# Structure ofn-Lie Algebras with Involutive Derivations

Authors: Ruipu Bai; Shuai Hou; Yansha Gao
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## Abstract

We study the structure ofn-Lie algebras with involutive derivations for n≥2. We obtain that a 3-Lie algebra A is a two-dimensional extension of Lie algebras if and only if there is an involutive derivation D on A=A1∔A-1 such that dimA1=2 or dimA-1=2, where A1 and A-1 are subspaces of A with eigenvalues 1 and -1, respectively. We show that there does not exist involutive derivations on nonabelian n-Lie algebras with n=2s for s≥1. We also prove that if A is a (2s+2)-dimensional (2s+1)-Lie algebra with dimA1=r, then there are involutive derivations on A if and only if r is even, or r satisfies 1≤r≤s+2. We discuss also the existence of involutive derivations on (2s+3)-dimensional (2s+1)-Lie algebras.


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## Body

## 1. Introduction

Derivation is an important tool in studying the structure of n-Lie algebras [1]. The derivation algebra Der(A) of an n-Lie algebra A over the field of real numbers is the Lie algebra of the automorphism group Aut(A), which is a Lie group if dimA<∞ [2]. Any n-Lie algebra-module (V,ρ) is a module of the inner derivation algebra ad(A), which is a linear Lie algebra [3]. Also, derivations have close relationship with extensions of n-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 D on A, then (A,{,,}D) is a 3-pre-Lie algebra, where {x,y,z}D=D(ad(x,y)D(z)), ∀x,y,z∈A, and the 3-Lie algebra A is a subadjacent 3-Lie algebra of (A,{,,}D), and r=∑iei∗⊗D(ei)-D(ei)⊗ei∗ is a skew-symmetric solution of the 3-Lie classical Yang-Baxter equation in the 3-Lie algebra A⋉ad∗A∗, where {e1,⋯,em} is a basis of A and {e1∗,⋯,em∗} is the dual basis of A∗.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 Section2, we discuss the properties of involutive derivations on n-Lie algebras. In Section 3, we study the existence of involutive derivations on (2s+2)-dimensional (2s+1)-Lie algebras. In Section 4, we consider the existence of involutive derivations on (2s+3)-dimensional (2s+1)-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 fieldF with characteristic zero, Id is the identity mapping, and Z is the set of integers. For λ∈F and an F-linear mapping D on a vector space A, Aλ denotes the subspace {x∈A∣D(x)=λx}.


## 2.n-Lie Algebras with Involutive Derivations

An n -Lie algebra [1] is a vector space A over a field F equipped with a linear multiplication [,⋯,]:∧nA→A satisfying, for all x1,⋯,xn,y2,⋯,yn∈A,(1)x1,⋯,xn,y2,⋯,yn=∑i=1nx1,⋯,xi,y2,⋯,yn,⋯,xn.Equation (1) is usually calledthe generalized Jacobi identity, orFilippov identity.The derived algebra of an n-Lie algebra A is a subalgebra of A generated by [x1,⋯,xn] for all x1,⋯,xn∈A, and is denoted by A1. We use Z(A) to denote the center of A; that is, Z(A)={x∣x∈A,[x,A,⋯,A]=0}.A derivation of A is an endomorphism of A satisfying(2)Dx1,⋯,xn=∑i=1nx1,⋯,Dxi,⋯,xn,∀x1,⋯,xn∈A.If a derivation D satisfies that D2=Id, then D is called aninvolutive derivation on A. Der(A) denotes the derivation algebra of A.Forx1,⋯,xn-1∈A, map ad(x1,⋯,xn-1): A→A, (3)adx1,⋯,xn-1x=x1,⋯,xn-1,x,∀x∈Ais calleda left multiplication defined by elements x1,⋯,xn-1. From (1), left multiplications are derivations.The following lemma can be easily verified.Lemma 1.
LetV be a finite dimensional vector space over F and D be an endomorphism of V with D2=Id. Then V can be decomposed into the direct sum of subspaces V=V1+˙V-1, where V1={v∈V∣Dv=v} and V-1={v∈V∣Dv=-v}.IfA is a finite dimensional n-Lie algebra with an involutive derivation D, then we have(4)A=A1+˙A-1.Lemma 2.
