JOURNAL OF MATHEMATICAL PHYSICS 51, 063505 共2010兲
Realizations of 3-Lie algebras Ruipu Bai,1,a兲 Chengming Bai,2,b兲 and Jinxiu Wang1,c兲 1
College of Mathematics and Computer, Hebei University, Baoding 071002, People’s Republic of China 2 Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, People’s Republic of China 共Received 17 February 2010; accepted 29 April 2010; published online 2 June 2010兲
3-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. In this paper, we provide a construction of 3-Lie algebras in terms of Lie algebras and certain linear functions. Moreover, with the construction from ␥-matrices and two-dimensional extensions of metric Lie algebras, all the complex 3-Lie algebras in dimension ⱕ5 are obtained along this approach. As a special case, we study the structure of the 3-Lie algebras constructed from the general linear Lie algebras with trace forms and prove that they are semisimple and local. © 2010 American Institute of Physics. 关doi:10.1063/1.3436555兴
I. INTRODUCTION
The notion of n-Lie algebra 共or Lie n-algebra, Filippov algebra, Nambu–Poisson algebra, and so on兲 was introduced by Filippov in 1985.1 An n-Lie algebra A is a vector space A endowed with an n-ary skew-symmetric multiplication satisfying the n-Jacobi identity, n
关关x1, . . . ,xn兴,y 2, . . . ,y n兴 = 兺 关x1, . . . ,关xi,y 2, . . . ,y n兴, . . . ,xn兴.
共1.1兲
i=1
On the other hand, in 1973, motivated by some problems of quark dynamics, Nambu2 introduced an n-ary generalization of Hamiltonian dynamics by means of the n-ary Poisson bracket,
冉 冊
关f 1, . . . , f n兴 = det
fi . xj
共1.2兲
Much latter, it was noted by some physicists that the n-bracket 共1.2兲 satisfies Eq. 共1.1兲. Due to this fact, Takhtajan3 developed the foundations of the theory of n-Poisson or Nambu–Poisson manifolds systematically. In fact, n-Lie algebras are a kind of multiple algebraic systems appearing in many fields in mathematics and mathematical physics. Recently, the study of n-Lie algebras attracts more and more attention due to their close relationships with the string theory 共cf. Refs. 4–12兲. For example, the structure of 3-Lie algebras is applied to the study of the supersymmetry and gauge symmetry transformations of the world-volume theory of multiple coincident M2-branes; the Bagger– Lambert theory has a novel local gauge symmetry, which is based on a metric 3-Lie algebra; the identity 共1.1兲 for a 3-Lie algebra is essential to define the action with N = 8 supersymmetry; the n-Jacobi identity can be regarded as a generalized Plucker relation in the physics literature, and so on. The theory of n-Lie algebras has been widely studied.13–17
a兲
Electronic mail:
[email protected]. Electronic mail:
[email protected]. Electronic mail:
[email protected].
b兲 c兲
0022-2488/2010/51共6兲/063505/12/$30.00
51, 063505-1
© 2010 American Institute of Physics
063505-2
J. Math. Phys. 51, 063505 共2010兲
Bai, Bai, and Wang
In this paper, we pay our main attention to the 3-Lie algebras. It is quite necessary to get more examples as a guide for further study. However, it is not easy here due to the 3-ary operation. In order to avoid the complicated equations involving the structural constants like in the classification 共cf. Ref. 17兲, it is natural to consider to use some well known structures to obtain some 3-Lie algebras. Such an approach is called a “realization” theory. We have already obtained some experiences. • 共Commutative associative algebras with derivations兲 (Reference 1) Let 共A , ·兲 be a commutative associative algebra and 兵D1 , D2 , D3其 be three pairwise commuting derivations. Then there is a 3-Lie algebra structure on A which is called a Jacobian algebra defined by
冢
冣
D1共x1兲 D1共x2兲 D1共x3兲 关x1,x2,x3兴 = det D2共x1兲 D2共x2兲 D2共x3兲 . D3共x1兲 D3共x2兲 D3共x3兲
共1.3兲
• 共Dirac ␥-matrices兲 (Reference 9) Let A be a vector space spanned by the four-dimensional ␥-matrices 共␥兲 and let ␥5 = ␥1 , . . . , ␥4. Then the 3-ary operation 关x,y,z兴 = 关关x,y兴␥5,z兴
∀ x,y,z 苸 A,
共1.4兲 14
defines a 3-Lie algebra, which is isomorphic to the unique simple 3-Lie algebra. • 共Two-dimensional extensions of metric Lie algebras兲 (Reference 18) Let 共g , B兲 be a metric Lie algebra over a field F, that is, B is a nondegenerate symmetric bilinear form on g satisfying B共关x , y兴 , z兲 = −B共y , 关x , z兴兲 for every x , y , z 苸 g. Suppose 兵x1 , . . . , xm其 is a basis of g m and 关xi , x j兴 = 兺k=1 akijxk, 1 ⱕ i , j ⱕ m. Set g0 = g 丣 Fx0 丣 Fx−1
共the direct sum of vector space兲.
共1.5兲
Then there is a 3-Lie algebra structure on g0 given by 关x0,xi,x j兴 = 关xi,x j兴,
1 ⱕ i, j ⱕ m,
关x−1,xi,x j兴 = 0,
0 ⱕ i, j ⱕ m,
共1.6兲
m
关xi,x j,xk兴 = 兺 asijB共xs,xk兲x−1,
1 ⱕ i, j,k ⱕ m.
共1.7兲
s=1
• 共The general linear Lie algebras with trace forms兲 (Reference 19) Let g = gl共m , F兲 be the general linear Lie algebra. Then there is a 3-Lie algebra structure on g defined by 关A,B,C兴 = 共tr A兲关B,C兴 + 共tr B兲关C,A兴 + 共tr C兲关A,B兴
∀ A,B,C 苸 g.
