ap-5-10.dvi


Acta Polytechnica Vol. 50 No. 5/2010

On Representations of sl(n, C) Compatible with a Z2-grading

M. Havlíček, E. Pelantová, J. Tolar

Abstract

This paper extends existing Lie algebra representation theory related to Lie algebra gradings. The notion of a repre-
sentation compatible with a given grading is applied to finite-dimensional representations of sl(n, C) in relation to its
Z2-gradings. For representation theory of sl(n, C) the Gel’fand-Tseitlin method turned out very efficient. We show that
it is not generally true that every irreducible representation can be compatibly graded.

1 Introduction

Contractions of Lie algebras, of interest in connecting physical theories, are traditionally understood as limit
procedures throughwhichLie algebras aremodified into different, non-isomorphic Lie algebras [5, 8]. Neverthe-
less, formanyphysical applications, especially in quantumtheory, representations of Lie algebras are important.
It should be noted that contractions usually produce non-compact Lie algebras whose unitary representations
are infinite-dimensional.
Remaining inside the framework of Lie algebras, a completely different notion of graded contractions was

proposed in [11]. In a seminal paper [13], R. V. Moody and J. Patera pushed the theory of graded contrac-
tions of Lie algebras further with graded contractions of representations of Lie algebras. By considering along
with graded Lie algebras their compatibly graded finite-dimensional representations, they obtained a theory of
contractions of representations that contains the Lie algebra contractions as a special case for adjoint repre-
sentation. This is unfortunately the only existing mathematical theory of this matter, and moreover it is not
concerned with the question of which representations can be compatibly graded. Namely, compatibly graded
finite-dimensional representations were assumed throughout the paper [13], Eqs. (2.10) and (2.11) which is a
valid assumption if the grading is induced by an inner automorphism. In this respect they also provided a
recipe for finding the corresponding grading of vector space V on which the representation is acting (5). One
should also mention a short note [15] on the subject, but up to now nobody has gone ahead with a further
study of representations of Lie algebras related to their gradings, especially when the gradings are induced by
outer automorphisms. We are aware that finite-dimensional representations of contracted Lie algebras can only
be non-unitary. However, such representations, usually indecomposable, also have some interest in physics.
Our paper, as a starting point for such an investigation, gives answers under the restrictive assumptions

used in [13]. Thus we restrict our consideration to

1. complex Lie algebras of type A,

2. finite-dimensional representations,

3. group gradings with the grading group Z2.

Z2-gradings are closely related to involutive (second order) automorphisms of Lie algebras. In physical
applications they are especially useful as generalized parity transformations. In this connection our earlier
paper [12] dealt with the well-known space-time parity transformations — space inversion and time reversal—
and the associated graded contractions for the de Sitter Lie algebras (type B). Here we start with the simplest
case of finite-dimensional representations of classical Lie algebras of type A.
Thepaper is organizedas follows. Section 2 is devoted to representationscompatiblewith agrading. Explicit

results are obtained in Section 3 for finite-dimensional representations of sl(n, C) compatible with Z2-gradings
generated either by an inner automorphism of order 2 or by an outer automorphism of order 2. Our concrete
results are illustrated on the simple Lie algebra sl(3, C).

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Acta Polytechnica Vol. 50 No. 5/2010

2 Representations compatible with grading

2.1 Graded contractions of Lie algebras

A grading of a Lie algebra L is a decomposition Γ of the vector space L into vector subspaces Lj, j ∈ J , such
that L is a direct sum of these subspaces Lj, and, for any pair of indices j, k ∈ J , there exists l ∈ J such that
[Lj, Lk] ⊆ Ll. We denote the grading by

Γ:L =
⊕
j∈J

Lj;

(let us note that in our definition of grading we do not exclude trivial subspaces Li = {0}). It follows di-
rectly from the definition that for any grading Γ:

⊕
j∈J

Lj and any automorphism g ∈ Aut L the decomposition

Γ′:
⊕
j∈J

g(Lj) is also a grading. Gradings Γ and Γ
′ are called equivalent.

