1. Lie algebras, Killing forms and signatures
Vladimir Cuesta †
Instituto de Ciencias Nucleares, Universidad Nacional Aut´ noma de M´ xico, 70-543, Ciudad de
o e
M´ xico, M´ xico
e e
Abstract. The killing form is a basic ingredient in abstract Lie algebra theory. If I make a linear
transformation in the set of generators of a Lie algebra, I can show that the Killing form is a contravariant
tensor, using Sylvester’s law of inertia I can diagonalize it and with this I can ﬁnd all the Lie algebras
of a given dimension according to the signature or equivalently the number of negative, zero or positive
eigenvalues.
I show that there is one and only one Lie algebra with two generators, using the same method I ﬁnd
that there are ﬁve possible Lie algebras with three generators, I show three of them.
With this method I can ﬁnd all the Lie algebras that are isomorphic of a given dimension because
the signature is independent of the base, I present some important examples showing isomorphic Lie
algebras.
1. Introduction
Lie algebras is a branch of abstract mathematics with intense and constant development, in the
literature we can ﬁnd a lot of great and concise books like [1], [2] and so on. We can study different
applications (see [3] and [4] for example), even more, we can study the subject as a inﬁnitesimal
version of a Lie group (see [5] for instance).
Like the reader can note almost all the books are interested in a speciﬁc kind of Lie algebras:
semisimple Lie algebras. However, according to the Levi decomposition, we take an abstract real
Lie algebra with a ﬁnite number of generators and it can be divided in a solvable subalgebra and a
semisimple subalgebra and this is a ﬁrst example to make research of the subject, I mean, if I take a
Lie algebra I could be interested in the Levi decomposition for different applications.
In the case of pure Lie algebras one of the problems that the researcher can ﬁnd in this area of
knowledge is to determine and to classify all the semisimple subalgebras of a speciﬁc Lie algebra (see
[6] and [7] for instance), its matricial representations and so on.
As a third example, if I take a Lie algebra I can study it as a subalgebra of another algebra
with more generators (see [1]), which is the opposite of a decomposition problem. Previously I have
†
vladimir.cuesta@nucleares.unam.mx
2. Lie algebras, Killing forms and signatures 2
presented a brief list of a longer one, in the present paper I will follow the following line of reasoning:
I present basic deﬁnitions on Lie algebras, I show that the Killing form is a contravariant tensor and
like a linear algebra or gravitation specialist knows the signature is an invariant of a bilinear form
(like the Killing metric) and so that, if I ﬁnd the signature of a diagonal Killing form I will obtain an
invariant of the theory. Even more, with this it is possible to characterize isomorphic Lie algebras.
That is the purpose of the present paper.
2. Lie algebras: deﬁnitions and basics
2.1. Lie algebras
I present the formal deﬁnition of a Lie algebra, along all the present work I will use the following
deﬁnition (see [2] and [5] for instance):
Let L be a vector space over the real ﬁeld , L is a real Lie algebra when there exists a binary
operation denoted as [ , ] and is called commutator or Lie bracket
[ , ]:L×L→L (1)
and when the commutator obeys the following three properties
• Antisymmetry.- For all x, y ∈ L
[x, y] = −[y, x], (2)
• Bilinearity.- For all x, y ∈ L and all α, β ∈
[αx + βy, z] = α[x, z] + β[y, z], (3)
• Jacobi identity.- For all x, y, z ∈ L
[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0, (4)
Now, Let L1 and L2 be two real Lie algebras, then L1 and L2 are isomorphic if there exist a
bijection
φ : L1 → L2 , (5)
such that
φ(αx + βy) = αφ(x) + βφ(y), φ([x, y]) = [φ(x), φ(y)], (6)
where x, y ∈ L1 and α, β ∈ .
3. Lie algebras, Killing forms and signatures 3
2.2. The Killing form as a tensor
We have the deﬁnition
[ei , ej ] = Cij k ek , (7)
for the structure constants of a general Lie algebra (see [2] and [3], for instance). With the previous
equations we can deﬁne the set of matrices Ti (adjoint representation for the ei generator) as follows:
k
(Ti )j = −Cij k , (8)
where i runs from 1 to the dimension of the Lie algebra.
Now, taking the linear transformation
ei = Ai m em , (9)
I will ﬁnd the transformation law for the adjoint representations matrices. In fact, these must obey
[ei , ej ] = Cij k ek , (10)
replacing the transformation law for the Lie algebra generators in the previous equation I ﬁnd
Ai m Aj n Cmn r − Cij k Ak r er = 0, (11)
and the result is
Tij s = Ai m An A−1
j
s
Tmn r , (12)
r
the following step is to replace Tij k in
gij = Tim n Tjn m , (13)
and the result is
gij = Ai m Aj n gmn , (14)
I mean, g is a tensor. In fact,
gij = T r (Ti · Tj ) = T r (Tj · Ti ) = gji , (15)
and g is a Symmetric bilinear form (see [8] for details), in the case when the Killing form is non-
degenerate we call this kind of Lie algebras as semisimple.