共1.8兲
Note that the latter three cases are more or less related to Lie algebras. On the other hand, there are many interesting connections between 3-Lie algebras and Lie algebras. So it is natural to consider “how to get 3-Lie algebras from Lie algebras,” which is the main aim of this paper. Motivated by the construction of 3-Lie algebras from the general linear Lie algebras with trace forms, we provide a construction of 3-Lie algebras from Lie algebras and certain linear functions. More importantly, we will show that with the construction from ␥-matrices and two-dimensional extensions of metric Lie algebras, all the complex 3-Lie algebras in dimension ⱕ5 are obtained along this approach. This paper is organized as follows. Section II introduces some basic notions. Section III provides a construction of 3-Lie algebras from Lie algebras and certain linear functions. Section IV devotes to realize the 3-Lie algebras in dimension ⱕ5 in terms of Lie algebras. Section V studies the structure of the 3-Lie algebras constructed from the general linear Lie algebras with trace forms. Throughout this paper, all algebras are of finite dimension and over the complex field C. Any bracket which is not listed in the multiplication table of an n-Lie algebra or a Lie algebra is assumed to be zero.
063505-3
J. Math. Phys. 51, 063505 共2010兲
Realizations of 3-Lie algebras
II. FUNDAMENTAL NOTIONS
Let A be an n-Lie algebra. The n-ary skew symmetry of the operation 关x1 , . . . , xn兴 means that 关x1, . . . ,xn兴 = sgn共兲关x共1兲, . . . ,x共n兲兴
∀ x1, . . . ,xn 苸 A
共2.1兲
for any permutation 苸 Sn. The n-Jacobi identity 共1.1兲 is also called the generalized Jacobi identity, or simply the Jacobi identity. A subspace B of A is called a subalgebra if 关B , . . . , B兴 債 B. In particular, the subalgebra generated by the vectors 关x1 , . . . , xn兴 for any x1 , . . . , xn 苸 A is called the derived algebra of A, which is denoted by A1. An ideal of an n-Lie algebra A is a subspace I such that 关I , A , . . . , A兴 債 I. If A1 ⫽ 0 and A has no ideals except for 0 and itself, then A is called a simple n-Lie algebra. Due to Ling,14 up to isomorphisms there exists only one finite-dimensional simple n-Lie algebra over an algebraically closed field of characteristic 0, that is, the 共n + 1兲-dimensional n-Lie algebra with dim A1 = n + 1 If an n-Lie algebra A contains a unique minimal proper ideal, then A is called a local n-Lie algebra. An ideal I of an n-Lie algebra A is called 2-solvable if I共s兲 = 0 for some s ⱖ 0, where I共0兲 = I and 共s兲 I is defined as I共s+1兲 = 关I共s兲,I共s兲,A, . . . ,A兴
共2.2兲
for s ⱖ 0. If the maximal 2-solvable ideal of A is zero, then A is called a 2-semisimple n-Lie algebra. In particular, a 2-semisimple 3-Lie algebra is also called semisimple. An ideal I of an n-Lie algebra A is called nilpotent if Is = 0 for some s ⱖ 0, where I0 = I and Is is defined as Is = 关Is−1,I,A, . . . ,A兴
共2.3兲
for s ⱖ 1. An n-Lie algebra is called Abelian if A2 = 0. The subset Z共A兲 = 兵x 苸 A兩关x,y 1, . . . ,y n−1兴 = 0
∀ y 1, . . . ,y n−1 苸 A其
共2.4兲
is called the center of A. Obviously, Z共A兲 is an Abelian ideal of A. Note that an ideal I of an n-Lie algebra A may not be a nilpotent ideal, although it is a nilpotent subalgebra. Such an ideal is called a hyponilpotent ideal of A.16 If a hyponilpotent ideal I is not properly contained in any hyponilpotent ideals, then I is called a maximal hyponilpotent ideal of A. In fact, the sum of two hyponilpotent ideals of A may not be hyponilpotent.16 A subalgebra H of an n-Lie algebra A is called a Cartan subalgebra of A if H is a nilpotent subalgebra and H is equal to its normalizer, that is, H = 兵x 苸 A兩关x,H, . . . ,H兴 債 H其.
共2.5兲
An n-Lie algebra A is called metric if A is endowed with a nondegenerate symmetric bilinear form  satisfying
共关x1, . . . ,xn−1,y 1兴,y 2兲 = − 共关x1, . . . ,xn−1,y 2兴,y 1兲
∀ x1, . . . ,xn−1,y 1,y 2 苸 A.
共2.6兲
A bilinear form  on an n-Lie algebra A is called invariant if  satisfies Eq. 共2.6兲. III. CONSTRUCTION OF 3-LIE ALGEBRAS FROM LIE ALGEBRAS AND LINEAR FUNCTIONS
Theorem 3.1: Let g be a Lie algebra and gⴱ be the dual space of g. Suppose that f 苸 gⴱ satisfying f共关x , y兴兲 = 0 for all x , y 苸 g. Then there is a 3-Lie algebra structure on g given by 关x,y,z兴 f = f共x兲关y,z兴 + f共y兲关z,x兴 + f共z兲关x,y兴 We denote it by g f . Proof: It is obvious that for any x1 , x2 , x3 苸 g,
∀ x,y,z 苸 g.
共3.1兲
063505-4
J. Math. Phys. 51, 063505 共2010兲
Bai, Bai, and Wang
关x1,x2,x3兴 f = sgn共兲关x共1兲,x共2兲,x共3兲兴 f
∀ 苸 S3 .