Now we describe a specific type of grading, namely a group grading. A grading Γ:L = ⊕j∈J Lj is called
a group grading if the index set J can be embedded into a semigroup G (whose binary operation is denoted
by +), and, for any pair of indices j, k ∈ J , it holds that

[Lj, Lk] ⊆ Lj+k. (1)

Since even trivial subspaces are generally allowed in the decomposition of L, the semigroup G may be used as
the index set of the group grading. In this case we will speak about a G-grading Γ. We will focus in this paper
on group gradings only andwe assume in the sequel that the indices of the grading subspaces belong to a group
G, i.e. Γ is a G-grading of L.
A grading Γ:L = ⊕i∈J Li of a Lie algebra L is a starting point for the study of graded contractions of the

Lie algebra. This method for finding contractions of Lie algebras was introduced in [11, 13]. In this type of
contraction, we define new Lie brackets by the prescription

[x, y]new := εj,k[x, y], where x ∈ Lj, y ∈ Lk. (2)

The complex or real parameters εj,k for j, k ∈ G must be determined in such away that the vector space L with
the binary operation [., .]new again forms a Lie algebra. Antisymmetry of Lie brackets demands that εj,k = εk,j.
If, moreover, the coefficients εj,k fulfill the first basic set of contraction equations [14]:

εi,j εi+j,k = εj,kεj+k,i = εk,iεk+i,j for all i, j, k ∈ G, (3)

then the vector space L with new brackets [x, y]new satisfies the Jacobi identities as well. This new Lie algebra
will be denotedby Lε. Note that the equations (3) involve only relevantparameters forwhich the corresponding
commutators [Lj, Lk] do not vanish.

Example 1 Z2-grading. The most notorious case of group grading is Z2-grading.
1 Here a Lie algebra L over

C is decomposed into two non-zero grading subspaces L0 and L1, where

0 
= [L0, L0] ⊆ L0, 0 
= [L0, L1] ⊆ L1, 0 
= [L1, L1] ⊆ L0. (4)

Here we have applied the generic condition that in each class of commutators there exists at least one non-
vanishing commutator. For a Z2-grading of a Lie algebra L, the generic system of equations (3) has a very
simple form

(ε00 − ε01)ε01 =0= (ε00 − ε01)ε11, ε10 = ε01.
There exist infinitely many solutions ε = (εjk) of this system. However for many solutions, the contracted
algebras Lε are isomorphic. It can be shown that only four solutions

(εjk)=

(
1 1

1 0

)
,

(
1 0

0 0

)
,

(
0 0

0 1

)
, and

(
0 0

0 0

)
give mutually non-isomorphic Lie algebras Lε over C. (The original Lie algebra is obtainedwith all parameters
εij =1.) The contracted algebra obtained by the first solution is the semidirect sum of L0 with a commutative
algebra L1 and corresponds to the Inönü-Wigner contraction. The second solution is the direct sum of L0 and
the commutative algebra L1. The third solution corresponds to the central extension of L1 (considered as a
commutative algebra) by the commutative algebra L0. The fourth solution is an Abelian Lie algebra.

1Note that special Z2-graded contractions are closely related to Inönü–Wigner contractions [12].

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Acta Polytechnica Vol. 50 No. 5/2010

2.2 Representations of graded contractions

Let us focus on the question of a representation of the contracted Lie algebra Lε. We will reformulate the
method proposed in [13], which enables us to find a representation of Lε by modifying a given representation of
the original algebra L. It involves a simultaneous grading of the Lie algebra L and the representation space V .

Definition 2.1 Let r:L �→ EndV be a representation of Lie algebra L and let Γ:L = ⊕i∈GLi be its G-grading.
We say that the representation r is compatible with the G-grading, if there exists a decomposition of the vector
space V into a direct sum V = ⊕i∈GVi such that

r(Xi)Vj ⊂ Vi+j for each i, j ∈ G and any Xi ∈ Li . (5)

Remark 2.2 Let r be a representation of L compatible with the grading L = ⊕i∈GLi and h ∈ Aut L be any
automorphismof L. Then r◦h−1 is a representationof L compatiblewith the equivalentgrading L = ⊕i∈Gh(Li),
since (r ◦ h−1)h(Xi)Vj = r(Xi)Vj ⊂ Vi+j.

