabelian extensions, double extensions, and Kac-Moody extensions. Abelian extensions and nonabelian extensions of Hom-Lie algebras are, respectively, researched in [21, 22]. The general extensions of -Hom-Lie algebras is researched in [23]. In 1997, Bordemann introduced the notion of -extensions of Lie algebras in [24]. The method of -extension was used in [25] and was generalized to many other algebras recently in [26, 27]. This paper researches general extensions and -extensions of -ary multiplicative Hom-Nambu-Lie superalgebras. In addition, the paper also discusses representations of -ary multiplicative Hom-Nambu-Lie superalgebras as a generalization of the notions of representations for -ary multiplicative Hom-Nambu-Lie algebras.

This paper is organized as follows. In Section 2, we give the representation and the cohomology for an -ary multiplicative Hom-Nambu-Lie superalgebra. In Section 3, we give a one-to-one correspondence between extensions of an -ary multiplicative Hom-Nambu-Lie superalgebras by an abelian one and In Section 4, we introduce the notion of -extensions of -ary multiplicative Hom-Nambu-Lie superalgebras and prove that every finite-dimensional nilpotent metric -ary multiplicative Hom-Nambu-Lie superalgebra over an algebraically closed field of characteristic not 2 such that is isometric to (a nondegenerate ideal of codimension of) a -extension of a nilpotent -ary multiplicative Hom-Nambu-Lie superalgebra whose nilpotent length is at most a half of the nilpotent length of .

2. -ary Hom-Nambu-Lie Superalgebras

In the paper, let be a finite-dimensional -graded vector space. The degree of an element in will be denoted by and in what follows appearance of will mean that is a homogeneous element and stands for its degree, where and

Definition 1. An -ary Hom-Nambu-Lie superalgebra is a triple consisting of a -graded vector space a multilinear mapping and a family of even linear maps satisfying where denotes the degree of a homogeneous element

An -ary Hom-Nambu-Lie superalgebra is multiplicative, if and the following equality is satisfied:

Moreover, the multiplicative -ary Hom-Nambu-Lie superalgebra is also denoted by

For a multiplicative -ary Hom-Nambu-Lie superalgebra equation (4) can be read:

It is clear that -ary Hom-Nambu-Lie algebras and Hom-Lie superalgebras are particular cases of -ary Hom-Nambu-Lie superalgebras.

Definition 2. Let and be two -ary Hom-Nambu-Lie superalgebras. A linear map is an -ary Hom-Nambu-Lie superalgebra homomorphism if it satisfies

Example 3. Let be an -ary Nambu-Lie superalgebra and let be an -ary Nambu-Lie superalgebra endomorphism. Then, is an -ary multiplicative Hom-Nambu-Lie superalgebra.

Definition 4. Let be an -ary Hom-Nambu-Lie superalgebra. A graded subspace is a Hom-subalgebra of if and is closed under the bracket operation i.e., for all A graded subspace is a Hom-ideal of if and for all and

Definition 5. Let be an -ary Hom-Nambu-Lie superalgebra. A Hom-ideal of is abelian if for all and

3. Cohomology for -ary Multiplicative Hom-Nambu-Lie Superalgebras

Definition 6. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra. is called a fundamental object of For all . It is clear that .
Let and be two fundamental objects of . A bilinear map defined by A linear map defined by Then, .

Proposition 7. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra. Suppose that , and are fundamental objects of and is an arbitrary element in . Then,

Proof. It is easy to see that (9) is equivalent to (6). Using (9), by exchanging and , we have Comparing (9) with (11), we obtain (10).

Definition 8. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra and be a -graded vector space over a field and A graded representation of on is a linear map such that for and where the sign indicates that the element below must be omitted. The -graded representation space is said to be a graded -module.

We use a supersymmetric notation (like (3)) to denote Set and for all and then becomes an -ary multiplicative Hom-Nambu-Lie superalgebra such that is a -graded abelian ideal of , that is,

In the sequel, we will usually abbreviate with .

Example 9. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra. Then, defined by is a graded representation of it is also called the adjoint graded representation of

Definition 10. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra and be a graded -module. An -cochain is an -linear map such that for all and We denote the set of -cochain by

Definition 11. For we define an -coboundary operator of the -ary multiplicative Hom-Nambu-Lie superalgebra which is an even linear map, by where and the last term is defined by where

Theorem 12. Let be an -cochain. Then,

Proof. See the appendix.

Remark 13. The -coboundary operator as above is a generalization of the one defined for -ary multiplicative Hom-Nambu-Lie algebras in [16] and for first-class -Lie superalgebras in [20].