Furthermore, for any x1 , x2 , x3 , y 1 , y 2 苸 g, 关关x1,x2,x3兴 f ,y 1,y 2兴 f = 关f共x1兲关x2,x3兴 + f共x2兲关x3,x1兴 + f共x3兲关x1,x2兴,y 1,y 2兴 f = f共x1兲关关x2,x3兴,y 1,y 2兴 f + f共x2兲关关x3,x1兴,y 1,y 2兴 f + f共x3兲关关x1,x2兴,y 1,y 2兴 f = f共x1兲f共y 1兲关y 2,关x2,x3兴兴 + f共x1兲f共y 2兲关关x2,x3兴,y 1兴 + f共x2兲f共y 1兲关y 2,关x3,x1兴兴 + f共x2兲f共y 2兲关关x3,x1兴,y 1兴 + f共x3兲f共y 1兲关y 2,关x1,x2兴兴 + f共x3兲f共y 2兲关关x1,x2兴,y 1兴, 关关x1,y 1,y 2兴 f ,x2,x3兴 f + 关x1,关x2,y 1,y 2兴 f ,x3兴 f + 关x1,x2,关x3,y 1,y 2兴 f 兴 f = 关f共x1兲关y 1,y 2兴 + f共y 1兲关y 2,x1兴 + f共y 2兲关x1,y 1兴,x2,x3兴 f + 关x1, f共x2兲关y 1,y 2兴 + f共y 1兲关y 2,x2兴 + f共y 2兲关x2,y 1兴,x3兴 f + 关x1,x2, f共x3兲关y 1,y 2兴 + f共y 1兲关y 2,x3兴 + f共y 2兲关x3,y 1兴兴 f = f共x2兲f共x1兲关x3,关y 1,y 2兴兴 + f共x2兲f共y 1兲关x3,关y 2,x1兴兴 + f共x2兲f共y 2兲关x3,关x1,y 1兴兴 + f共x3兲f共x1兲关关y 1,y 2兴,x2兴 + f共x3兲f共y 1兲关关y 2,x1兴,x2兴 + f共x3兲f共y 2兲关关x1,y 1兴,x2兴 + f共x1兲f共x2兲关关y 1,y 2兴,x3兴 + f共x1兲f共y 1兲关关y 2,x2兴,x3兴 + f共x1兲f共y 2兲关关x2,y 1兴,x3兴 + f共x3兲f共x2兲关x1,关y 1,y 2兴兴 + f共x3兲f共y 1兲关x1,关y 2,x2兴兴 + f共x3兲f共y 2兲关x1,关x2,y 1兴兴 + f共x1兲f共x3兲关x2,关y 1,y 2兴兴 + f共x1兲f共y 1兲关x2,关y 2,x3兴兴 + f共x1兲f共y 2兲关x2,关x3,y 1兴兴 + f共x2兲f共x3兲关关y 1,y 2兴,x1兴 + f共x2兲f共y 1兲关y 2,x3兴,关x1兴 + f共x2兲f共y 2兲关关x3,y 1兴,x1兴. By the Jacobi identities for the Lie algebra g, we show that 关关x1,x2,x3兴 f ,y 1,y 2兴 f = 关关x1,y 1,y 2兴 f ,x2,x3兴 f + 关x1,关x2,y 1,y 2兴 f ,x3兴 f + 关x1,x2,关x3,y 1,y 2兴 f 兴 f . Hence, the conclusion holds. 䊐 The following two conclusions are obvious. Corollary 3.2: Let g be a Lie algebra. If 关g , g兴 = g, then any 3-Lie algebra g f constructed through Theorem 3.1 is Abelian. In particular, any 3-Lie algebra g f constructed from a semisimple Lie algebra g through Theorem 3.1 is Abelian. Corollary 3.3: If a 3-Lie algebra g f is constructed from a Lie algebra g and a linear function f 苸 gⴱ through Theorem 3.1, then 关g1f , g1f , g1f 兴 f = 0. Proposition 3.4: Let g f be a 3-Lie algebra constructed from a Lie algebra g and a linear function f 苸 gⴱ through Theorem 3.1. If g is nilpotent, then the 3-Lie algebra g f is nilpotent. s−1 . Since Proof: By Eq. 共3.1兲, we show that g1f = 关g f , g f , g f 兴 f 債 关g , g兴 = g1. Suppose that gs−1 f 債g s−1 f共g 兲 = 0 for s ⬎ 1, we have s−1 ,g,g兴 f 債 关gs−1,g兴 = gs . gsf = 关gs−1 f ,g f ,g f 兴 f 債 关g
Since g is nilpotent, there exists s ⱖ 0 such that gs = 0. Thus, gsf = 0. Therefore, g f is a nilpotent 3-Lie algebra. 䊐 Similarly, we have the following conclusion. Proposition 3.5: Let g f be a 3-Lie algebra constructed from a Lie algebra g and a linear function f 苸 gⴱ through Theorem 3.1. If g is solvable, then g f is a 2-solvable 3-Lie algebra. Example 3.6: Let g be a Lie algebra with a basis 兵x1 , . . . , xm , . . . , xs其, where 兵xm+1 , . . . , xs其 is a basis of the derived algebra 关g , g兴. For 1 ⱕ i ⱕ m, let f i 苸 gⴱ such that f i共x j兲 = ␦ij. Then there is a family of 3-Lie algebras 兵g f i其 given by 关xi,x j,xk兴 f i = 关x j,xk兴
and
关x j1,x j2,x j3兴 f i = 0
Moreover, I = 兺 j⫽iCx j is a maximal ideal of g f i.
if i 苸 兵j1, j2, j3其.
063505-5
J. Math. Phys. 51, 063505 共2010兲
Realizations of 3-Lie algebras
Example 3.7: Let g be the m-dimensional simplest filiform Lie algebra with a basis 兵x1 , . . . , xm其 satisfying 关x1,xi兴 = xi−1,
i = 3, . . . ,m.
Let f 苸 gⴱ such that f共xm兲 = 1, f共x j兲 = 0, j ⫽ m. Then the 3-Lie algebra g f constructed through Theorem 3.1 is the m-dimensional simplest filiform 3-Lie algebra in the sense of Ref. 16. Moreover, there exist exactly five classes of solvable 3-Lie algebras in dimension ⱕm + 2 with the maximal hyponilpotent ideal g f . Example 3.8: In general, for a fixed Lie algebra g, one can obtain the nonisomorphic 3-Lie algebras g f from the different linear functions f 苸 gⴱ through Theorem 3.1. For example, let g be an eight-dimensional Lie algebra with a basis 兵e1 , . . . , e8其 satisfying 关e2,e8兴 = e3,
关e3,e8兴 = e4,
关e4,e8兴 = e5,
关e5,e8兴 = e6,
关e6,e8兴 = e7,
关e2,e3兴 = 2e6 ,
关e2,e4兴 = 2e7 . Let f 1 , f 2 , f 3 苸 gⴱ such that f 1共e1兲 = 1,
f 1共e2兲 = − 1,
f 2共e2兲 = 1,
f 2共e8兲 = 1, f 3共e8兲 = 1
and and
and
f 1共ei兲 = 0,
f 2共ei兲 = 0, f 3共ei兲 = 0,
3 ⱕ i ⱕ 8,
i = 1,3,4,5,6,7, i ⫽ 8.