Suppose we are given a representation r of L compatible with the G-grading. We are looking for a repre-
sentation rε of a contracted Lie algebra Lε. According to [13] we define

rε(Xi)vj := ψi,j r(Xi)vj (for each i, j ∈ G , any Xi ∈ Li and any vj ∈ Vj), (6)

where ψi,j are unknown parameters. The requirement that r
ε is a representation of Lε formally means

rε
(
[Xi, Xj]new

)
vk = [r

ε(Xi), r
ε(Xj)]vk =

(
rε(Xi)r

ε(Xj) − rε(Xj)rε(Xi)
)
vk

for any Xi ∈ Li, Xj ∈ Lj, and vk ∈ Vk. Using equations (2) and (6) and relation (5) we obtain

ψj,kψi,j+kr(Xi)r(Xj) − ψi,kψj,i+kr(Xj)r(Xi)= εi,j ψi+j,kr([Xi, Xj])

Since r is a representation of L, we know that r(Xi)r(Xj)− r(Xj)r(Xi)= r([Xi, Xj]). Therefore, the choice of
parameters ψi,j satisfying the second basic set of contraction equations [14]

ψj,kψi,j+k = ψi,kψj,i+k = εi,j ψi+j,k (7)

implies that rε defined by (6) is a representation of the contracted Lie algebra Lε. Solutions of (7) determine
the contractions of the chosen representations. Let us stress that, if r([Xi, Xj] = 0 for all Xi ∈ Li, Xj ∈ Lj,
conditions (7) are not necessary.
Comparing (7) and (3) we see that the system of quadratic equations for parameters ψi,j has at least

one solution, namely ψi,j = εi,j for each pair i, j (adjoint representation of L
ε). Therefore the mapping

rε:Lε �→ EndV defined by (6) is a representation of the graded Lie algebra Lε. Usually, there also exist other
solutions of the system (7), and therefore more representations of the same contracted algebra Lε.

Example 2 Z2-graded representation. Consider a Z2-grading of a Lie algebra L and its representation r
which is compatible with the grading. For the corresponding decomposition of the vector space V = V0 ⊕ V1 we
may construct a basis B of V composed of the basis of V0 and the basis of V1. In such a basis B, the grading
relations (4) acquire the block form explicitly

r(X0)=

(
A(X0) 0

0 B(X0)

)
and r(X1)=

(
0 C(X1)

D(X1) 0

)
.

In the sequel, we will illustrate all notions on the Lie algebra Lε obtained by contraction from a Z2-grading
of a Lie algebra L by the first solution

(εjk)=

(
1 1

1 0

)
given in Example 1. For this Lie algebra Lε the commutation relations have the form

[x, y]new = [x, y], if x, y ∈ L0 or if x ∈ L0, y ∈ L1 and [x, y]new =0, if x, y ∈ L1.
In this case the system of equations (7) is

ψ00ψ00 = ψ00, ψ10ψ01 = ψ00ψ10 = ψ10 ,

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Acta Polytechnica Vol. 50 No. 5/2010

ψ01ψ01 = ψ01, ψ11ψ00 = ψ01ψ11 = ψ11 ,

ψ10ψ11 =0 .

All solutions (up to equivalence of representations) of this system are

(ψjk)=

(
1 1

1 0

)
,

(
1 1

0 1

)
,

(
1 1

0 0

)
,

(
1 0

0 0

)
,

(
0 1

0 0

)
and

(
0 0

0 0

)

The representations rε of the contracted Lie algebra Lε in the chosen basis B of the vector space V have the
block form

rε(X0)=

(
ψ00A(X0) 0

0 ψ01B(X0)

)
and rε(X1)=

(
0 ψ11C(X1)

ψ10D(X1) 0

)
,

where for parameters (ψij) one may choose one of the six solutions. Let us mention that only the first two
solutions are interesting since the elements of subalgebra L1 are represented by zero operators in the remaining
solutions.

2.3 Group gradings and automorphisms

The simplestway to find a groupgradingof a Lie algebra is to decompose the vector space L into eigensubspaces
of a diagonalizable automorphism g ∈ Aut L [6]. For any pair of its eigenvectors xλ and xμ corresponding to
eigenvalues λ and μ, respectively, we have

g([xλ, xμ])= [g(xλ), g(xμ)]= λμ[xλ, xμ] .

Thus the commutator [xλ, xμ] is either zero or an eigenvector corresponding to the eigenvalue λμ. Let us
denote by σ(g) the spectrum of automorphism g and by Lλ the eigensubspace corresponding to λ ∈ σ(g). The
decomposition

Γ:L =
⊕

λ∈σ(g)
Lλ (8)

is a group grading, where the multiplicative semigroup generated by the spectrum of g can be taken as a
semigroup G.

Remark 2.3 If h ∈ Aut L, then the decomposition of L into eigensubspaces of the automorphism hgh−1 is
L =

⊕
λ∈σ(g)

h(Lλ), i.e. the gradings given by conjugated automorphisms g and hgh
−1 are equivalent. There-

fore, the automorphisms g and hgh−1 are called equivalent as well. Note however that different inequivalent
automorphisms may even give the same grading.