Now, Sylvester’s law of inertia says that when a symmetric bilinear form is written in diagonal
form the number of negative, zero and positive eigenvalues (signature) is independent of the original
set of generators (see [9] for a detailed discussion).
Let A be a Lie algebra of dimension n, then if I have the set of structure constants, I can obtain
the adjoint representation, the Killing form and I can ﬁnd the set of eigenvalues and in this way I can
identify and classify a speciﬁc Lie algebra according to the signature of the diagonal Killing form.
I mean, all the Lie algebras with the same number of negative, zero and positive eigenvalues in the
diagonal Killing form are isomorphic.
4. Lie algebras, Killing forms and signatures 4
3. Examples
3.1. Lie algebras with two generators
I present the bi-dimensional Lie algebra, the general form is
[e1 , e2 ] = αe1 + βe2 , (16)
with the adjoint representations
0 0 −α −β
T1 = , T2 = ,
α β 0 0
with Killing metric
β 2 −αβ
(gij ) = , (17)
−αβ α2
and the eigenvalues that I found are λ1 = 0 and λ2 = α2 + β 2 . Then, this is the unique Lie algebra
of dimension 2 and like the reader can see, in a base where the Killing form is diagonal the number
of negative and zero eigenvalues is zero and the number of positive eigenvalues is one and the Lie
algebra is non-semisimple.
3.2. Lie algebras with three generators
In the present subsection I present three of the ﬁve possible different Lie algebras with three
generators.
Case I
The generators for this ﬁrst case are (where I take the commutator between matrices)
0 0 0 0 0 1 −1 −2 0
0 0 −1 , (e2ij ) = 0
(e1ij ) = , (e3ij ) = 1
0 0 1 0 ,
1 2 0 −1 −1 0 0 0 0
the commutation relations between the three generators are,
[e1 , e2 ] = e3 , [e1 , e3 ] = −e1 − 2e2 , [e2 , e3 ] = e1 + e2 , (18)
using the Killing form deﬁnition, I ﬁnd the result
−4 2 0
2 −2 0 ,
(gij ) =
0 0 −2
5. Lie algebras, Killing forms and signatures 5
√ √
and I ﬁnd the eigenvalues λ1 = −3 − 5, λ2 = −2 and λ3 = −3 + 5. I mean, I have a Lie algebra
with three negative eigenvalues and it is semisimple (the Lie algebra of this example is equivalent to
a Lie algebra with three positive eigenvalues because there is a difference of global sign).
Case II
I will study the Lie algebra with commutators
[e1 , e2 ] = 2e2 , [e1 , e3 ] = 4e2 − 2e3 , [e2 , e3 ] = e1 , (19)
using this set I ﬁnd the adjoint representations
0 0 0 0 2 0 0 4 −2
(e1ij ) = 0 −2 0 , (e2ij ) = 0 0 0 , (e3ij ) = 1 0 0 ,
0 −4 2 −1 0 0 0 0 0
after a straightforward calculation I ﬁnd the killing form
8 0 0
(gij ) =
0 0 4 ,
0 4 8
√ √
and it has the set of eigenvalues λ1 = 4(1 + 2), λ2 = 8 and λ3 = 4(1 − 2), the number of negative
eigenvalues is one and positive eigenvalues are two, the present Lie algebra is semisimple.
Case III
The set of matrices that I will use are the following, where the Lie bracket is the commutator between
matrices
0 0 0 1 0 0 −1 0 0
(e1ij ) = −1 0 0 , (e2ij ) = 0 0 0 , (e3ij ) = −1 0 0 ,
1 0 0 1 0 0 0 0 0
The set of commutators for the three generators is
[e1 , e2 ] = e1 , [e1 , e3 ] = −e1 , [e2 , e3 ] = −e1 , (20)
and the ﬁnal result for the Killing form is
0 0 0
0 1 −1 ,
(gij ) =
0 −1 1
and If I diagonalize the previous tensor I ﬁnd the eigenvalues λ1 = 2, λ2 = 0 and λ3 = 0, in this case
the number of positive eigenvalues is one and two null eigenvalues, the Lie algebra is non-semisimple.