LetA be an n-Lie algebra over F. If D∈Der(A) is an involutive derivation, then, for all x1,⋯,xn∈A,(5)x1,⋯,xn=-2n-1∑i<jx1,⋯,xi-1,Dxi,xi+1,⋯,xj-1,Dxj,xj+1,⋯,xn,(6)Dx1,⋯,Dxn=-2n-1∑i<jDx1,⋯,Dxi-1,xi,Dxi+1,⋯,Dxj-1,xj,Dxj+1,⋯,Dxn.Proof.
IfD is an involutive derivation on A, then, for all x1,⋯,xn∈A,(7)x1,⋯,xn=D2x1,⋯,xn=D∑i=1nx1,⋯,Dxi,⋯,xn=nx1,⋯,xn+2∑1≤i<j≤nx1,⋯,Dxi,⋯,Dxj,⋯,xn.Equation (5) follows. Equation (6) follows from (4) and D2=Id.Theorem 3.
LetA be a finite dimensional n-Lie algebra with n=2s, s≥1. Then there is an involutive derivation D on A if and only if A is abelian.Proof.
IfA is abelian, then the result is trivial.
Conversely, letD be an involutive derivation on A. By Lemma 1, A=A1+˙A-1. Then, for any i∈Z, 1≤i≤n, x1,⋯,xn∈A1, and y1,⋯,,yn∈A-1, (8)Dx1,⋯,xi,y1,⋯,yn-i=ix1,⋯,xi,y1,⋯,yn-i-n-ix1,⋯,xi,y1,⋯,yn-i=2i-2sx1,⋯,xi,y1,⋯,yn-i∈A2i-2s.Dx1,⋯,xn=2sx1,⋯,xn,Dy1,⋯,yn=-2sy1,⋯,yn.Thanks to ±2s≠±1 and 2i-2s≠±1, A2i-n=A±2s=0. Therefore, A is abelian.Theorem 4.
LetA be a finite dimensional n-Lie algebra with n=2s+1, s≥1, and D be an involutive derivation on A. Then A1 and A-1 are abelian subalgebras, and(9)A1,⋯,A1︸i,A-1,⋯,A-1︸2s+1-i=0,∀1≤i≤2s,i≠s,s+1,(10)A1,⋯,A1︸s,A-1,⋯,A-1︸s+1⊆A-1,(11)A1,⋯,A1︸s+1,A-1,⋯,A-1︸s⊆A1.Proof.
SinceD∈DerA, [A1,⋯,A1︸i,A-1,⋯,A-1︸2s+1-i]⊆A2i-2s-1,0≤i≤2s+1. If [A1,⋯,A1︸i,A-1,⋯,A-1︸2s+1-i]≠0, then 2i-2s-1=±1, that is, i=s+1, or i=s. Therefore, [A1,⋯,A1]=[A-1,⋯,A-1]=0. The result follows.Theorem 5.
LetA be an m-dimensional n-Lie algebra with n=2s+1, s≥1. Then there is an involutive derivation on A if and only if A has the decomposition A=B+˙C (as direct sum of subspaces), and(12)B,⋯,B︸i,C,⋯,C︸2s+1-i=0,0≤i≤2s+1,i≠s,s+1,(13)B,⋯,B︸s,C,⋯,C︸s+1⊆C,B,⋯,B︸s+1,C,⋯,C︸s⊆B.Proof.
If there is an involutive derivationD on A, then, by Theorem 4, B=A1 and C=A-1 satisfy (12) and (13).
Conversely, define an endomorphismD of A by D(x)=x,D(y)=-y, ∀x∈B,y∈C. Then D2=Id, B=A1 and C=A-1. By (12) and (13), D is a derivation.Corollary 6.
LetA be a (2s+1)-dimensional (2s+1)-Lie algebra with the multiplication [e1,⋯,e2s+1]=e1, where {e1,⋯,e2s+1} is a basis of A. Then the linear mapping D:A→A defined by D(ei)=ei, 1≤i≤s+1, D(ej)=-ej, s+2≤j≤2s+1 is an involutive derivation on A.Proof.