Similarly, if g1, g2, . . . , gr are mutually commuting automorphisms of L, then the decomposition of L into
commoneigensubspaces of all these automorphisms is a groupgradingof L. The semigroup suitable for indexing
this grading is G1×G2×. . .×Gr, where each Gi is the semigroupgeneratedby the spectrumof the automorphism
gi.
Furthermore, for Lie algebrasover the complexfield C, any groupgrading canbe obtainedby this procedure.

Let us emphasize that this is not the case for real Lie algebras. In the following, in order to study the
compatibility problem, we shall consider the simplest case of group grading determined by one automorphism.

2.4 Group grading determined by one automorphism

Let Γ be a grading of the form (8), i.e. obtained by decomposition of L into eigensubspaces of a single
automorphism g. We may assume that g has a finite order, say gk = Id. For its spectrum we have

σ(g) ⊂ {ei
2π
k

	 | � =0,1,2, . . . , k − 1} =: G. (9)

This means that Γ is a G-grading.
Let us consider an irreducible d-dimensional representation r of the Lie algebra L. Our aim is to discuss the

question of compatibility of r with G-grading. Let Rg be a non-singular matrix in C
d×d such that

r(g(x)) = Rgr(x)R
−1
g for all x ∈ L . (10)

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Acta Polytechnica Vol. 50 No. 5/2010

As gk = Id, the previous equality gives

r(x) = r(gk(x))= Rkg r(x)R
−k
g or [R

k
g , r(x)] = 0 for all x ∈ L .

Since the representation r is irreducible, by Schur’s lemma Rkg = α Id for some α ∈ C. Of course, any nonzero
multiple of Rg also satisfies the relation (10). Therefore without loss of generality, we may assume that

Rkg = Id , where k is the order of automorphism g. (11)

This normalization guarantees that the spectrum of matrix Rg and the spectrum of automorphism g belong to
the same group G. In particular, since Rkg is the identity, matrix Rg is diagonalizable. Let V = ⊕λ∈GVλ denote
the decomposition of column space Cd into eigensubspaces of matrix Rg, i.e. Rgvλ = λvλ for all vλ ∈ Vλ. We
will show that this decomposition is exactly the decomposition required in Definition 2.1.
Let us consider some μ ∈ σ(g) so that g(xμ) = μxμ for all xμ ∈ Lμ. Relation (10) for x = xμ leads to the

matrix relation
r(g(xμ))Rg = r(μxμ)Rg = μ r(xμ)Rg = Rgr(xμ)

which acts on a column vector vλ ∈ Vλ as

μ λ r(xμ)vλ = Rgr(xμ)vλ .

The last equalitymeans that the column r(xμ)vλ is either zero or it is an eigenvector ofmatrix Rg corresponding
to eigenvalue μ λ. Therefore

r(xμ)Vλ ⊂ Vμλ for any λ, μ ∈ G and any xμ ∈ Lμ .

This is relation (5) written in the multiplicative form. Of course, our multiplicative group G defined in (9) is
isomorphic to the additive group Zk.
We have seen that matrix Rg with the properties (10) and (11) guarantees the compatibility of grading

of L with the representation of the Lie algebra L. Such matrix Rg will be called the simulation matrix of
automorphism g. Matrix Rg depends on the chosen automorphism g and on the chosen representation r. The
idea for finding the simulation matrix is more straightforward if g ∈ Aut L is an inner automorphism. In this
case it is natural to search for Rg among matrices in the representation of the corresponding Lie group. This
idea was already presented in [11] and [15], where Rg was a representation of an element of finite order [9].
Nevertheless, we show that it is also possible to find the simulation matrix Rg even for an outer automorphism
g. In the sequel, we will concentrate on the Lie algebras sl(n, C). The reason is that these algebras (with the
exception of o(8, C)) are the only simple classical Lie algebras over C for which the group of automorphisms
contains an outer automorphism as well [7].

3 Representations of sl(n, C) compatible with Z2-grading

We will identify the Lie algebra sl(n, C) with {X ∈ Cn×n | trX =0}. Any Z2-grading of it is uniquely related
to an automorphism of order 2. Let us therefore recall the structure of Aut sl(n, C) as described in [7]:

1. for any inner automorphism g there exists a matrix A ∈ SL(n, C) := {A ∈ Cn×n | detA =1} such that

g(X)= AdAX = AXA
−1 for any X ∈ sl(n, C);

2. the mapping given by the prescription

OutI X := −X T for any X ∈ sl(n, C)

is an outer automorphism of order 2;

3. any outer automorphism g is a composition of an inner automorphism and the automorphism OutI.