Then there are the following eight-dimensional 3-Lie algebras g f 1, g f 2, and g f 3:
冦 冧 关e1,e2,e8兴 f 1 = e3
关e1,e3,e8兴 f 1 = e4
关e1,e4,e8兴 f 1 = e5
关e1,e5,e8兴 f 1 = e6
关e1,e6,e8兴 f 1 = e7
关e2,e3,e8兴 f 1 = − e4
关e2,e4,e8兴 f 1 = − e5
关e2,e5,e8兴 f 1 = − e6
冦
关e2,e3,e8兴 f 2 = 2e6 + e4 关e2,e4,e8兴 f 2 = 2e7 + e5 关e2,e5,e8兴 f 2 = e6 关e2,e6,e8兴 f 2 = e7 ,
再 冧
关e2,e4,e8兴 f 3 = 2e7 关e2,e3,e8兴 f 3 = 2e6 .
冎
关e1,e2,e3兴 f 1 = 2e6
关e2,e6,e8兴 f 1 = − e7
关e1,e2,e4兴 f 1 = 2e7 ,
Obviously, g f 1, g f 2, and g f 3 are not mutually isomorphic. Recall that an algebra 共A , ⴰ兲 is called a Lie-admissible algebra if the commutator 关x , y兴 = x ⴰ y − y ⴰ x for any x , y 苸 A is a Lie algebra. Obviously, an associative algebra is a Lie-admissible algebra. Corollary 3.9: Let 共A , ⴰ兲 be a Lie-admissible algebra and f 苸 Aⴱ. If f共x ⴰ y兲 = f共y ⴰ x兲 for any x , y 苸 A, then there exists a 3-Lie algebra structure on A given by 关x,y,z兴 = 共f共z兲x − f共x兲z兲 ⴰ y + 共f共x兲y − f共y兲x兲 ⴰ z + 共f共y兲z − f共z兲y兲 ⴰ x
∀ x,y,z 苸 A.
共3.2兲
Proof: It follows from Theorem 3.1. 䊐 Lemma 3.10: (Reference 20) Let 共A , ⴰ兲 be a commutative associative algebra and D be a derivation. Then the binary operation given by
063505-6
J. Math. Phys. 51, 063505 共2010兲
Bai, Bai, and Wang
x ⴱ y = x ⴰ D共y兲
∀ x,y 苸 A,
共3.3兲
defines a Lie-admissible algebra. Corollary 3.11: Let 共A , ⴰ兲 be a commutative associative algebra and D be a derivation. Let f 苸 Aⴱ such that f共x ⴰ D共y兲兲 = f共D共x兲 ⴰ y兲 for any x , y 苸 A. Then there exists a 3-Lie algebra structure on A given by (for any x , y , z 苸 A) 关x,y,z兴 = y ⴰ 共f共x兲D共z兲 − f共z兲D共x兲兲 + z ⴰ 共f共y兲D共x兲 − f共x兲D共y兲兲 + x ⴰ 共f共z兲D共y兲 − f共y兲D共z兲兲. 共3.4兲 Proof: It follows from Corollary 3.9 and Lemma 3.10. 䊐 Example 3.12: Let A be a linear space spanned by 兵sin mx , cos nx 兩 m , n are all integers其. Then A is a commutative associative algebra with the common multiplication of functions, that is,
冦
sin mx sin nx = − 21 共cos共m + n兲x − cos共m − n兲x兲 cos mx cos nx = 21 共cos共m + n兲x + cos共m − n兲x兲 cos mx sin nx = 21 共sin共m + n兲x − sin共m − n兲x兲.
冧
Set D0:A → A,
D0共f共x兲兲 =
df共x兲 dx
∀ f共x兲 苸 A.
Then D0 is a derivation of the associative algebra A and D0共sin mx兲 = m cos共mx兲, D0共cos mx兲 = − m sin共mx兲. Moreover, A is a Lie algebra with the bracket given by 关␣共x兲, 共x兲兴 = ␣共x兲
d共x兲 d␣共x兲 − 共x兲 dx dx
∀ ␣共x兲, 共x兲 苸 A.
It is obvious that 1 苸 关A , A兴. Set f 苸 Aⴱ by f共1兲 = 1
and
f共sin mx兲 = f共cos nx兲 = 0,
m,n ⫽ 0.
Then there is a 3-Lie algebra structure on A given by 共for any ␣共x兲 , 共x兲 , ␥共x兲 苸 A兲 关␣共x兲, 共x兲, ␥共x兲兴 = 共x兲共f共␣共x兲兲D0共␥共x兲兲 − f共␥共x兲兲D0共␣共x兲兲兲 + ␥共x兲共f共共x兲兲D0共␣共x兲兲 − f共␣共x兲兲D0共共x兲兲兲 + ␣共x兲共f共␥共x兲兲D0共共x兲兲 − f共共x兲兲D0共␥共x兲兲兲.
共3.5兲
Explicitly, we have
冦
关1,sin mx,sin nx兴 = n sin mx cos nx − m cos mx sin nx 关1,sin mx,cos nx兴 = − n sin mx sin nx − m cos mx cos nx 关1,cos mx,cos nx兴 = − n cos mx sin nx + m sin mx cos nx 关cos sx,cos mx,cos nx兴 = 关cos sx,cos mx,sin nx兴 = 关cos sx,sin mx,sin nx兴 =关sin sx,sin mx,sin nx兴 = 0,
m,n,s ⫽ 0.
冧
共3.6兲
IV. REALIZATIONS OF 3-LIE ALGEBRAS IN DIMENSION m ⱕ 5
In this section, we study the realizations of the complex 3-Lie algebras in dimensions 3–5. It is clear that any Abelian 3-Lie algebra g f can be obtained from an Abelian Lie algebra g and any
063505-7
J. Math. Phys. 51, 063505 共2010兲
Realizations of 3-Lie algebras
linear function f 苸 gⴱ through Theorem 3.1. So next we mainly study the non-Abelian 3-Lie algebras. Note that the classification of the complex 3-Lie algebras in dimensions 4 and 5 was given in Ref. 17. It is known that up to isomorphism, there is only one three-dimensional non-Abelian 3-Lie algebra with a basis 兵e1 , e2 , e3其 given by 关e1,e2,e3兴 = e1 .