The result follows from a direct computation.


## 3. Involutive Derivations on(n+1)-Dimensional n-Lie Algebras with n=2s+1

In this section, we study involutive derivations on(n+1)-dimensional n-Lie algebras over F. From Theorem 3, we only need to discuss the case of n=2s+1, s≥1.Lemma 7 (see [5]).
LetA be an (n+1)-dimensional nonabelian n-Lie algebra over F, n≥3. Then up to isomorphisms A is one and only one of the following possibilities:(14)b1e2,e3,⋯,en+1=e1.b2e1,e2,⋯,en=e1.c1e2,⋯,en+1=e1,e1,e3,⋯,en+1=e2.c2e2,⋯,en+1=αe1+e2,e1,e3,⋯,en+1=e2,α∈F,α≠0.c3e1,e3,⋯,en+1=e1,e2,⋯,en+1=e2.dre1,⋯,e^i,⋯,en+1=ei,1≤i≤r,where {e1,⋯,en+1} is a basis of A, 3≤r≤n+1, and e^i means that ei is omitted.Theorem 8.
LetA be a (2s+2)-dimensional (2s+1)-Lie algebra over F and dimA1=r. Then there exists an involutive derivation D on A if and only if r is even, or 0≤r≤s+2.Proof.
IfdimA1=r≤s+2, then, by Lemma 7, and a direct computation, the linear mapping D:A→A defined by D(ei)=ei,D(ej)=-ej,1≤i≤s+2,s+3≤j≤2s+2, is an involutive derivation on A.
Now we discuss the casedimA1=r≥s+3. Let {e1,⋯,e2s+2} be a basis of A and the multiplication in the basis be as follows:(15)ei=-12s+2+ie1,⋯,e^i,⋯,e2s+2=∑l=12s+2βilel,βil∈F,1≤i≤2s+2,where βil∈F,1≤i,l≤2s+2. Thanks to Theorem 3 in [1], A is a 3-Lie algebra if and only if the (2s+2)×(2s+2)-matrix B=(βil) is symmetric.
Ifr=2t>3, 2≤t≤s+1, then define the multiplication on A by(16)e1,⋯,e^i,⋯,e2s+2=-1ie2s+3-i,1≤i≤t,3-t<i-2s≤2,e1,⋯,e^j,⋯,e2s+2=0,t<j≤2s+3-t,that is, βi,2s+3-i=β2s+3-i,i=1 for 1≤i≤t, or 2s+3-t<i≤2s+2, and others are zero. Then, B=(βil) is symmetric. Therefore, A is a (2s+1)-Lie algebra with the multiplication (16).
Define an endomorphismD of A by Dei=ei, 1≤i≤s+1, and Dej=-ej for s+2≤j≤2s+2. Then D is an involutive derivation on A.
For the casedimA1=r=2t+1≥s+3. Suppose l=dimA1, l′=dimA-1.
If there is an involutive derivationD on A, then, by Theorem 4, l+l′=2s+2, s≤l≤s+2 and s≤l′≤s+2. Since dimA1=r=2t+1≥s+3, A1∩A1≠0 and A1∩A-1≠0. Therefore, dimA1=dimA-1=s+1. Without loss of generality, we can suppose {e1,⋯,es+1}⊆A1, and {es+2,⋯,x2s+2}⊆A-1. By (10) and (11), the (2s+2)×(2s+2)-matrix B=(βil) defined by (15) is nonsymmetric, which is a contradiction. Therefore, if dimA1=r=2t+1≥s+3, then there do not exist involutive derivations on A.By Theorem8, if A is a 10-dimensional 9-Lie algebra with dimA1=7, or 9, then there does not exist involutive derivation on A. If 1≤dimA1=r≤10 and r≠7,9, then there are involutive derivations on A.


## 4. Involutive Derivations on(n+2)-Dimensional n-Lie Algebras with n=2s+1

By Theorem3, we only need to discuss the case where n is odd. So we suppose that A is a (2s+3)-dimensional (2s+1)-Lie algebra over F, s≥1, and that Et=Diag(1,⋯1) is the (t×t)-unit matrix.Lemma 9 (see [6]).
