Generalization of Ky Fan-Amir-Mo?ez-Horn-Mirsky?s result on the
eigenvalues and real singular values of a matrix
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee.
This dissertation does not include proprietary or classifled information.
Wen Yan
Certiflcate of Approval:
Randall R. Holmes
Associate Professor
Mathematics and Statistics
Tin-Yau Tam, chair
Professor
Mathematics and Statistics
Ming Liao
Professor
Mathematics and Statistics
Peter M. Nylen
Professor
Mathematics and Statistics
Stephen L. McFarland
Dean
Graduate School
Generalization of Ky Fan-Amir-Mo?ez-Horn-Mirsky?s result on the
eigenvalues and real singular values of a matrix
Wen Yan
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulflllment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
December 16, 2005
Generalization of Ky Fan-Amir-Mo?ez-Horn-Mirsky?s result on the
eigenvalues and real singular values of a matrix
Wen Yan
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon the request of individuals or institutions and at their expense.
The author reserves all publication rights.
Signature of Author
Date
Copy sent to:
Name Date
iii
Vita
Wen Yan, son of Changyou Yan and Pingju Wu, was born in Xupu, Hunan,
P. R. China, on January 15, 1973. He attended the public school of Xupu, Hu-
nan, P.R.China. After graduating from Xupu High School, he entered Xiangtan
University in September, 1990, from which he received his B.S. (Math) in June,
1994. He started his graduate study in the Department of Mathematics of Xiang-
tan University in August, 1994 and received his M.S. (Math) in January, 1997.
He continued his graduate study in the Department of Mathematics of Auburn
University in August, 2000.
iv
Dissertation Abstract
Generalization of Ky Fan-Amir-Mo?ez-Horn-Mirsky?s result on the
eigenvalues and real singular values of a matrix
Wen Yan
Doctor of Philosophy, December 16, 2005
(M.S., Xiangtan University, 1997)
(B.S., Xiangtan University, 1994)
115 Typed Pages
Directed by Tin-Yau Tam
Ky Fan?s result states that the real part of the eigenvalues of an n?n com-
plex matrix A is majorized by the eigenvalues of the Hermitian part of A. The
converse was established by Amir-Mo?ez and Horn, and Mirsky, independently. We
extend the results in the context of complex semisimple Lie algebras. Inequalities
associated with the classical complex Lie algebras are given. The real case is also
discussed.
v
Acknowledgments
The author would like to thank Dr. Tin-Yau Tam for his excellent guidance
throughout the course of this research, his parents for their constant support and
encouragement, his wife Li Huang for her loving patience and support.
vi
Style manual or journal used Transactions of the American Mathematical
Society (together with the style known as \auphd"). Bibliography follows van
Leunen?s A Handbook for Scholars.
Computer software used The document preparation package TEX (speciflcally
LATEX) together with the departmental style-flle auphd.sty.
vii
Table of Contents
List of Figures ix
1 Introduction 1
2 The proof of the result of Ky Fan-Amir-Mo?ez-Horn-Mirsky 9
3 Preliminaries 17
4 The complex semisimple case 34
5 The inequalities associated with an and cn 47
6 The inequalities associated with bn and dn 60
7 The real semisimple case 81
8 The eigenvalues and the real and imaginary singular values
for sl(2;C) and sl(2;R) 101
Bibliography 104
viii
List of Figures
4.1 The Convex Hull convWfl For a2 . . . . . . . . . . . . . . . . . . . 43
4.2 The Convex Hull convWfl For b2 . . . . . . . . . . . . . . . . . . . 44
4.3 The Convex Hull convWfl For c2 . . . . . . . . . . . . . . . . . . . 45
4.4 The Convex Hull convWfl For d2 . . . . . . . . . . . . . . . . . . . 46
5.1 Z ?Y in C := dualit(it)+ for sl(3;C) . . . . . . . . . . . . . . . . . 51
6.1 The Union of the Convex Hulls: L1 [L2 . . . . . . . . . . . . . . . 80
ix
Chapter 1
Introduction
Let Cn?n be the space of n ? n complex matrices. Each A 2 Cn?n has the
Hermitian decomposition
A = 12(A?A?)+ 12(A+A?); (1.1)
where ? denotes the complex conjugate transpose. Clearly the matrix A1 := 12(A?
A?) is skew-Hermitian, i.e., A?1 = ?A1 (called the skew-Hermitian part of A) and
A2 := 12(A+A?) is Hermitian, i.e., A?2 = A2 (called the Hermitian part of A).