共4.1兲
Theorem 4.1: Let A be the non-Abelian three-dimensional 3-Lie algebra with a basis 兵e1 , e2 , e3其 given by Eq. (4.1). Then A is isomorphic to a 3-Lie algebra g f constructed from a three-dimensional Lie algebra g and a linear function f 苸 gⴱ through Theorem 3.1. Proof: Let g be the three-dimensional Lie algebra with the basis 兵e1 , e2 , e3其 defined by 关e1 , e2兴 = e1. Let f 苸 gⴱ such that f共e3兲 = 1, f共e1兲 = f共e2兲 = 0. Then it is straightforward to check that 䊐 A is isomorphic to g f . Lemma 4.2: (Reference 17) Let A be a four-dimensional complex 3-Lie algebra with a basis 兵e1 , e2 , e3 , e4其. Then A is isomorphic to one of the following (not mutually isomorphic) 3-Lie algebras: (a) (b)
If dim A1 = 0, then A is an Abelian 3-Lie algebra. If dim A1 = 1, then there are two cases, 共b1兲 关e2,e3,e4兴 = e1,
(c)
1
If dim A = 2, then there are three cases, 共c1兲
再
关e2,e3,e4兴 = e1 关e1,e3,e4兴 = e2 ,
冎 再 再 共c2兲
共c3兲 (d)
关e2,e3,e4兴 = ␣e1 + e2 关e1,e3,e4兴 = e2 ,
关e1,e3,e4兴 = e1
关e2,e3,e4兴 = e2 ,
冎
冎
共4.2兲
␣ ⫽ 0,
共4.3兲
共4.4兲
If dim A1 = 3, then there is one case, 共d兲 关e2,e3,e4兴 = e1,
(e)
共b2兲 关e1,e2,e3兴 = e1 .
关e1,e3,e4兴 = e2,
关e1,e2,e4兴 = e3 .
共4.5兲
If dim A1 = 4, then there is one case, 共e兲 关e2,e3,e4兴 = e1,
关e1,e3,e4兴 = e2,
关e1,e2,e4兴 = e3,
关e1,e2,e3兴 = e4 .
共4.6兲
Note that (e) gives the unique simple 3-Lie algebra.14 Theorem 4.3: Let A be a four-dimensional 3-Lie algebra. If A is not simple, then A is isomorphic to a 3-Lie algebra g f constructed from a four-dimensional Lie algebra g and a linear function f 苸 gⴱ through Theorem 3.1. Proof: With the notations in Lemma 4.2, we only need to consider cases 共b兲–共d兲. Case dim A1 = 1 . • 共b1兲 is isomorphic to g f , where the function f 苸 gⴱ satisfies f共ei兲 = ␦4i for • 共b2兲 is isomorphic to g f , where the function f 苸 gⴱ satisfies f共ei兲 = ␦3i for
Lie algebra g is given by 关e2 , e3兴 = e1 and the linear 1 ⱕ i ⱕ 4. Lie algebra g is given by 关e1 , e2兴 = e1 and the linear 1 ⱕ i ⱕ 4.
Case dim A1 = 2 . • 共c1兲 is isomorphic to g f , where the Lie algebra g is given by 关e2 , e3兴 = e1 , 关e1 , e3兴 = e2 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 4. • 共c2兲 is isomorphic to g f , where the Lie algebra g is given by 关e2 , e3兴 = ␣e1 + e2 , 关e1 , e3兴 = e2 , ␣ ⫽ 0, and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 4.
063505-8
J. Math. Phys. 51, 063505 共2010兲
Bai, Bai, and Wang
• 共c3兲 is isomorphic to g f , where the Lie algebra g is given by 关e1 , e3兴 = e1 , 关e2 , e3兴 = e2 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 4. Case dim A1 = 3 . 共d兲 is isomorphic to g f , where the Lie algebra g is given by 关e1 , e3兴 = e2, 关e2 , e3兴 = e1, 关e1 , e2兴 = e3 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 4. 䊐 Remark 4.4: As in Sec. I, the four-dimensional simple 3-Lie algebra 共e兲 given by Eq. 共4.6兲 can be constructed from ␥-matrices through Eq. 共1.4兲. In order to be self-contained, we give an explicit illustration. In fact, we choose the matrices ␥1, ␥2, ␥3, ␥4, and ␥5 as follows:
␥1 =
冢
0 0 0
冣 冢 冢 冣 0 1
0
1 0
−1 0 0
−1
␥4 =
0
0
1 0
0
0
0 1
0
0
0 0
,
0
0
0 0 −1
0
0 1
0
0
1 0
0
−1 0 0
0
␥2 =
,
0 0
0 0 −1
0
冣 冢 冣 冢 冣 ␥3 =
,
0
0 1
0
0 0 −1
−1 0 0 0
1 0
0
0
,
0
0 0 1 0
1 1 0 0 0 1 ␥ 5 = ␥ 1␥ 2␥ 3␥ 4 = 4 4 1 0 0 0 0 1 0 0
and
−1
.
Note that the above ␥ , ␥ , and ␥4 are the ordinary ␥-matrices and ␥2 is the corresponding ordinary ␥-matrix multiplied by −冑−1. Let V = C␥1 丣 C␥2 丣 C␥3 丣 C␥4 be a four-dimensional vector space. Then there is a 3-Lie algebra structure on V given by 1
3
关␥1, ␥2, ␥3兴 = 关关␥1, ␥2兴␥5, ␥3兴 = ␥4,
关␥1, ␥2, ␥4兴 = 关关␥1, ␥2兴␥5, ␥4兴 = ␥3 ,
关␥1, ␥3, ␥4兴 = 关关␥1, ␥3兴␥5, ␥4兴 = ␥2,
关␥2, ␥3, ␥4兴 = 关关␥2, ␥3兴␥5, ␥4兴 = ␥1 .
Moreover, it is obvious that V is isomorphic to 共e兲 by letting ␥i → ei for i = 1 , 2 , 3 , 4. Lemma 4.5: (Reference 17) Let A be a five-dimensional complex 3-Lie algebra with a basis 兵e1 , e2 , e3 , e4 , e5其. Then dim A1 ⱕ 4 and A is isomorphic to one of the following (not mutually isomorphic) 3-Lie algebras: (a) (b)
If dim A1 = 0, then A is an Abelian 3-Lie algebra. If dim A1 = 1, then there are two cases, 共b1兲 关e2,e3,e4兴 = e1,
(c)
共b2兲 关e1,e2,e3兴 = e1 .