LetA be a (2s+3)-dimensional (2s+1)-Lie algebra over F with a basis {e1,⋯,e2s+3}. Then A is isomorphic to one and only one of the following possibilities:(17)aAisanabelian.bdimA1=1:b1e2,⋯,e2s+2=e1;b2e1,⋯,e2s+1=e1.cdimA1=2:c1e2,⋯,e2s+2=e1,e3,⋯,e2s+3=e2;c2e2,⋯,e2s+2=e1,e2,e4,⋯,e2s+3=e2,e1,e4,⋯,e2s+3=e1;c3e2,⋯,e2s+2=αe1+e2,e1,e3,⋯,e2s+2=e2,e2,e4,⋯,e2s+3=e2,e1,e4⋯,e2s+3=e1;c4e2,⋯,e2s+2=e1,e2,e3,⋯,e2s+2=e2,e2,e4,⋯,e2s+3=e2,e1,e4,⋯,e2s+3=e1;c5e2,⋯,e2s+2=e1,e1,e3,⋯,e2s+2=e2;c6e2,⋯,e2s+2=αe1+e2,e1,e3,⋯,e2s+2=e2;α∈F,α≠0.c7e1,e3,⋯,e2s+2=e1,e2,e3,⋯,e2s+2=e2;ddimA1=3:d1e2,⋯,e2s+2=e1,e2,e4,⋯,e2s+3=e2,e3,⋯,e2s+3=e3;d2e2,⋯,e2s+2=e1,e3,⋯,e2s+3=e3+αe2,e2,e4,⋯,e2s+3=e3,e1,e4,⋯,e2s+3=e1;d3e2,⋯,e2s+2=e1,e3,e4,⋯,e2s+3=e3,e2,e4,⋯,e2s+3=e2,e1,e4,⋯,e2s+3=2e1;d4e2,⋯,e2s+2=e1,e1,e3,⋯,e2s+2=e2,e1,e2,e4,⋯,e2s+2=e3;d5e1,e4,⋯,e2s+3=e1,e2,e4,⋯,e2s+3=e3,e3,e4,⋯,e2s+3=βe2+1+βe3,;d6e1,e4,⋯,e2s+3=e1,e2,e4,⋯,e2s+3=e2,e3,e4,⋯,e2s+3=e3;d7e1,e4,⋯,e2s+3=e2,e2,e4,⋯,e2s+3=e3,e3,e4,⋯,e2s+3=se1+te2+ue3,β,s,t,u∈F,β≠0,1,s≠0.And n-Lie algebras corresponding to the case (d7) with coefficients s,t,u and s′,t′,u′ are isomorphic if and only if there exists a nonzero element λ∈F such that s=λ3s′,t=λ2t′, u=λu′, s,s′,t,t′, u,u′∈F.(18)rdimA1=r,4≤r<2s+3,for2≤j≤r,1≤i≤r,r1e2,⋯,e2s+2=e1,e2,⋯,e^j,⋯,e2s+3=ej;r2e1,⋯,e^i,⋯,e2s+2=ei.Theorem 10.
IfA is a (2s+3)-dimensional (2s+1)-Lie algebra over F with dimA1=r<s+3, then there are involutive derivations on A.Proof.
Define linear mappingsDj:A→A, 1≤j≤6 by(19)D1ei=ei,1≤i≤s+2,ori=2s+3,-ei,otherwise;D2ei=ei,1≤i≤s+2,-ei,otherwise;D3ei=-ei,s+2≤i≤2s+1,ei,otherwise;D4ei=ei,1≤i≤s+1,ori=2s+2,-ei,otherwise;D5ei=ei,1≤i≤s+1,ori=2s+3,-ei,otherwise;D6ei=ei,1≤i≤s+3,-ei,otherwise.
