Structure of -Lie Algebras with Involutive Derivations

Abstract

We study the structure of -Lie algebras with involutive derivations for . We obtain that a -Lie algebra is a two-dimensional extension of Lie algebras if and only if there is an involutive derivation on such that or , where and are subspaces of with eigenvalues and , respectively. We show that there does not exist involutive derivations on nonabelian -Lie algebras with for . We also prove that if is a -dimensional -Lie algebra with , then there are involutive derivations on if and only if is even, or satisfies . We discuss also the existence of involutive derivations on -dimensional -Lie algebras.

1. Introduction

Derivation is an important tool in studying the structure of n-Lie algebras [1]. The derivation algebra of an -Lie algebra over the field of real numbers is the Lie algebra of the automorphism group , which is a Lie group if [2]. Any -Lie algebra-module is a module of the inner derivation algebra , which is a linear Lie algebra [3]. Also, derivations have close relationship with extensions of -Lie algebras.

The concept of 3-Lie classical Yang-Baxter equations is introduced in [4]. It is known that if there is an involutive derivation on , then is a 3-pre-Lie algebra, where , , and the 3-Lie algebra is a subadjacent 3-Lie algebra of , and is a skew-symmetric solution of the 3-Lie classical Yang-Baxter equation in the 3-Lie algebra , where is a basis of and is the dual basis of .

Due to this importance of involutive derivations on 3-Lie algebras, we investigate in this paper the existence of involutive derivations on finite dimensional n-Lie algebras. More specifically, in Section 2, we discuss the properties of involutive derivations on n-Lie algebras. In Section 3, we study the existence of involutive derivations on -dimensional -Lie algebras. In Section 4, we consider the existence of involutive derivations on -dimensional -Lie algebras. In Section 5, we investigate a class of 3-Lie algebras with involutive derivations which are two-dimensional extension of Lie algebras.

In the following, we assume that all algebras are over an algebraically closed field with characteristic zero, is the identity mapping, and is the set of integers. For and an -linear mapping on a vector space , denotes the subspace .

2. -Lie Algebras with Involutive Derivations

An -Lie algebra [1] is a vector space over a field equipped with a linear multiplication satisfying, for all ,Equation (1) is usually called the generalized Jacobi identity, or Filippov identity.

The derived algebra of an -Lie algebra is a subalgebra of generated by for all , and is denoted by . We use to denote the center of ; that is, .

A derivation of is an endomorphism of satisfyingIf a derivation satisfies that , then is called an involutive derivation on . denotes the derivation algebra of .

For , map ad: , is called a left multiplication defined by elements . From (1), left multiplications are derivations.

The following lemma can be easily verified.

Lemma 1. Let be a finite dimensional vector space over and be an endomorphism of with . Then can be decomposed into the direct sum of subspaces , where and

If is a finite dimensional -Lie algebra with an involutive derivation , then we have

Lemma 2. Let be an -Lie algebra over . If is an involutive derivation, then, for all ,

Proof. If is an involutive derivation on , then, for all ,Equation (5) follows. Equation (6) follows from (4) and .

Theorem 3. Let be a finite dimensional -Lie algebra with , . Then there is an involutive derivation on if and only if is abelian.

Proof. If is abelian, then the result is trivial.
Conversely, let be an involutive derivation on . By Lemma 1, . Then, for any , , , and , Thanks to and , . Therefore, is abelian.

Theorem 4. Let be a finite dimensional -Lie algebra with , , and be an involutive derivation on . Then and are abelian subalgebras, and

Proof. Since , If , then , that is, , or Therefore, . The result follows.

Theorem 5. Let be an -dimensional -Lie algebra with , . Then there is an involutive derivation on if and only if has the decomposition (as direct sum of subspaces), and

Proof. If there is an involutive derivation on , then, by Theorem 4, and satisfy (12) and (13).
Conversely, define an endomorphism of by , . Then , and . By (12) and (13), is a derivation.