共4.7兲
If dim A1 = 2, then there are seven cases: 共␣ ⫽ 0兲,
共c 兲 1
共c4兲
冦
再
关e2,e3,e4兴 = e1 关e3,e4,e5兴 = e2 ,
关e2,e3,e4兴 = e1 关e1,e3,e4兴 = e2 关e2,e4,e5兴 = e2 关e1,e4,e5兴 = e1
冧
冎 冦
共c5兲
共c 兲 2
再
关e2,e3,e4兴 = e1 关e2,e4,e5兴 = e2 关e1,e4,e5兴 = e1 ,
关e2,e3,e4兴 = ␣e1 + e2 关e1,e3,e4兴 = e2 ,
共c7兲
再
关e1,e3,e4兴 = e1 关e2,e3,e4兴 = e2 .
冧 再 共c3兲
冎 冎
共c6兲
冦
关e2,e3,e4兴 = e1 关e1,e3,e4兴 = e2 ,
冎
共4.8兲
关e2,e3,e4兴 = ␣e1 + e2 关e1,e3,e4兴 = e2 关e2,e4,e5兴 = e2 关e1,e4,e5兴 = e1 ,
冧
共4.9兲 共4.10兲
063505-9
(d)
J. Math. Phys. 51, 063505 共2010兲
Realizations of 3-Lie algebras
If dim A1 = 3, then there are seven cases: 共␣ ⫽ 0兲,
共d1兲
冦
关e2,e3,e4兴 = e1 关e2,e4,e5兴 = − e2 关e3,e4,e5兴 = e3 ,
共d 兲 4
冦
冧
共d2兲
关e2,e3,e4兴 = e1 关e1,e3,e4兴 = e2 关e1,e2,e4兴 = e3 ,
冦
关e3,e4,e5兴 = e3 + ␣e2 关e2,e4,e5兴 = e3 关e1,e4,e5兴 = e1 ,
冧 冦 共d 兲 5
关e1,e4,e5兴 = e1
共d 兲 关e2,e4,e5兴 = e2 关e3,e4,e5兴 = e3 , 6
冦
关e2,e3,e4兴 = e1
冧 冦 共d3兲
关e2,e3,e4兴 = e1 关e3,e4,e5兴 = e3 关e2,e4,e5兴 = e2 关e1,e4,e5兴 = 2e1 ,
共4.11兲
关e1,e4,e5兴 = e1 关e2,e4,e5兴 = e3 关e3,e4,e5兴 = e2 + 共1 + 兲e3,
冧 冦 共d 兲 7
 ⫽ 0,1,
关e1,e4,e5兴 = e2 关e2,e4,e5兴 = e3 关e3,e4,e5兴 = se1 + te2 + ue3,
冧
s ⫽ 0.
冧
冧
共4.12兲
共4.13兲
Note that any two 3-Lie algebras of the type 共d7兲 with the parameters 兵s , t , u其 and 兵s⬘ , t⬘ , u⬘其 are isomorphic if and only if there exists r 苸 C such that r ⫽ 0 and s = r 3s ⬘, (e)
t = r 2t ⬘,
u = ru⬘ .
共4.14兲
If dim A1 = 4, then there are two cases,
共e1兲
冦
关e2,e3,e4兴 = e1 关e3,e4,e5兴 = e2 关e2,e4,e5兴 = e3 关e2,e3,e5兴 = e4 ,
冧 冦 共e2兲
关e2,e3,e4兴 = e1 关e1,e3,e4兴 = e2 关e1,e2,e4兴 = e3 关e1,e2,e3兴 = e4 .
冧
共4.15兲
Theorem 4.6: Let A be a five-dimensional 3-Lie algebra. If dim A1 ⱕ 3, then A is isomorphic to a 3-Lie algebra g f constructed from a five-dimensional Lie algebra g and a linear function f 苸 gⴱ through Theorem 3.1. Proof: Case dim A1 = 1. • 共b1兲 is isomorphic to g f , where the function f 苸 gⴱ satisfies f共ei兲 = ␦4i for • 共b2兲 is isomorphic to g f , where the function f 苸 gⴱ satisfies f共ei兲 = ␦3i for
Lie algebra g is given by 关e2 , e3兴 = e1 and the linear 1 ⱕ i ⱕ 5. Lie algebra g is given by 关e1 , e2兴 = e1 and the linear 1 ⱕ i ⱕ 5.
Case dim A1 = 2. • 共c1兲 is isomorphic to g f , where the Lie algebra g is given by 关e2 , e3兴 = e1, 关e3 , e5兴 = −e2 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 5. • 共c2兲 is isomorphic to g f , where the Lie algebra g is given by 关e2 , e3兴 = e1, 关e2 , e5兴 = −e2, 关e1 , e5兴 = −e1 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 5. • 共c3兲 is isomorphic to g f , where the Lie algebra g is given by 关e2 , e3兴 = e1, 关e1 , e3兴 = e2 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 5. • 共c4兲 is isomorphic to g f , where the Lie algebra g is given by 关e2 , e3兴 = e1, 关e1 , e3兴 = e2, 关e2 , e5兴 = −e2, 关e1 , e5兴 = −e1 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 5. • 共c5兲 is isomorphic to g f , where the Lie algebra g is given by 关e2 , e3兴 = ␣e1 + e2, ␣ ⫽ 0, 关e1 , e3兴 = e2 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 5.