SincedimA1=r≤s+2, it is easy to verify that D1 is an involutive derivation on the 3-Lie algebras of the cases of (b1), (ci), (dj), and (rk), where 1≤i≤7, 1≤j≤4 and 1≤k≤2.D2 is an involutive derivation on the 3-Lie algebras of the cases of (b1), (d4), and (ci), where 5≤i≤7. D3,D4, and D5 are involutive derivations on the 3-Lie algebras of the case of (b2). And D6 is an involutive derivation on the 3-Lie algebras of the cases of (d5),(d6), and (d7). Also Di are involutive derivations on abelian algebras for 1≤i≤6.Next, we discuss the case ofdimA1=r≥s+3. Let D be an endomorphism of A, (20)Dei=∑j=12s+3bijej,bij∈F,1≤i≤2s+3,and B=(bij) be the (2s+3)×(2s+3)-matrix. Then(21)De1,⋯,e2s+3T=Be1,⋯,e2s+3T=B1B0B2B3e1,⋯,e2s+3T,where B1B0B2B3 is the block matrix of B. First we discuss (2s+3)-dimensional (2s+1)-Lie algebras of the case (r1) in Lemma 9.Lemma 11.
IfA is a (2s+3)-dimensional (2s+1)-Lie algebra of the case (r1) with dimA1=r≥s+3,s≥1. Then the linear mapping D is an involutive derivation on A if and only if the block matrix B=B1B0B2B3 satisfies that B0=Or×(2s+3-r) (which is the zero (r×(2s+3-r))-matrix), and(22)B12=Er,B32=E2s+3-r,B2B1+B3B2=0,(23)∑j=22s+2bjj=b11,∑j=2,j≠i2s+3bjj=bii,2≤i≤r,b2s+3,i=-1i+1bi,1,2≤i≤r,bj,i=-1j-i-1bij,2≤i,j≤r,i≠j.Proof.
By (2), and a direct computation, D is a derivation of A if and only if matrix B has the property:(24)∑j=22s+2bjj=b11,∑j=2,j≠i2s+3bjj=bii,2≤i≤r,b1l=0,2≤l≤2s+3,b2s+3,i=-1i+1bi,1,bil=0,2≤i≤r,l≥r+1,bj,i=-1j-i-1bij,2≤i,j≤r,i≠j.Therefore, matrix B satisfies (23) and B0=Or×(2s+3-r). And D2=Id if and only if(25)B2=B12B1B0+B0B3B2B1+B3B2B2B0+B32=ErOOE2s+3-r.Thanks to B0=Or×(2s+3-r), (22) holds.Theorem 12.
LetA be a (2s+3)-dimensional (2s+1)-Lie algebra of the case (r1) with dimA1=r≥s+3, s≥1. If r is odd, then there are involutive derivations on A.Proof.
Letr=2t+1≥s+3. Then t≥2 and r≥5. Suppose D is an endomorphism of A and the matrix of D with respect to the basis {e1,⋯e2s+3} is B=(bij)=B1B0B2B3 which satisfies (22) and (23), and B0=Or×(2s+3-r). Then (26)B1=b1100⋯00b21b22b23⋯b2,r-1b2,rb31b23b33⋯b3,r-1b3,r⋮⋮⋮⋱⋮br-1,1-1rb2,r-1-1r-1b3,r-1⋯br-1,r-1br-1,rbr,1-1r+1b2,r-1rb3,r⋯br-1,rbr,r,B2=br+1,1br+1,2br+1,3⋯br+1,r⋮⋮⋮⋱⋮b2s+2,1b2s+2,2b2s+2,3⋯b2s+2,rb2s+3,1-b2,1b3,1⋯-1r+1br,1.Since ∑j=22s+2bjj=b11,∑j=2,j≠i2s+3bjj=bii,2≤i≤r, we have -b11+2b22-b2s+3,2s+3=0, (r-3)b22+∑2s+3r+1bii=0, b22=bii,3≤i≤r. Therefore,(27)b11=-1r-3r-1k1+2∑j=22s+3-rkj,bii=-1r-3∑j=12s+3-rkj,2≤i≤r,bjj=k2s+3-j+1,r+1≤j≤2s+3,k2s+3-j+1∈F.Suppose(28)B12=c11c12⋯c1i⋯c1j⋯c1r⋮⋮⋱⋮⋱⋮⋱⋮ci1ci2⋯cii⋯cij⋯cir⋮⋮⋱⋮⋱⋮⋱⋮cj1cj2⋯cji⋯cjj⋯cjr⋮⋮⋱⋮⋱⋮⋱⋮cr1cr2⋯cri⋯crj⋯crr.By (23), (29)c11=b112,c1l=0,2≤l≤r,cii=∑l=1i-1i+l-1bli2+∑l=i+1r-1i+l-1bil2,2≤i≤r,cij=∑l=1i-1l+i+1bliblj+∑l=i+1jbilblj+∑l=j+1r-1l+j+1bilbjl,2≤i<j≤r.cij=∑l=1j-1l+i+1bljbli+∑l=j+1i-1i+jbjlbli+∑l=i+1r-1l+j+1bjlbil,1≤j<i≤r.Therefore, the endomorphisms D of A, which are defined by (30)De1=e1orDe1=-e1,De2=∑k=3rek,Dei=-1i-1e2+∑k=3i-1-1iek+-1i-1∑k=i+1rek,3≤i≤r,Dej=-1jej,r+1≤j≤2s+3,De1=e1orDe1=-e1,De2=∑k=3rek,Dei=-1i-1e2+∑k=3i-1-1iek+-1i-1∑k=i+1rek,3≤i≤r,Dej=-1j-1ej,r+1≤j≤2s+3,are involutive derivations on A.Theorem 13.