Corollary 6. Let be a -dimensional -Lie algebra with the multiplication , where is a basis of . Then the linear mapping defined by , , , is an involutive derivation on .

Proof. The result follows from a direct computation.

3. Involutive Derivations on -Dimensional -Lie Algebras with

In this section, we study involutive derivations on -dimensional -Lie algebras over . From Theorem 3, we only need to discuss the case of , .

Lemma 7 (see [5]). Let be an -dimensional nonabelian -Lie algebra over , . Then up to isomorphisms is one and only one of the following possibilities:where is a basis of , , and means that is omitted.

Theorem 8. Let be a -dimensional -Lie algebra over and . Then there exists an involutive derivation on if and only if is even, or .

Proof. If , then, by Lemma 7, and a direct computation, the linear mapping defined by , is an involutive derivation on .
Now we discuss the case . Let be a basis of and the multiplication in the basis be as follows:where Thanks to Theorem 3 in [1], is a -Lie algebra if and only if the -matrix is symmetric.
If , , then define the multiplication on bythat is, for , or , and others are zero. Then, is symmetric. Therefore, is a -Lie algebra with the multiplication (16).
Define an endomorphism of by , , and for Then is an involutive derivation on .
For the case . Suppose , .
If there is an involutive derivation on , then, by Theorem 4, , and Since , and . Therefore, . Without loss of generality, we can suppose , and . By (10) and (11), the -matrix defined by (15) is nonsymmetric, which is a contradiction. Therefore, if , then there do not exist involutive derivations on .

By Theorem 8, if is a -dimensional -Lie algebra with , or , then there does not exist involutive derivation on . If and , then there are involutive derivations on .

4. Involutive Derivations on -Dimensional -Lie Algebras with

By Theorem 3, we only need to discuss the case where is odd. So we suppose that is a -dimensional -Lie algebra over , , and that is the -unit matrix.

Lemma 9 (see [6]). Let be a -dimensional -Lie algebra over with a basis . Then is isomorphic to one and only one of the following possibilities:And n-Lie algebras corresponding to the case with coefficients and are isomorphic if and only if there exists a nonzero element such that , , ,

Theorem 10. If is a -dimensional -Lie algebra over with , then there are involutive derivations on .

Proof. Define linear mappings , bySince , it is easy to verify that is an involutive derivation on the 3-Lie algebras of the cases of , , , and , where , and is an involutive derivation on the 3-Lie algebras of the cases of , , and , where . , and are involutive derivations on the 3-Lie algebras of the case of . And is an involutive derivation on the 3-Lie algebras of the cases of , and . Also are involutive derivations on abelian algebras for .

Next, we discuss the case of Let be an endomorphism of , and be the -matrix. Thenwhere is the block matrix of . First we discuss -dimensional -Lie algebras of the case in Lemma 9.

Lemma 11. If is a -dimensional -Lie algebra of the case with Then the linear mapping is an involutive derivation on if and only if the block matrix satisfies that (which is the zero -matrix, and

Proof. By (2), and a direct computation, is a derivation of if and only if matrix has the property:Therefore, matrix satisfies (23) and . And if and only ifThanks to , (22) holds.

Theorem 12. Let be a -dimensional -Lie algebra of the case with , . If is odd, then there are involutive derivations on .

Proof. Let . Then and . Suppose is an endomorphism of and the matrix of with respect to the basis is which satisfies (22) and (23), and . Then Since , we have , , Therefore,SupposeBy (23), Therefore, the endomorphisms of , which are defined by are involutive derivations on .

Theorem 13. Let be a -dimensional -Lie algebra of the case with , then there does not exist an involutive derivation on .

Proof. If is an involutive derivation on , then, by Lemma 11 and (23), Thanks to (22), Therefore, , which is a contradiction.

Now we discuss case .

Theorem 14. Let be a -dimensional -Lie algebra of the case with . Then there exist involutive derivations on if and only if is even.