063505-10
J. Math. Phys. 51, 063505 共2010兲
Bai, Bai, and Wang
• 共c6兲 is isomorphic to g f , where the Lie algebra g is given by 关e2 , e3兴 = ␣e1 + e2, ␣ ⫽ 0, 关e2 , e5兴 = −e2, 关e1 , e3兴 = e2, 关e1 , e5兴 = −e1 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 5. • 共c7兲 is isomorphic to g f , where the Lie algebra g is given by 关e1 , e3兴 = e1, 关e2 , e3兴 = e2 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 5. Case dim A1 = 3 . • 共d1兲 is isomorphic to g f , where the Lie algebra g is given by 关e2 , e3兴 = e1, 关e2 , e5兴 = e2, 关e3 , e5兴 = −e3 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 5. • 共d2兲 is isomorphic to g f , where the Lie algebra g is given by 关e3 , e5兴 = −␣e2 − e3, ␣ ⫽ 0, 关e2 , e5兴 = −e3, 关e2 , e3兴 = e1, 关e1 , e5兴 = −e1 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 5. • 共d3兲 is isomorphic to g f , where the Lie algebra g is given by 关e2 , e3兴 = e1, 关e1 , e5兴 = −2e1, 关e3 , e5兴 = −e3, 关e2 , e5兴 = −e2 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 5. • 共d4兲 is isomorphic to g f , where the Lie algebra g is given by 关e2 , e3兴 = e1, 关e1 , e3兴 = e2, 关e1 , e2兴 = e3 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦4i for 1 ⱕ i ⱕ 5. • 共d5兲 is isomorphic to g f , where the Lie algebra g is given by 关e3 , e4兴 = e2 + 共1 + 兲e3 ,  ⫽ 0 , 1, 关e1 , e4兴 = e1, 关e2 , e4兴 = e3 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦5i for 1 ⱕ i ⱕ 5. • 共d6兲 is isomorphic to g f , where the Lie algebra g is given by 关e1 , e4兴 = e1, 关e2 , e4兴 = e2, 关e3 , e4兴 = e3 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦5i for 1 ⱕ i ⱕ 5. • 共d7兲 is isomorphic to g f , where the Lie algebra g is given by 关e3 , e4兴 = se1 + te2 + ue3 , s ⫽ 0, 䊐 关e1 , e4兴 = e2, 关e2 , e4兴 = e3 and the linear function f 苸 gⴱ satisfies f共ei兲 = ␦5i for 1 ⱕ i ⱕ 5. Proposition 4.7: Let A be a five-dimensional 3-Lie algebra. If dim A1 = 4, then A is not isomorphic to any 3-Lie algebra g f constructed from a five-dimensional Lie algebra g and a linear function f 苸 gⴱ through Theorem 3.1. On the other hand, 共e1兲 is obtained as a two-dimensional extension of the simple Lie algebra sl共2 , C兲 (with a metric) and 共e2兲 is the direct sum of the simple 3-Lie algebra (e) (in Lemma 4.2) and one-dimensional center, whereas the ideal (e) is constructed by ␥ -matrices. Proof: By Lemma 4.5, it is obvious that 关A1 , A1 , A1兴 ⫽ 0 when dim A1 = 4. Hence, A is not isomorphic to any 3-Lie algebra g f constructed from a five-dimensional Lie algebra g and a linear function f 苸 gⴱ through Theorem 3.1 due to Corollary 3.3. Let g1 = Cx1 丣 Cx2 丣 Cx3 be the simple Lie algebra sl共2 , C兲 satisfying 关x2,x3兴 = x1,
关x1,x2兴 = x3,
关x1,x3兴 = x2 ,
and B : g1 丢 g1 → C be a bilinear form on g1 defined by B共xi,x j兲 = 共− 1兲ij␦ij,
1 ⱕ i, j ⱕ 3.
It is obvious that B is a metric 共a scalar multiple of the Killing form兲. Then the 3-Lie algebra g0 = g1 丣 Cx0 丣 Cx−1 obtained through Eqs. 共1.6兲 and 共1.7兲 satisfies 3
k 关x1,x2,x3兴 = 兺 a12 B共xk,x3兲x−1 = x−1,
关x1,x2,x0兴 = 关x1,x2兴 = x3 ,
k=1
关x1,x3,x0兴 = 关x1,x3兴 = x2,
关x2,x3,x0兴 = 关x2,x3兴 = x1 .
Moreover, g0 is isomorphic to 共e1兲 by a linear transformation of basis, x 0 → e 5,
x 1 → e 2,
The conclusion involving 共e2兲 is obvious.
x 2 → e 3,
x 3 → e 4,
x−1 → e1 . 䊐
063505-11
J. Math. Phys. 51, 063505 共2010兲
Realizations of 3-Lie algebras
V. THE 3-LIE ALGEBRAS CONSTRUCTED FROM THE GENERAL LINEAR LIE ALGEBRAS WITH TRACE FORMS
Let g = gl共m , C兲 be the general linear Lie algebra and a linear function f 苸 gⴱ given by f共A兲 = tr共A兲 for any A 苸 g. Then there is the 3-Lie algebra gtr constructed by Eq. 共1.8兲. It is a special case of the study in Sec. III. Let Eij , 1 ⱕ i , j ⱕ m, be the 共m ⫻ m兲-matrix with 1 at the ith row and jth column and 0 otherwise. Then 兵Eij , 1 ⱕ i , j ⱕ m其 is a basis of gtr and Eij = 关Ekk,Eik,Ekj兴,
Eii − E jj = 关Ekk,Eij,E ji兴
for 1 ⱕ k ⱕ m.
共5.1兲
The derived algebra of gtr is gtr1 = 兵A兩A 苸 gtr,tr共A兲 = 0其
and
共as the direct sum of vector spaces兲,
gtr = gtr1 丣 CIm
共5.2兲 where Im is the 共m ⫻ m兲-identity matrix. Theorem 5.1: The 3-Lie algebra gtr is semisimple and local and the derived algebra gtr1 is the minimal ideal of gtr. Proof: Let J be a nonzero ideal of gtr. Then we have 关Im,J,gtr兴 = 关J,L兴 債 J, that is, J is a nonzero ideal of Lie algebra g. Therefore, J = 兵A 兩 A 苸 g , tr共A兲 = 0其 or J = CIm since g = gl共m , C兲 = sl共m , C兲 丣 CIm 共as the direct sum of ideals兲. Since the subalgebra CIm is not an ideal of gtr, we have J = 兵A兩A 苸 gtr,tr共A兲 = 0其 = gtr1 . Therefore, gtr is a local 3-Lie algebra. By Eq. 共1.8兲 again, we have J共1兲 = 关J,J,gtr兴 傶 关J,J,Im兴 = 关J,J兴 = J, and J共1兲 = J. Hence, J共i兲 = J for every i ⱖ 1. Therefore, J is not a solvable ideal of gtr and gtr is a 2-semisimple 3-Lie algebra. 䊐 Remark 5.2: To our knowledge, the 共n + 1兲-dimensional n-Lie algebras with dim A1 ⱖ 3 given by Filippov1 are the only known examples of finite-dimensional 2-semisimple n-Lie algebras. By Theorem 5.1, for any m, gtr provides a 2-semisimple 3-Lie algebra in dimension m2. At the end of this paper, we give a structure theory of gtr. Theorem 5.3: Let gtr be the 3-Lie algebra constructed from the general linear Lie algebra g = gl共m , C兲 with the trace form through Eq. (1.8). (1) (2)
gtr has a unique 共m2 − 1兲-dimensional hyponilpotent ideal J = gtr1, which is an Abelian subalgebra of gtr. m CEii is a Cartan subalgebra of gtr and h = 兺i=1 gtr = h+˙
兺
ij苸共h ∧ h兲ⴱ
共5.3兲
gij ,
where weight ij 苸 共h ∧ h兲ⴱ is antisymmetric and ij共Ekk,Ell兲 = ␦li − ␦lj + ␦kj − ␦ki
∀ 1 ⱕ k,l ⱕ m,
gij = 兵x 苸 gtr兩关h1,h2,x兴 = ij共h1,h2兲x, ∀ h1,h2 苸 h其 = CEij, (3)
1 ⱕ i ⫽ j ⱕ m.