LetA be a (2s+3)-dimensional (2s+1)-Lie algebra of the case (r1) with dimA1=r=2s+2(s≥1), then there does not exist an involutive derivation on A.Proof.
IfD is an involutive derivation on A, then, by Lemma 11 and (23), (31)b11=-2s+1k12s-1,b2s+3,2s+3=k1,bii=-k12s-1,2≤i≤r,k1∈F.Thanks to (22), b2s+3,2s+32=b112=k12=1. Therefore, (-(2s+1)k1/(2s-1))2=((2s+1)/(2s-1))2=1, which is a contradiction.Now we discuss case(r2).Theorem 14.
LetA be a (2s+3)-dimensional (2s+1)-Lie algebra of the case (r2) with dimA1=r≥s+3. Then there exist involutive derivations on A if and only if r is even.Proof.
By Lemma7, A=A1+˙Fe2s+3, where e2s+3∈Z(A), and A1 is a (2s+2)-dimensional (2s+1)-Lie subalgebra of A with dimA1=dimA11=r. Then there exist involutive derivations on A if and only if there exist involutive derivations on A1.
By Theorem 3 in [1], there is a basis {e1,⋯,e2s+2} of A1 such that(32)e1,⋯,e^j,⋯,e2s+2=0,t<j≤2s+3-t,e1,⋯,e^i,⋯,e2s+2=-1ie2s+3-i,1≤i≤t,or2s+3-t<i≤2s+2.
Ifr is even, then r=2t≥4. By Theorem 8 and (32), endomorphism D1 of A1 defined by(33)D1ei=ei,i=1,⋯,s+1,-ei,i=s+2,⋯,2s+2is an involutive derivation on A1. Therefore, the endomorphism D of A defined by (34)Dei=ei,1≤i≤s+1,Dej=-ej,s+2≤j≤2s+2,De2s+3=±e2s+3is involutive derivation on A.
IfdimA1=r is odd and endomorphism D of A is an involutive derivation on A, then r=2t+1>4. Suppose D(ei)=∑j=12s+3aijej, 1≤i≤2s+3. Then (35)De2s+3,ei1,⋯,ei2s=De2s+3,ei1,⋯,ei2s-∑j=12se2s+3,⋯,Deij,⋯,ei2s=0.We get D(e2s+3)∈Z(A)=Fe2s+3. Since D2=Id, D(e2s+3)=±e2s+3. By (32), A1=Fe1+⋯+Fet+Fe2s+1+⋯+Fe2s+3-t, and (36)-1iDe2s+3-i=∑k=1,k≠i2s+2e1,⋯,e^i,⋯,Dek,⋯,e2s+2,where 1≤i≤t, and 2s+3-t<i≤2s+2. Then ai,2s+3=0, for 1≤i≤t, or 2s+3-t<i≤2s+2, and DA1⊆A1. Then the endomorphism D2 of A1 defined by (37)D2ei=Dei,1≤i≤t,or2s+3-t<i≤2s+2,D2ej=Dej-aj,2s+3e2s+3=∑j=12s+2aijej,t<j≤2s+3-tis an involutive derivation on the (2s+2)-dimensional (2s+1)-Lie algebra A1, contradiction (Theorem 8). Therefore, there does not exist involutive derivation on A.