4. to any outer automorphism OutA of order two there exists an inner automorphism AdP such that
Ad−1P OutAAdP = OutP T AP = OutI, i.e. OutA and OutI are equivalent (see [6], Lemma A.1).

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Acta Polytechnica Vol. 50 No. 5/2010

The next ingredient for the construction of simulationmatrices of automorphisms is the knowledge of finite-
dimensional irreducible representationsof sl(n, C). These representationsarewell describedbyGel’fand-Tseitlin
formalism [4, 10, 2]. Any irreducible representation r of sl(n, C) is in one-to-one correspondencewith an n-tuple
(m1,n, m2,n, . . . , mn,n) of non-negative integer parameters m1,n ≥ m2,n ≥ . . . ≥ mn,n = 0. The dimension of
the representation space of r = r(m1,n, m2,n, . . . , mn,n) is given by the number of triangular patterns

m=

⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

m1,n m2,n m3,n . . . mn,n

m1,n−1 m2,n−1 m3,n−1 . . . mn−1,n−1

m1,n−2 m2,n−2 . . . mn−2,n−2
...

...
...

m1,2 m2,2

m1,1

⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

in which the numbers mi,j ∈ Z satisfy mi,j+1 ≥ mi,j ≥ mi+1,j+1 for all 1 ≤ i ≤ j ≤ n − 1. To any such pattern
m, we assign the basis vector ξ(m). The representation r is fully determined by the action r(Ek	) on all basis
vectors ξ(m) for any k, � = 1,2, . . . , n. (We have adopted the notation Ek	 for n × n matrices with elements(
Ek	
)
ij
= δikδ	j.) This action can be found e.g. in [4], but for the reader’s convenience the representation of

gl(n, C) is described in the Appendix.

3.1 Inner automorphisms of order two

Any innerautomorphism g oforder two is associatedby the equality g = AdA withagroupelement A ∈ SL(n, C)
such that A does not belong to the center Z[SL(n, C)] and A2 belongs to the center. If we denote ω = e

iπ
n ,

then the center can bewritten explicitly Z[SL(n, C)] = {ω2	In | � =0,1, . . . , n −1}. A simple calculation shows
that any such element A ∈ SL(n, C) is up to equivalence one of the matrices

An,s := ω
η(s)

(
In−s 0

0 −Is

)
where s =1, . . . , �

n

2
� and η(s)=

{
0 if s is even

1 if s is odd

(Note that An,0 = In belongs to Z[SL(n, C)].) These matrices may be rewritten by using elements of the Lie
algebra sl(n, C) as follows:

An,s =exp(Xn,s) with Xn,s = iπ

⎛
⎜⎝

η(s)
n

In−s 0

0
η(s)
n

Is + Ms

⎞
⎟⎠ ,

where Ms = diag(−1,1, −1, . . . ,(−1)s) ∈ Cs×s. One can use the notation of Ekk and write

Xn,s = iπ

(
η(s)
n

n∑
k=1

Ekk +
n∑

k=n−s+1
(−1)n−s+1−kEkk

)
. (12)

If r is any representation of the Lie algebra sl(n, C), then RAn,s := exp(r(Xn,s)) satisfies

RAn,s r(X)
(
RAn,s

)−1
= r(An,sXA

−1
n,s)= r(AdAn,s X) and

(
RAn,s

)2
= Id .

Therefore, matrix RAn,s is the simulation matrix of the inner automorphism g = AdAn,s. We have shown

Theorem 3.1 Any Z2-grading of the Lie algebra sl(n, C) obtained by an inner automorphism and any irre-
ducible representation of sl(n, C) are compatible.

Using (12) and the explicit form of the Gel’fand-Tseitlin representationwe obtain for any basis vector ξ(m)

r(Xn,s)ξ(m)= iπ

(
η(s)
n

rn(m)+2
s−1∑
k=1

(−1)k−1rn−s+k(m) − rn−s(m) − (−1)η(s)rn(m)
)

ξ(m) .