The map is called an -supercocycle if . We denote by the graded subspace spanned by -supercocycles. Since for all , is a graded subspace of . Therefore, we can define a graded cohomology space of as the graded space

4. Extensions of -ary Multiplicative Hom-Nambu-Lie Superalgebras

Definition 14. Let be a family of -ary multiplicative Hom-Nambu-Lie superalgebras over . is a morphism of -ary multiplicative Hom-Nambu-Lie superalgebras. The sequence is called an exact sequence of -ary multiplicative Hom-Nambu-Lie superalgebras, if it satisfies

Definition 15. Let and be -ary multiplicative Hom-Nambu-Lie superalgebras over . is called an extension of by if there is an exact sequence of -ary multiplicative Hom-Nambu-Lie superalgebras:

Let and be two -ary multiplicative Hom-Nambu-Lie superalgebras over . Suppose that is an abelian graded ideal of , i.e., is a graded ideal such that We consider the case that is an extension of by an abelian graded ideal of . Let be a homogeneous even linear map with and Let and let . Then, becomes a graded -module. Let us write and then denote the elements of by for all and . Then, the bracket in is defined by where and . It is easy to see that . Let and . Then,

Therefore, .

Conversely, suppose that an abelian -ary multiplicative Hom-Nambu-Lie superalgebras is a graded -module, , and . Let where Then is an -ary multiplicative Hom-Nambu-Lie superalgebra with the bracket defined by (24). Then we can define an exact sequence where . Thus is an extension of by and is an abelian graded ideal of .

Therefore, we get the following theorem.

Theorem 16. Suppose that and are two -ary multiplicative Hom-Nambu-Lie superalgebras over and is abelian. Then, there is a one-to-one correspondence between extensions of by and .

5. -Extensions of -ary Multiplicative Hom-Nambu-Lie Superalgebras

Let be an -ary multiplicative Hom-Nambu-Lie superalgebra and be its dual space. Since and are -graded vector space, the direct sum is a -graded vector space. In the sequel, whenever appears, it means that is homogeneous and .

Lemma 17. Let be the dual -graded vector space of an -ary multiplicative Hom-Nambu-Lie superalgebra . Let us consider the even linear map defined by for all and . Then, is a representation of on if and only if the following conditions hold: for all

Proof. We firstly prove that the necessity holds. Then, by the definition of one gets and Moreover, we have By (13), we have By (14), we obtain It is easy to see that the sufficiency holds. The proof is complete.

The representation as defined in Lemma 17 is called the coadjoint representation of Let be a homogeneous -linear map from into of degree 0. Now, we define a bracket on :

Theorem 18. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra. Assume that the coadjoint representation exists. Then, is an -ary multiplicative Hom-Nambu-Lie superalgebra if and only if where

Proof. It is clear that satisfies (3) if and only if . Let and . Then, we have Since satisfies (4) and satisfies (14), it can be concluded that satisfies (4) if and only if i.e., .

Definition 19. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra. A bilinear form on is said to be nondegenerate if invariant if supersymmetric if consistent if is called -symmetric, if a subspace of is called isotropic if

In this section, we only consider consistent bilinear forms. If admits a nondegenerate invariant supersymmetric bilinear form such that is -symmetric, then we call a metric -ary multiplicative Hom-Nambu-Lie superalgebra. In particular, a metric vector space is a pair consisting of a -graded vector space and an endomorphism of admitting a nondegenerate invariant supersymmetric bilinear form such that is -symmetric.

Lemma 20. With notations of Theorem 18, define a bilinear form by Then, is nondegenerate and is -symmetric, where Moreover, is metric if and only if the following identity holds

Proof. If is orthogonal to all elements of then for arbitrary element we have and which implies that and so is nondegenerate. Moreover, we have In addition, one gets Hence,
Furthermore, is metric if and only if i.e., (45) holds.

Now, we give the definition of -extensions.

Definition 21. For a -supercocycle satisfying (45) we shall call the metric -ary multiplicative Hom-Nambu-Lie superalgebra the -extension of (by ) and denote it by

Theorem 22. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra over a field . Let is called solvable (nilpotent) of length if and only if there is a smallest integer such that (). Then (1)If is solvable of length , then is solvable of length or .(2)If is nilpotent of length , then is nilpotent of length at least and at most . In particular, the nilpotent length of is .(3)If can be decomposed into a direct sum of two Hom-ideals of , then can be too.

Proof. (1) Suppose that is solvable of length . Since and , we have , which implies because is abelian, and it follows that is solvable of length or .
(2) Suppose that is nilpotent of length . Since and , we have . Let , , . Then This proves that . Hence is nilpotent of length at least and at most .
Now consider the case of trivial -extension of . Note that Then, , as required.
(3) Suppose that , where and are two nonzero Hom-ideals of . Let and . Then, (resp. ) can canonically be identified with the dual space of (resp. ) and .
Note that since Moreover, for we have since and that is, Then, is a Hom-ideal of and so is in the same way. Hence, can be decomposed into the direct sum of two nonzero Hom-ideals of .