## 5. Structure of 3-Lie Algebras with Involutive Derivations

Let(L,[,]) be a Lie algebra over F, and p be an element which is not contained in L. Then A=L+˙Fp is a 3-Lie algebra in the multiplication(38)x,y,z=0,p,x,y=x,y,for  allx,y,z∈L.And the 3-Lie algebra (A,[,,]) is calledone-dimensional extension ofL.Theorem 15.
LetA be a 3-Lie algebra, then A is one-dimensional extension of a Lie algebra if and only if there exists an involutive derivation D on A such that dimA1=1, or dimA-1=1.Proof.
IfA is an one-dimensional extension of a Lie algebra L, then A=L+˙Fp. Define the endomorphism D of A by D(p)=-p (or p), and D(x)=x (or -x), ∀x∈L. Thanks to (38), D2=Id, and D([x,y,z])=0=[Dx,y,z]+[x,Dy,z]+[x,y,Dz], D([p,x,y])=[p,x,y]=[Dp,x,y]+[p,Dx,y]+[p,x,Dy], for all x,y,z∈L. Therefore, D is an involutive derivation on A, and dimA-1=1 (or dimA1=1).
Conversely, letD be an involutive derivation on a 3-Lie algebra A, and dimA-1=1 (or dimA1=1). Let A-1=Fp, and A1=L (or A-1=L, A1=Fp), where p∈A-L. Thanks to Theorem 3, L is a Lie algebra with the multiplication [x,y]=[p,x,y], for all x,y∈L, and A is one-dimensional extension of L.Let(L,[,]1) and (L,[,]2) be Lie algebras and {x1,⋯,xm} be a basis of L. For convenience, denote Lie algebras (L,[,]k) by Lk, k=1,2, respectively. Suppose p1 and p2 are two distinct elements which are not contained in L, and 3-Lie algebras (B,[,,]1) and (C,[,,]2) are one-dimensional extensions of Lie algebras L1 and L2, respectively, where B=L+˙Fp1, C=L+˙Fp2. Then Der(L1) and Der(L2) are subalgebras of gl(L).Definition 16.
LetL1=(L,[,]1) and L2=(L,[,]2) be two Lie algebras, and p1,p2 be two distinct elements which are not contained in L, and A=L+˙Fp1+˙Fp2. Then 3-algebra (A,[,,]) is called a two-dimensional extension of Lie algebras Lk, k=1,2, where [,,]:A∧A∧A→A defined by(39)x,y,p1=x,y1,x,y,p2=x,y2,x,y,z=0,p1,p2,x=λxp1+μxp2,∀x,y,z∈L,λx,μx∈F. If A is a 3-Lie algebra, then A is called a two-dimensional extension 3-Lie algebra of Lie algebras Lk, k=1,2.LetA=L+˙W be a two-dimensional extension of Lie algebras Lk, k=1,2, where W=Fp1+˙Fp2. Define linear mappings D1,D2:L→End(L) and D:L→W by(40)D1x=adp1,x,D2x=adp2,x,Dx=adp1,p2x,∀x∈L,that is, for all y∈L, D1(x)(y)=[p1,x,y]=[x,y]1, D2(x)(y)=[p2,x,y]=[x,y]2, D(x)=[p1,p2,x]. We have the following result.Theorem 17.
Let 3-algebraA be a two-dimensional extension of Lie algebras L1 and L2. Then A is a 3-Lie algebra if and only if linear mappings D1, D2, and D satisfy that D1:L1→Der(L1), D2:L2→Der(L2) are Lie homomorphisms, and(41)D1x3x1,x22=D1x3x1,x22+x1,D1x3x22-λx3x1,x21-μx3x1,x22,(42)D2x3x1,x21=D2x3x1,x21+x1,D2x3x21+λx3x1,x21+μx3x1,x22,(43)Dx1,x21=μx1λx2-λx1μx2p1,Dx1,x22=μx1λx2-λx1μx2p2,(44)λx1,x21=μx1,x22=μx1λx2-λx1μx2,μx1,x21=λx1,x22=0,(45)Dkx1x2=-Dkx2x1,forallx1,x2∈L,k=1,2,where x1,x2,x3∈L, D(xi)=λxip1+μxip2,i=1,2,3.Proof.