6. Lie algebras, Killing forms and signatures 6
4. Isomorphic Lie algebras
First example: su(2)
In this case the three matrices are
0 0 0 0 0 1 0 −1 0
(Sxij ) = 0 0 −1 , (Syij ) = 0 0 0 , (Szij ) = 1 0 0 ,
0 1 0 −1 0 0 0 0 0
and the commutator is the commutator between matrices, the commutation relations between the three
generators are (see [10] for instance),
[Sx , Sy ] = Sz , [Sy , Sz ] = Sx , [Sz , Sx ] = Sy , (21)
using the Killing form deﬁnition, I ﬁnd the result
−2 0 0
(gij ) = 0 −2 0 ,
0 0 −2
and is a diagonal Killing form with eigenvalues λ1 = −2, λ2 = −2 and λ3 = −2. In this case all
the eigenvalues are negative and it means that this Lie algebra is isomorphic to the ﬁrst Lie algebra
of the previous subsection with three positive eigenvalues because the difference between this pair of
diagonal Killing forms is a global sign.
Second example: sl(2, )
I will study the Lie algebra with generators
0 0 0 0 0 1 −2 0 0
0 0 −1 , (fij ) = 0 0 0 , (hij ) = 0 2 0 ,
(eij ) =
2 0 0 0 −2 0 0 0 0
in this case the set of commutators is (see [11] and [4] for instance)
[h, e] = 2e, [h, f ] = −2f, [e, f ] = h, (22)
and after a straightforward calculation I ﬁnd the killing form
0 4 0
(gij ) =
4 0 0 ,
0 0 8
7. Lie algebras, Killing forms and signatures 7
and the set of eigenvalues for the Killing form is λ1 = 8, λ2 = −4 and λ3 = 4, the number of negative
eigenvalues is one and the number of positive eigenvalues is two and the Lie algebra is isomorphic to
the second Lie algebra of the previous subsection.
Third example: Antisymmetric 3 × 3 matrices
I will study the Lie algebra with generators
0 0 0 0 0 −1 0 1 0
0 0 1 , (e2ij ) = 0 0 0 , (e3ij ) = −1 0 0 ,
(e1ij ) =
0 −1 0 1 0 0 0 0 0
in this case the set of commutators is
[e1 , e2 ] = −e3 , [e1 , e3 ] = e2 , [e2 , e3 ] = −e1 , (23)
and after a straightforward calculation I ﬁnd the killing form
−2 0 0
(gij ) = 0 −2 0 ,
0 0 −2
and the set of eigenvalues for the Killing form is λ1 = −2, λ2 = −2 and λ3 = −2, the number of
negative eigenvalues is three and the Lie algebra is isomorphic to the ﬁrst Lie algebra of the previous
subsection (the difference is a global sign).
5. Conclusions and perspectives
In the present paper I have shown a general method for ﬁnding isomorphic Lie algebras (see [4], too),
all the Lie algebras with the same number of negative, null and positive eigenvalues for the Killing
form are isomorphic (see [12] for a precise discussion on signatures). I have shown that there is
one and only one Lie algebra with two generators and there are ﬁve possible Lie algebras with three
generators, altough I show only three of all. Later, I show two examples to illustrate the method to
recognize isomorphic Lie algebras in the case of three generators, I present the following table where
I resume my results
N egative Zero P ositive Lie algebra
eigenvalues eigenvalues eigenvalues type
Case I 0 0 3 semisimple
Case II 1 0 2 semisimple
Case III 0 1 2 non − semisimple
Case IV 1 1 1 non − semisimple
Case V 0 2 1 non − semisimple
8. Lie algebras, Killing forms and signatures 8
To ﬁnish the paper I show the following table, where I ﬁnd all the possible signatures for the diagonal
Killing form of a Lie algebra with four generators. In fact, I found nine Lie algebras and like I said, If
I could diagonalize the Killing form I can identify isomorphic Lie algebras If the number of negative,
null and positive eigenvalues for the diagonal Killing form are the same,
N egative Zero P ositive Lie algebra
eigenvalues eigenvalues eigenvalues type
Case I 0 0 4 semisimple
Case II 1 0 3 semisimple
Case III 0 1 3 non − semisimple
Case IV 0 2 2 non − semisimple
Case V 2 0 2 semisimple
Case VI 1 1 2 non − semisimple
Case V II 0 3 1 non − semisimple
Case V III 1 2 1 non − semisimple
Case IX 0 4 0 non − semisimple
for future work, I can identify isomorphic Lie algebras with four generators or another option is to
ﬁnd a classiﬁcation when the generators are four, ﬁve and so on.
References
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[3] Francesco Iachello, Lie Algebras and Applications, Springer, (2006),
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Press, (1995),
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