There are three important sets of scalars associated with A, known as the
eigenvalues, denoted by ?1;?2;:::;?n 2 C, the real singular values, denoted by
fi1 ? fi2 ????? fin and the imaginary singular values of A, denoted by fl1 ? fl2 ?
??? ? fln. An eigenvalue ? of A is a scalar such that there exists a nonzero vector
z 2Cn such that
Az = ?z:
The real singular values of A are the eigenvalues of the Hermitian part 12(A+A?)
of A. The imaginary singular values of A are the eigenvalues of the Hermitian
matrix 12i(A?A?).
There is a nice result of Ky Fan [7], [23, p.239] relating the eigenvalues of A
and the real singular values of A. We need the following important notion called
1
majorization [23] in order to state the result of Ky Fan. Majorization has a lot of
applications in difierent branches of mathematics [3, 16, 23].
Deflnition 1.1 Let a;b 2Rn. We say that a is majorized by b, denoted by a ` b,
if
kX
i=1
a[i] ?
kX
i=1
b[i]; k = 1;:::;n?1;
nX
i=1
a[i] =
nX
i=1
b[i];
where a[1] ? a[2] ? ??? ? a[n] and b[1] ? b[2] ? ??? ? b[n] are the rearrangements of
the entries of a and b, respectively, in nonincreasing order.
Theorem 1.2 (KyFan)GivenA 2Cn?n, therealpartsoftheeigenvalues(?1;:::;?n)T 2
Cn of A is majorized by the real singular values (fi1;:::;fin)T 2 Rn of A, i.e.,
Re? ` fi.
The converse was established by Amir-Mo?ez and Horn [1], and independently
by Mirsky [24]. It was later rediscovered by Sherman and Thompson [29]. The
study can be traced back to some old results of Bendixson [2], Hirsch [13], and
Bromwich [4]. Also see [23, p.237-239]. We state the result in the following theo-
rem.
Theorem 1.3 (Amir-Mo?ez-Horn and Mirsky) If ? 2 Cn and fi 2 Rn such that
Re? ` fi, then there exists A 2Cn?n such that ??s are the eigenvalues of A and
fi?s are the real singular values of A.
2
Similar results hold for the imaginary part of the eigenvalues of A and the
imaginary singular values of A.
The following is the result of Sherman and Thompson [29].
Theorem 1.4 (Sherman and Thompson) If H is a given Hermitian matrix with
eigenvalues fl 2 Rn and if fi 2 Rn satisfles fi ` fl, then there exists a skew
Hermitian matrix K such that fi is the real part of the eigenvalues of K +H.
Remark 1.5 The result of Sherman and Thompson is indeed equivalent to Amir-
Mo?ez-Horn-Mirsky?s result. We now establish their equivalence.
We will use the fact that the eigenvalues, real singular values, and the imagi-
nary singular values of any A 2Cn?n are invariant under unitary similarity.
Theorem 1.3 =) Theorem 1.4: Given fi ` fl, where fl is the eigenvalues of a
Hermitian matrix H, by Amir-Mo?ez-Horn-Mirsky?s result (Theorem 1.3), there is
a matrix A1 such that fi is the real part of the eigenvalues of A1 and fl is the eigen-
values of the Hermitian part of A1. Since H and A1+A?12 are both Hermitian with
the same eigenvalues fl?s, by the spectral decomposition for Hermitian matrices [16,
p.171], they are both unitarily similar to the diagonal matrix diagfl, hence unitarily
similar to each other. Let U be the unitary matrix such that H = U?1A1+A?12 U. Let
K := U?1A1?A?12 U. Then K is skew Hermitian, and A := K + H = U?1A1U. By
the fact that eigenvalues and real singular values are invariant under unitary simi-
larity, A and A1 have the same eigenvalues and real singular values. Therefore K is
the required skew Hermitian matrix. This proves that Amir-Mo?ez-Horn-Mirsky?s
result implies the result of Sherman and Thompson.
3
Theorem 1.4 =) Theorem 1.3: Conversely, suppose Re? ` fl for ? 2 Cn
and fl 2 Rn. Let H = diagfl. By Sherman and Thompson?s result there is a
skew Hermitian matrix K such that Re? is the real part of the eigenvalues ??s of
A1 := K+H and the real singular values of A1 are fl?s. By Schur triangularization
theorem (Theorem 2.1), there exists a unitary matrix U such that B := U?1A1U
is upper triangular and ? = diagB. Then A := B + idiag(Im? ? Im?) has
eigenvalues ??s and the Hermitian part of A is B+B?2 = U?1A1+A?12 U. Thus A and
A1 have the same real singular values fl?s. Therefore A has eigenvalues ??s and
real singular values fl?s. So the result of Sherman and Thompson implies Amir-
Mo?ez-Horn-Mirsky?s result.