IfA is a two-dimensional extension 3-Lie algebra, then, by Definition 16, linear mappings Dk satisfy that Dk(Lk)⊆Der(Lk), and Dk are Lie homomorphisms, k=1,2. Thanks to (39),(46)D1x3x1,x22=p2,p1,x3,x1,x2+p2,x1,p1,x3,x2+p1,x3,p2,x1,x2=D1x3x1,x22+x1,D1x3x22-λx3x1,x21-μx3x1,x22,for all x1,x2,x3∈L, (41) holds. Similarly, we have (42).
Thanks to (39) and (40),(47)Dx1,x21=adp1,p2x1,x21=λx1,x21p1+μx1,x21p2=μx1λx2-λx1μx2p1,Dx1,x22=adp1,p2x1,x22=λx1,x22p1+μx1,x22p2=μx1λx2-λx1μx2p2,Equations (43) and (44) hold. Equation (45) follows from (39) and (40), directly.
Conversely, by (39), ∀x1,x2,x3,x∈L,(48)x1,x2,x3=0,p1,x1,x2=D1x1x2=x1,x21,p2,x1,x2=D2x1x2=x1,x22,p1,p2,x=Dx=λxp1+μxp2.Since Dk(L)⊆Der(Lk) and Dk are Lie homomorphisms, B=L+˙Fp1 and C=L+˙Fp2 are 3-Lie algebras, which are one-dimensional extension 3-Lie algebras of Lie algebras Lk, k=1,2, respectively.
Next we only need to prove that the multiplication onA defined by (39) satisfies (1). For all xi∈L, 1≤i≤5, that products [[x1,x2,x3],x4,x5], [[pj,x2,x3], x4,x5], [[x1,x2,x3], x4,pj] and [[x1,x2,pj], x4,pj] satisfy (1), j=1,2 follow from that B and C are one-dimensional extension 3-Lie algebras of Lk and (39), directly.
From (41) and (42), it follows that products [[pi,x1,x2],pj,x3], 1≤i≠j≤2, satisfy (1). It follows from (43)–(45) that products [p1,p2,[pi,x1,x2]], [x1,x2,[pi,p2,x3]], and [pi,x1,[p1,p2,x2], i=1,2, satisfy (1). We omit the computation process.Theorem 18.
Let(A,[,,]) be a 3-Lie algebra. Then A is a two-dimensional extension 3-Lie algebra of Lie algebras if and only if there is an involutive derivation T on A such that dimA1=2 or dimA-1=2.Proof.
IfA is a two-dimensional extension 3-Lie algebra of Lie algebras. Then by Theorem 15, there are Lie algebras L1=(L,[,]1) and L2=(L,[,]2), such that A=L+˙W and the multiplication of A is defined by (39), where W=Fp1+Fp2.
Define the endomorphismT of A by T(x)=x,T(p1)=-p1,T(p2)=-p2, or T(x)=-x,T(p1)=p1,T(p2)=p2,∀x∈L. Then T2=Id, and A1=L, A-1=W, or A-1=L, A1=W. Thanks to (38) and (41)-(45), T is a derivation of A.
Conversely, if there is an involutive derivationT on the 3-Lie algebra A such that dimA-1=2 (or dimA1=2). By Theorem 4, [A1,A1,A1]=0, [A1,A1,A-1]⊆A1, [A1,A-1,A-1]⊆A-1. Let L=A1 and A-1=Fp1+˙Fp2. Then [L,L,p1]⊆L,[L,L,p2]⊆L, and (L,[,]1) and (L,[,]2) are Lie algebras, where [x,y]1=[x,y,p1], [x,y]2=[x,y,p2], ∀x,y∈L. Thanks to Theorem 17, the 3-Lie algebra A is a two-dimensional extension 3-Lie algebra of Lie algebras L1 and L2.



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