Thus we have arrived at the explicit form of the simulation matrix of the automorphism g = AdAn,s

RAn,s ξ(m)= e
iπ

((
η(s)

n
−1
)
rn(m)−rn−s(m)

)
ξ(m)

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Acta Polytechnica Vol. 50 No. 5/2010

3.2 Outer automorphism of order two

As explained at the beginning of Section 3, any outer automorphismof order two on sl(n, C) is up to equivalence
the automorphism OutI(X)= −X T , and thus we will focus only on it without loss of generality.
It is well known that for an irreducible representation r characterized in the Gel’fand-Tseitlin formalism

by the n-tuple (m1,n, m2,n, . . . , mn,n), the mapping −rT (to minus transposed matrices) is also an irreducible
representation. This representation is equivalent to the contragredient representation rc, which is characterized
by the n-tuple (m′1,n, m

′
2,n, . . . , m

′
n,n), where

m′i,n = m1,n − mn−i+1,n for i =1,2, . . . , n .

Let us consider a triangular patternm filled by indices mi,j, 1 ≤ i ≤ j ≤ n, and associatedwith the basis vector
ξ(m) of representation r. To any such pattern m, we may assign the unique triangular patternm′ with indices
m′i,j := m1,n − mj−i+1,j. It is easy to check that m

′
i,j satisfies the necessary inequalities for m

′ to be a correct
pattern of the contragredient representation rc. Let us define the linear mapping J of the representation space
of r onto the representation space of rc by

J ξ(m) := (−1)
∑

i,j
mi,j

ξ(m′) .

On the other hand, from the formulae in the Appendix one sees that

rT(Eij)= r(Eji)= r(E
T

ij ) . (13)

Using this fact one can prove by direct verification that the mapping J satisfies

− J rT(X)= rc(X)J for any X ∈ sl(n, C) . (14)

Let us return to our original task. We are looking for the simulation matrix of the automorphism g = OutI,
i.e., we are looking for a matrix Rg of order two such that

r(OutI(X))= −r(X T)= Rgr(X)R−1g .

According to (13), we have r(X T) = rT(X) and therefore the existence of the simulation matrix Rg means
equivalence of the representations r and −rT , i.e. equivalence of r and its contragredient representation
rc. The Gel’fand-Tseitlin result states that this is possible if and only if n-tuples (m1,n, m2,n, . . . , mn,n) and
(m′1,n, m

′
2,n, . . . , m

′
n,n) coincide. In this case the simulation matrix Rg is equal to J. We have deduced

Theorem 3.2 A Z2-grading of the Lie algebra sl(n, C) obtained by an outer automorphism OutI is compatible
with an irreducible representation r of sl(n, C) if and only if the representation is self-contragredient.

If we do not insist on the irreducibility of representation r, the class of representations compatible with the Z2-
grading obtained by the automorphism OutI is larger. Of course, if for a representation r1 it is possible to find
a simulationmatrix R(1) and for a representation r2 a simulationmatrix R

(2), then the direct sum R(1) ⊕ R(2) is
the simulationmatrix for the direct sum r1 ⊕ r2. To avoid a discussion of all such obvious cases, wewill describe
only those representations r with simulationmatrices R for which the operator set {R} ∪ {r(X) | X ∈ sl(n, C)}
is irreducible, whereas the set {r(X) | X ∈ sl(n, C)} is reducible.
If r0 is a d-dimensional irreducible representation of sl(n, C) then the 2d-dimensional representation r :=

r0 ⊕
(
−r T0

)
assigns to X the matrix

r(X)=

(
r0(X) 0

0 −
(
r0(X)

)T
)

and therefore

r(OutI(X))=

(
−
(
r0(X)

)T
0

0 r0(X)

)
=

(
0 Id

Id 0

)
r(X)

(
0 Id

Id 0

)
.

The matrix

(
0 Id

Id 0

)
is the simulationmatrix of OutI. It is easy to see that the simulationmatrix together

with all r(X) form an irreducible set.

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Acta Polytechnica Vol. 50 No. 5/2010

3.3 Z2-grading of sl(3, C)

Let us illustrate the conclusions of the previous sections on the Lie algebra sl(3, C). On this algebra there exist
only two inequivalent automorphisms of order two. In our notation g1 = AdA3,1 with

A3,1 = ω

⎛
⎜⎜⎝
1 0 0

0 1 0

0 0 −1

⎞
⎟⎟⎠ , where ω = e iπ3

and g2 = OutI. The corresponding Z2-gradings are

Γ1:sl(3, C)=
{⎛⎜⎜⎝

a b 0

c d 0

0 0 −a − d

⎞
⎟⎟⎠∣∣∣ a, b, c, d ∈ C} ⊕ {

⎛
⎜⎜⎝
0 0 a

0 0 b

c d 0

⎞
⎟⎟⎠∣∣∣ a, b, c, d ∈ C} ,

Γ2:sl(3, C)=
{⎛⎜⎜⎝

0 a b

−a 0 c
−b −c 0

⎞
⎟⎟⎠∣∣∣ a, b, c ∈ C} ⊕ {

⎛
⎜⎜⎝

a b c

b d e

c e −a − d

⎞
⎟⎟⎠∣∣∣ a, b, c, d, e ∈ C} .