How would Ky Fan-Amir-Mo?ez-Horn-Mirsky?s result be if we restrict our at-
tention to complex skew symmetric matrices? If A 2 Cn?n is skew symmetric,
then its eigenvalues occur in pair but opposite in sign, since A and AT = ?A have
the same characteristic polynomial. We will see in Chapter 6 that majorization
remains to be the key, except the statements are stronger and we will separately
consider the even and odd cases. Similarly we consider the symplectic case as well
in Chapter 5.
We will have semisimple Lie algebra as a unifled framework and develop our
main result in Chapter 4. The following is the motivation for our study. A transla-
tion of A, that is, A+?I for some ? 2C, would translate the eigenvalues by ? and
the real singular values by Re?. Thus it is su?cient to consider those A 2Cn?n
such that tr A = 0 in Ky Fan-Amir-Mo?ez-Horn-Mirsky?s result. Recall that
sl(n;C) := fA 2Cn?n : tr A = 0g
4
is the Lie algebra of the special linear group
SL(n;C) := fA 2Cn?n : detA = 1g:
The special unitary group
SU(n) := fU 2Cn?n : U?U = In; detU = 1g
is a maximal compact subgroup of SL(n;C). The diagonal matrices in sl(n;C) form
a Cartan subalgebra h of sl(n;C), and those in h with purely imaginary diagonal
entries form a Cartan subalgebra t of the Lie algebra
su(n) := fA 2 sl(n;C) : A+A? = 0g
of SU(n). As a real SU(n)-module, sl(n;C) is just the direct sum of two copies of
the adjoint module su(n):
sl(n;C) = su(n)'isu(n)
which in our case is essentially the well known Hermitian decomposition of a com-
plex matrix (1.1). By the Schur triangularization theorem (See Theorem 2.1) for
complex matrices, the eigenvalues of a matrix A 2 sl(n;C) may be viewed as
the image of an element Y 2 AdSU(n)(A) \ b under the orthogonal projection
5
? : sl(n;C) ! h with respect to the inner product
hX;Yi = Retr XY ?; X;Y 2 sl(n;C);
where SU(n) acts on sl(n;C) via the the adjoint representation, AdSU(n)(A) is
the orbit of A under the action of SU(n), and b is the Borel subalgebra of sl(n;C)
consisting of n ? n upper triangular matrices. Thus taking the real part of the
eigenvalues of A amounts to sending Y via the orthogonal projection
? : sl(n;C) ! it
with respecttoh?;?i. The majorization relation fi ` fl isequivalenttofi 2 convSnfl
for fi;fl 2Rn by Theorem 2.7, where convSnfl denotes the convex hull of the orbit
of fl under the action of Sn. It is a direct result of Theorem 2.3 (Hardy-Littlewood-
Poly?a) and Theorem 2.6 (Birkhofi). So the result of Ky Fan (Theorem 1.2) may
be stated as
?(AdSU(n)(X +Z)\b) ? convSnZ; (1.2)
where Z 2 it, X 2 su(n) and Sn is the full symmetric group on f1;:::;ng. The
result of Amir-Mo?ez-Horn and Mirsky (in the version of Sherman and Thompson)
may be written as
convSnZ ?[X2su(n) ?(AdSU(n)(X +Z)\b): (1.3)
6
Combining (1.2) and (1.3) we have
[X2su(n) ?(AdSU(n)(X +Z)\b) = convSnZ: (1.4)
We will extend (1.4) in the context of flnite dimensional complex semisimple Lie
algebras (Theorem 4.6), as one of our main results in this dissertation. Notice that
(1.4) may be stated as
?((su(n)+AdSU(n)(Z))\b) = convSnZ:
In particular, for each U 2 sl(n;C),
?(AdSU(n)(U)\b) ? convSnz;
where Z 2 AdSU(n)(12(U ? U))\it, where
(X) = ?X?; X 2 sl(n;C):
Before we prove the extension of (1.4) in Theorem 4.6 in Chapter 4, we give
a detailed proof of Ky Fan-Amir-Mo?ez -Horn-Mirsky?s result in Chapter 2. We
then introduce some preliminary materials of complex semisimple Lie algebras in
Chapter 3 to pave the road for Chapter 4. In Chapter 4, we use Kostant?s result
[22, Theorem 8.2] to show that (1.4) remains true for all complex semisimple Lie
algebras. The interesting inequalities corresponding to the classical Lie algebras,
alike majorization, are discussed in Chapter 5 and Chapter 6. In Chapter 7, we
7
discuss the case for the real Lie algebra sl(n;R) and obtain a result for sl(n;R)
which is similar to Theorem 4.6. In particular we consider su(1;1), a real form of
sl(2;C). In Chapter 8, we consider some inequalities relating the eigenvalues, the
real and imaginary singular values for sl(2;C) and sl(2;R).