Proof. By Lemma 7, , where , and is a -dimensional -Lie subalgebra of with . Then there exist involutive derivations on if and only if there exist involutive derivations on .
By Theorem 3 in [1], there is a basis of such thatIf is even, then . By Theorem 8 and (32), endomorphism of defined byis an involutive derivation on . Therefore, the endomorphism of defined by is involutive derivation on .
If is odd and endomorphism of is an involutive derivation on , then . Suppose , . Then We get . Since , By (32), , and where , and Then , for , or , and Then the endomorphism of defined by is an involutive derivation on the -dimensional -Lie algebra , contradiction (Theorem 8). Therefore, there does not exist involutive derivation on .

5. Structure of 3-Lie Algebras with Involutive Derivations

Let be a Lie algebra over , and be an element which is not contained in . Then is a -Lie algebra in the multiplicationAnd the -Lie algebra is called one-dimensional extension of .

Theorem 15. Let be a -Lie algebra, then is one-dimensional extension of a Lie algebra if and only if there exists an involutive derivation on such that , or .

Proof. If is an one-dimensional extension of a Lie algebra , then . Define the endomorphism of by (or ), and (or ), . Thanks to (38), , and , , for all . Therefore, is an involutive derivation on , and (or ).
Conversely, let be an involutive derivation on a -Lie algebra , and (or ). Let , and (or , ), where . Thanks to Theorem 3, is a Lie algebra with the multiplication , for all , and is one-dimensional extension of .

Let and be Lie algebras and be a basis of . For convenience, denote Lie algebras by , , respectively. Suppose and are two distinct elements which are not contained in , and -Lie algebras and are one-dimensional extensions of Lie algebras and , respectively, where , . Then and are subalgebras of .

Definition 16. Let and be two Lie algebras, and be two distinct elements which are not contained in , and . Then 3-algebra is called a two-dimensional extension of Lie algebras , , where defined by If is a -Lie algebra, then is called a two-dimensional extension -Lie algebra of Lie algebras , .

Let be a two-dimensional extension of Lie algebras , , where . Define linear mappings and bythat is, for all , , , We have the following result.

Theorem 17. Let 3-algebra be a two-dimensional extension of Lie algebras and . Then is a -Lie algebra if and only if linear mappings , , and satisfy that , are Lie homomorphisms, andwhere ,

Proof. If is a two-dimensional extension -Lie algebra, then, by Definition 16, linear mappings satisfy that , and are Lie homomorphisms, . Thanks to (39),for all , (41) holds. Similarly, we have (42).
Thanks to (39) and (40),Equations (43) and (44) hold. Equation (45) follows from (39) and (40), directly.
Conversely, by (39), ,Since and are Lie homomorphisms, and are -Lie algebras, which are one-dimensional extension -Lie algebras of Lie algebras , , respectively.
Next we only need to prove that the multiplication on defined by (39) satisfies (1). For all , , that products , , , , and , satisfy (1), follow from that and are one-dimensional extension -Lie algebras of and (39), directly.
From (41) and (42), it follows that products , , satisfy (1). It follows from (43)–(45) that products , , and , , satisfy (1). We omit the computation process.

Theorem 18. Let be a -Lie algebra. Then is a two-dimensional extension -Lie algebra of Lie algebras if and only if there is an involutive derivation on such that or .

Proof. If is a two-dimensional extension -Lie algebra of Lie algebras. Then by Theorem 15, there are Lie algebras and , such that and the multiplication of is defined by (39), where .
Define the endomorphism of by , or Then , and , , or , Thanks to (38) and (41)-(45), is a derivation of .
Conversely, if there is an involutive derivation on the -Lie algebra such that (or ). By Theorem 4, , , Let and . Then , and and are Lie algebras, where , , Thanks to Theorem 17, the -Lie algebra is a two-dimensional extension -Lie algebra of Lie algebras and .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first named author was supported in part by the Natural Science Foundation (11371245) and the Natural Science Foundation of Hebei Province (A2018201126).