Lemma 23. Let be a metric -ary multiplicative Hom-Nambu-Lie superalgebra of even dimension over a field and be an isotropic -dimensional Hom-ideal of . Then, is abelian.

Proof. Since dim+dim and , we have .
By is a Hom-ideal of , one gets which implies .

Definition 24. Let and be two -ary Hom-Nambu-Lie superalgebras. A bijective homomorphism is called an isomorphism of -ary Hom-Nambu-Lie superalgebras.

Definition 25. Two metric -ary multiplicative Hom-Nambu-Lie superalgebras and is said to be isometric if there exists an -ary multiplicative Hom-Nambu-Lie superalgebra isomorphism such that

Theorem 26. Let be a metric -ary multiplicative Hom-Nambu-Lie superalgebra of dimension over a field of characteristic not 2. Suppose that is a -extension of Then, is isometric to if and only if is even and contains an isotropic Hom-ideal of dimension . In particular, .

Proof. () Since dim = dim, dim = dim is even. Moreover, for all It is clear that is a Hom-ideal of dimension and by the definition of , we have , i.e., is isotropic.
() Suppose that is an -dimensional isotropic graded ideal of . By Lemma 23, is abelian. Let and be the canonical projection. Since , we can choose a complement graded subspace such that and . Then, since dim.
Denote by (resp. ) the projection (resp. ) and let denote the homogeneous linear map , where .
If , then , hence and so , which implies is well defined. Moreover, is bijective and for all .
In addition, has the following property: where , .
Define a homogeneous -linear map where Then, is well defined since is a linear isomorphism and .
Now, define the bracket on by (35), then, is a metric -ary multiplicative Hom-Nambu-Lie superalgebra. Let be a linear map defined by Since and are linear isomorphisms, is also a linear isomorphism. Note that where we use the definitions of and and (56). Moreover, In fact, for then, Moreover, Therefore, one gets Then, is an isomorphism of -ary multiplicative Hom-Nambu-Lie superalgebras, hence, is an -ary multiplicative Hom-Nambu-Lie superalgebra. Furthermore, we have then, is isometric. The relation implies that on is invariant.
For then, there exist such that and Hence, we have Therefore, is a metric -ary multiplicative Hom-Nambu-Lie superalgebra. In this way, we get a -extension of and consequently, and are isometric as required.

Suppose that is an -ary multiplicative Hom-Nambu-Lie superalgebra and , satisfying (45). and are said to be equivalent if there exists an isomorphism of -ary multiplicative Hom-Nambu-Lie superalgebras such that and the induced map is the identity, i.e., . Moreover, if is also an isometry, then, and are said to be isometrically equivalent.

Proposition 27. Suppose that is an -ary multiplicative Hom-Nambu-Lie superalgebra over a field of characteristic not and , satisfying (45). Then, we have (1) is equivalent to if and only if and for all Moreover,becomes a supersymmetric invariant bilinear form on and is -symmetric. (2) is isometrically equivalent to if and only if there is such that and the bilinear form induced by in (64) vanishes

Proof. (1) Let be an isomorphism of -ary multiplicative Hom-Nambu-Lie superalgebras satisfying and . Set . Then and By we may obtain for all
For the converse, suppose that satisfies and for all Let be defined by . Then and Moreover, In fact, By one gets Therefore, is an isomorphism of -ary multiplicative Hom-Nambu-Lie superalgebras, that is, is equivalent to .
It is clear that defined by (64) is supersymmetric. Note that where we make use of and satisfying (45). Then, is invariant. In addition, since for all That is, is -symmetric.
(2) Let the isomorphism be defined as in (1). Then for all , we have Thus, is an isometry if and only if .

Lemma 28. Let be a metric -graded vector space of dimension over an algebraically closed field of characteristic not and be a Lie superalgebra consisting of nilpotent homogeneous endomorphisms of such that for each , the map defined by is contained in , too. Suppose that is an isotropic graded subspace of which is stable under and i.e., for all and then is contained in a maximally isotropic graded subspace of which is also stable under and moreover, . If is even, then . If is odd, then , and for all .

Proof. The proof is by induction on . The base step is obviously true. For the inductive step, we consider the following two cases.

Case 1. or there is a nonzero -stable vector (that is, ) such that .

Case 2. and every nonzero -stable vector satisfies

In the first case, is a nonzero isotropic -stable graded subspace, and is also -stable since . Now, consider the bilinear form on the factor graded space defined by , then is metric. Denote by the canonical projection and define by , then is well defined since and are -stable. Let . Then, is a Lie superalgebra. For each there is a positive integer such that , which implies that . Hence, also consists of nilpotent homogeneous endomorphisms of . Note that satisfies the same conditions of . In fact, let and be two arbitrary elements in . Then, by the definition of we have for arbitrary , which shows that for all .