共5.4兲 共5.5兲
Every invariant symmetric bilinear form  on the 3-Lie algebra gtr satisfies 共gtr , gtr1兲 = 0. So gtr is not a metric 3-Lie algebra.
063505-12
(4)
J. Math. Phys. 51, 063505 共2010兲
Bai, Bai, and Wang
Every non-Abelian 3-Lie algebra g f constructed from the Lie algebra g = gl共m , C兲 and any linear function f 苸 gⴱ through Theorem 3.1 is isomorphic to gtr.
Proof: By Eq. 共1.8兲 and Theorem 5.1, gtr1 is an Abelian subalgebra of gtr. Moreover, it is a maximal ideal of gtr since the codimension of gtr1 is 1. Therefore, conclusion 共1兲 holds. m CEii is an Abelian subalgebra. If 关X , h , h兴 債 h for some X 苸 gtr, then X By Eq. 共5.1兲, h = 兺i=1 苸 h, that is, h = N共h兲. Hence, h is a Cartan subalgebra of gtr. Again by a direct computation according to Eq. 共5.1兲, we obtain Eqs. 共5.3兲–共5.5兲. Suppose  is an invariant symmetric bilinear form on gtr. For any A 苸 gtr , D 苸 gtr1, by Theorem 5.1, there exist Bi , Ci 苸 gtr1 , i = 1 , 2 , . . . , k such that k
k
1 D = 兺 关Bi,Ci兴 = 兺 关Im,Bi,Ci兴. m j=1 j=1 Note that A = 共A − tr共A兲Im兲 + tr共A兲Im. By the invariant property of  and conclusion 共1兲, we have
冉
k
1 共A,D兲 =  共A − tr共A兲Im兲 + tr共A兲Im, 兺 关Im,Bi,Ci兴 m j=1 k
冊
k
1 tr共A兲 = 兺 共关A − tr共A兲Im,Bi,Ci兴,Im兲 + 兺 共关Im,Im,Bi兴,Ci兲 = 0. m j=1 m j=1 Therefore, conclusion 共3兲 holds. Let us prove conclusion 共4兲. Since g f is non-Abelian, we have f共Im兲 ⫽ 0, and f共A兲 = 0 if tr共A兲 = 0. Set a linear map : g f → gtr by
共Im兲 =
f共Im兲 I m, m
共Eij兲 = Eij
∀ i ⫽ j.
䊐 It is obvious that is an isomorphism of 3-Lie algebras. Remark 5.4: Note that when m = 2, gtr is isomorphic to the 3-Lie algebra 共d兲 in Lemma 4.2. ACKNOWLEDGMENTS
The first author 共R.B.兲 was supported in part by the National Natural Science Foundation of China 共Grant No. 10871192兲 and the Natural Science Foundation of Hebei Province 共Grant No. A2010000194兲. The second author 共C.B.兲 was supported in part by NSFC 共Grant No. 10621101 and 10921061兲, NKBRPC 共Grant No. 2006CB805905兲, and SRFDP 共Grant No. 200800550015兲. V. T. Filippov, Sib. Mat. Zh. 26, 126 共1985兲. Y. Nambu, Phys. Rev. D 7, 2405 共1973兲. 3 L. Takhtajan, Commun. Math. Phys. 160, 295 共1994兲. 4 J. Bagger and N. Lambert, Phys. Rev. D 77, 065008 共2008兲. 5 P. Ho, R. Hou, and Y. Matsuo, J. High Energy Phys. 0806, 020 共2008兲; e-print arXiv:0804.2110. 6 P. Ho, M. Chebotar, and W. Ke, Proc. - R. Soc. Edinburgh, Sect. A: Math. 113, 1273 共2003兲. 7 J.M. Figueroa-O’Farrill, J. Math. Phys. 49, 113509 共2008兲. 8 J. M. Figueroa-O’Farrill and G. Papadopoulos, J. Geom. Phys. 49, 294 共2004兲. 9 A. Gustavsson, Nucl. Phys. B 811, 66 共2009兲; e-print arXiv:0709.1260. 10 G. Papadopoulos, J. High Energy Phys. 0805, 054 共2008兲; e-print arXiv:0804.2662. 11 P. Medeiros, J. M. Figueroa-O’Farrill, and E. Méndez-Escobar, J. High Energy Phys. 0808, 045 共2008兲. 12 P. Medeiros, J. M. Figueroa-O’Farrill, and E. Méndez-Escobar, J. High Energy Phys. 0807, 111 共2008兲. 13 S. Kasymov, Algebra i Logika 26, 277 共1987兲. 14 W. Ling, “On the structure of n -Lie algebras,” Ph.D. thesis, University-GHS-Siegen, 1993. 15 R. Bai and D. Meng, Commun. Algebra 31, 5331 共2003兲. 16 R. Bai, C. Shen, and Y. Zhang, Linear Algebr. Appl. 431, 673 共2009兲. 17 R. Bai and G. Song, J. Phys. A: Math. Theor. 42, 035207 共2009兲. 18 P. Ho, Y. Imamura, and Y. Matsuo, J. High Energy Phys. 0807, 003 共2008兲; e-print arXiv:0805.1202. 19 H. Awata, M. Li, D. Minic, and T. Yoneya, J. High Energy Phys. 0102, 013 共2001兲. 20 I. Gelfand and I. Dorfman, Funct. Anal. Appl. 13, 13 共1979兲. 1 2