The first grading Γ1 is compatible with any irreducible representation. The simulation matrix Rg1 of the
automorphism g1 = AdA31 acts on the Gel’fand-Tseitlin triangular patterns as follows

Rg1

⎛
⎜⎜⎝

m1,3 m2,3 0

m1,2 m2,2

m1,1

⎞
⎟⎟⎠ = e−2iπ3 (m1,3+m2,3)e−iπ(m1,2+m2,2)

⎛
⎜⎜⎝

m1,3 m2,3 0

m1,2 m2,2

m1,1

⎞
⎟⎟⎠ .

The irreducible representations compatible with the second grading are only self-contragredient representa-
tions, i.e., representations r = r(2�, �,0). In such representation, the operator J is defined by

J

⎛
⎜⎜⎝
2� � 0

m1,2 m2,2

m1,1

⎞
⎟⎟⎠ =(−1)	+m1,2+m2,2+m1,1

⎛
⎜⎜⎝
2� � 0

2� − m2,2 2� − m1,2
2� − m1,1

⎞
⎟⎟⎠ .

The lowest-dimensional non-trivial self-contragredient representation is r = r(2,1,0), in fact, the adjoint
representation. Its dimension is 8 and has the following explicit form on the basis vectors:

Rg2

⎛
⎜⎜⎝
2 1 0

2 1

2

⎞
⎟⎟⎠=

⎛
⎜⎜⎝
2 1 0

1 0

0

⎞
⎟⎟⎠ , Rg2

⎛
⎜⎜⎝
2 1 0

1 0

0

⎞
⎟⎟⎠=

⎛
⎜⎜⎝
2 1 0

2 1

2

⎞
⎟⎟⎠ , Rg2

⎛
⎜⎜⎝
2 1 0

2 1

1

⎞
⎟⎟⎠ = −

⎛
⎜⎜⎝
2 1 0

1 0

1

⎞
⎟⎟⎠ ,

Rg2

⎛
⎜⎜⎝
2 1 0

1 0

1

⎞
⎟⎟⎠ = −

⎛
⎜⎜⎝
2 1 0

2 1

1

⎞
⎟⎟⎠ , Rg2

⎛
⎜⎜⎝
2 1 0

2 0

2

⎞
⎟⎟⎠= −

⎛
⎜⎜⎝
2 1 0

2 0

0

⎞
⎟⎟⎠ , Rg2

⎛
⎜⎜⎝
2 1 0

2 0

0

⎞
⎟⎟⎠ = −

⎛
⎜⎜⎝
2 1 0

2 0

2

⎞
⎟⎟⎠ ,

Rg2

⎛
⎜⎜⎝
2 1 0

1 1

1

⎞
⎟⎟⎠ =

⎛
⎜⎜⎝
2 1 0

1 1

1

⎞
⎟⎟⎠ , Rg2

⎛
⎜⎜⎝
2 1 0

2 0

1

⎞
⎟⎟⎠ =

⎛
⎜⎜⎝
2 1 0

2 0

1

⎞
⎟⎟⎠ .

If the representation r = r(m13, m23,0) is not self-contragredient, then grading Γ2 is compatible with the
reducible representation

r⊕(X) :=

(
r(X) 0

0 −
(
r(X)

)T
)

and the corresponding simulationmatrix on the double-dimensional space is J = σ1 ⊗ I, where I is the identity
operator on the representation space of representation r and σ1 denotes the first Pauli matrix.