Since , we can use the inductive hypothesis to get a maximally isotropic -stable subspace in and Clearly, = = = . For all , the relation implies that is isotropic. Note that , then, is maximally isotropic. Moreover, for all and , we have , which implies . It follows that is -stable and This proves the first assertion of the lemma in this case.

In the second case, by Engel’s Theorem of Lie superalgebras, there is a nonzero -stable vector such that for all . Clearly, is a nondegenerate -stable graded subspace of , then and is also -stable since . Now, if , then and , hence and so 0 is the maximally isotropic -stable subspace, then the lemma follows. If , then again by Engel’s Theorem of Lie superalgebras, there is a nonzero -stable vector such that for all . It follows that vanishes on the two-dimensional nondegenerate subspace of . Without loss of generality, we can assume that . Set , then it is easy to check that the nonzero vector is isotropic and -stable. This contradicts the assumption of Case 2.

Therefore, the existence of a maximally isotropic -stable graded subspace containing is proved. If is even, then dim=dim; if is odd, then dim and dim. Since is nilpotent, there exists a nonzero such that . Note that dim=1, which implies , so .

Theorem 29. Let be a nilpotent metric -ary multiplicative Hom-Nambu-Lie superalgebra of dimension over an algebraically closed field of characteristic not . If is an isotropic Hom-ideal of , then contains a maximally Hom-ideal of dimension containing . Moreover, if is even, then is isometric to some -extension of . If is odd, then is abelian and is isometric to a nondegenerate graded ideal of codimension 1 in some -extension of .

Proof. Consider . Then, is a Lie superalgebra. For any , is nilpotent since is nilpotent. Then, the following identity implies . By is an isotropic graded ideal of then, is an isotropic -stable graded subspace and by Lemma 28 so there is a maximally isotropic -stable graded subspace of containing such that and is also an isotropic graded ideal of Moreover, if is even, then, is isometric to some -extension of by Theorem 26.
If is odd, then and by Lemma 28. Note that which implies that , hence is abelian.
Take any nonzero element we define by Then, is a 1-dimensional abelian -ary multiplicative Hom-Nambu-Lie superalgebra. Define a bilinear map by . Then, is a nondegenerate supersymmetric invariant bilinear form on . Let . Define Then, is a nilpotent metric -ary multiplicative Hom-Nambu-Lie superalgebra since for all and is a nondegenerate Hom-ideal of codimension 1 of Since is not isotropic and is algebraically closed there exists and In addition, we have since for and Let and . Then, is an -dimensional isotropic graded ideal of .
In fact, for all , In light of Theorem 26, we conclude that is isometric to some -extension of .
Define . Then where we use the fact that is abelian and . Moreover, In fact, for we have It’s clear that is surjective and , so , hence the theorem follows.

Now, we show that there exists an isotropic Hom-ideal in every finite-dimensional metric -ary multiplicative Hom-Nambu-Lie superalgebra and investigate the nilpotent length of .

Proposition 30. Suppose that is a finite-dimensional metric -ary multiplicative Hom-Nambu-Lie superalgebra. (1)For any graded subspace , (2), where (3)If is nilpotent of length , then

Proof. The relation shows that . Notice that which implies , i.e., . Hence, (1) follows.By induction, (2) and (3) can be proved easily.

Theorem 31. Every finite-dimensional nilpotent metric -ary multiplicative Hom-Nambu-Lie superalgebra over an algebraically closed field of characteristic not such that is isometric to (a nondegenerate ideal of codimension of) a -extension of a nilpotent -ary multiplicative Hom-Nambu-Lie superalgebra whose nilpotent length is at most a half of the nilpotent length of .

Proof. Define . Since is nilpotent, the sum is finite. Proposition 30 (2) says , then is isotropic for all . Since we have It follows that Therefore, is an isotropic graded ideal of . Let denote the nilpotent length of . Using Proposition 30 (3) we can conclude that . This implies that is contained in . By Theorem 29, there is a maximally isotropic graded ideal of containing . It means that has nilpotent length at most , and the theorem follows.

Remark 32. Most results concerning -extensions in [20, 24, 26, 27] are contained in this section as special cases.

Appendix

Proof of Theorem 33. We now check that In fact, for one gets
where It can be verified that the sum of terms labeled with the same letter vanishes. For example, , in fact, and Moreover, we have Since one gets Then Therefore, the proof of Theorem 12 is completed.

Data Availability

There is no data in my manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the NNSF of China (Nos. 11771069 and 11801066), the NSF of Jilin province (No. 20170101048JC), the project of Jilin province department of education (No. JJKH20180005K), the Fundamental Research Funds for the Central Universities (No. 130014801), the Special Project of Basic Business for Heilongjiang Provincial Education Department (no. 135409317), and the Postdoctoral Scientific Research Developmental Fund of Heilongjiang (no. LBH-Q17175).