37



Acta Polytechnica Vol. 50 No. 5/2010

4 Conclusions

Since basic concepts connected with gradings of Lie algebras were laid down already in the work of J. Patera
and H. Zassenhaus [16], including the notion of compatibly graded representation, it is really surprising that
there does not yet exist a theory of representations of graded Lie algebras compatible with a given grading.
This work is devoted to first steps in investigating which irreducible representations of a Lie algebra L are
compatible with its G-grading, at least in a rather restricted framework. The main contribution of the paper
consists in elucidatingwhich representationsof classicalLie algebrasof type A are compatiblewith a Z2-grading.
Concretely, the results are as follows: if the involutive automorphism producing the Z2-grading is inner, then
every irreducible finite-dimensional representation is compatible with the grading, but if the automorphism
producing the grading is not inner, then the only irreducible finite-dimensional representations compatible with
the grading are the self-contragredient ones. For the outer automorphism there is also a possibility of reducible
representations involving pairs of mutually contragredient irreducible representations. Thus it is not generally
true that every irreducible representation can be compatibly graded. The sl(3, C)-case is enclosed to illustrate
the process.
One of our future goals is to enlarge the family of gradings of L for which one can decide about compati-

bility with representations of L. Another goal is to study representations of physical interest of the so-called
kinematical groups of space-times. The possible Lie algebras L of these groups were classified in [1]. A rather
remarkable fact was found there that very simple conditions imposed by space inversion and time reversal on
the generators very severely constrain the possible Lie algebras. This result was confirmed in [12] from the
corresponding Z2 × Z2-contractions of the de Sitter Lie algebras. From this point of view it would be useful
to identify the gradings implicitly present for instance in [3, 18] and in other papers where contractions of
representations are studied.

Acknowledgement

The authors would like to express their gratitude to the referees for their careful reading of the manuscript
and for their valuable critical comments, which have helped to improve the presentation. We are grateful to
Vyacheslav Futorny for fruitful discussions on relations between the notions of grading and group grading, and
to Jiří Patera for introducing us to the problems connectedwith representations of contractedLie algebras. We
acknowledge financial support from the grants MSM6840770039 and LC06002 of the Ministry of Education,
Youth, and Sports of the Czech Republic.

Appendix. The Gel’fand-Tseitlin formalism

Let us give an explicit description of the irreducible representations of gl(n, C) in the Gel’fand-Tseitlin for-
malism [4, 10, 2]. Any irreducible representation r of gl(n, C) is in one-to-one correspondence with an n-tuple
(m1,n, m2,n, . . . , mn,n) of non-negative integer parameters m1,n ≥ m2,n ≥ . . . ≥ mn,n ≥ 0. Since any Ek	 can
be obtained by commutation relations from Ek,k, Ek,k−1 and Ek−1,k, only formulas for r(Ek,k), r(Ek,k−1) and
r(Ek−1,k) are needed:

r(Ek,k)ξ(m)= (rk − rk−1)ξ(m) ,
where rk = m1,k + . . . + mk,k for k =1,2, . . . , n and r0 =0,

r(Ek,k−1)ξ(m)= a
1
k−1ξ(m

1
k−1)+ . . . + a

k−1
k−1ξ(m

k−1
k−1) ,

where mjk−1 denotes the triangular pattern obtained from m replacing mj,k−1 by mj,k−1 − 1,

a
j
k−1 =

[
−
∏k

i=1(mi,k − mj,k−1 − i + j +1)
∏k−2

i=1 (mi,k−2 − mj,k−1 − i + j)∏
i�=j(mi,k−1 − mj,k−1 − i + j +1)(mi,k−1 − mj,k−1 − i + j)

]1/2
and

r(Ek−1,k)ξ(m)= b
1
k−1ξ(m

1
k−1)+ . . . + b

k−1
k−1ξ(m

k−1
k−1) ,

where mjk−1 denotes the triangular pattern obtained from m replacing mj,k−1 by mj,k−1 +1, and

b
j
k−1 =

[
−
∏k

i=1(mik − mj,k−1 − i + j)
∏k−2

i=1 (mi,k−2 − mj,k−1 − i + j − 1)∏
i�=j(mi,k−1 − mj,k−1 − i + j)(mi,k−1 − mj,k−1 − i + j − 1)

]1/2
.

38



Acta Polytechnica Vol. 50 No. 5/2010

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Prof. Ing. Miloslav Havlíček, DrSc.
E-mail: miloslav.havlicek@fjfi.cvut.cz
Prof. Ing. Edita Pelantová, CSc.
E-mail: edita.pelantova@fjfi.cvut.cz
Prof. Ing. Jiří Tolar, DrSc.
E-mail: jiri.tolar@fjfi.cvut.cz
Doppler Institute, Faculty of Nuclear Sciences and Physical Engineering
Czech Technical University in Prague, Czech Republic

39