8
Chapter 2
The proof of the result of Ky Fan-Amir-Mo?ez-Horn-Mirsky
In this chapter we review the proof of the result of Ky Fan, Amir-Mo?ez and
Horn, and Mirsky. We shall then point out the key elements of the proof, which
will be essential for the extension in Chapter 4. The proof makes use of the well
known Schur?s triangularization theorem [16, p.79], a result of Schur [30] and a
result of A. Horn [15] on the diagonal and the eigenvalues of a Hermitian matrix.
We denote by U(n) = fU 2Cn?n : U?U = Ing the unitary group.
Theorem 2.1 (Schur?s triangularization theorem) Given A 2Cn?n with eigenval-
ues ?1;:::;?n 2Cin any prescribed order, there is a unitary matrix U 2 U(n) such
that UAU?1 is upper triangular and diag(UAU?1) = ?, where ? = (?1;:::;?n)T.
Remark 2.2 Indeed U(n) in Theorem 2.1 can be replaced by the special unitary
group SU(n).
We will use the following result of Hardy, Littlewood and Poly?a [11, p.49] to
establish Schur?s result (Theorem 2.4). A nonnegative matrix D is called a doubly
stochastic if the row sums and column sums of D are 1.
Theorem 2.3 (Hardy-Littlewood-Poly?a) Let fi;fl 2Rn. Then fi ` fl if and only
if there exists a n?n doubly stochastic matrix D such that fi = Dfl.
9
Clearly the diagonal entries of a Hermitian matrix are real. Schur [30], [16,
p.193] obtained the following nice result relating the diagonal and the eigenvalues
of a Hermitian matrix A 2Cn?n.
Theorem 2.4 (Schur) Let A 2 Cn?n be a Hermitian matrix. The diagonal d =
(d1;:::;dn)T 2Rn of A is majorized by the vector ? = (?1;:::;?n)T 2Rn of the
eigenvalues of A, i.e., d ` ?.
Proof: Let A = (aij). By the spectral decomposition [16, p.171] there exists
U 2 U(n) such that A = U(diag?)U?1. By direct computation,
aij =
nX
s=1
uis?ujs?s:
Thus d = D?, where D = (dij) = (juijj2) is called an orthostochastic matrix.
An orthostochastic matrix is clearly a doubly stochastic matrix. By Theorem 2.3
d ` ?.
The converse of Theorem 2.4 was obtained by A. Horn [15]. The original proof
is a long and intricate argument. The following simple proof was flrst obtained by
Chan and Li [5] and later rediscovered by Zha and Zhang [35].
Theorem 2.5 (A. Horn) If d;? 2Rn and d ` ?, then there exists a real symmetric
matrix A 2Rn?n such that diagA = d and ??s are the eigenvalues of A.
Proof: Suppose d ` ?. Since permutation similarity would not change the
eigenvalues of a matrix but permute the diagonal entries, we may assume that
d1 ? d2 ????? dn and ?1 ? ?2 ????? ?n. We use induction on n. When n = 2,
10
the real symmetric matrix
A =
?d
1 ?
? d2
!
has the desired property if we choose ? =
p2
2 (?
2
1 + ?
2
2 ?d
2
1 ?d
2
2)
1=2. Suppose the
statement is true when n = m. Let n = m + 1. Suppose 2 ? j ? m + 1 be the
largest index such that ?j?1 ? d1 ? ?j. Clearly ?1 ? maxfd1;?1 + ?j ? d1g ?
minfd1;?1 + ?j ?d1g ? ?j. Then there exists a 2?2 orthogonal matrix U1 such
that
U1
??
1
?j
!
U?11 =
?d
1 ?
? ?1 +?j ?d1
!
:
Set U2 := U1 'Im?1. Then
A1 := U2 diag(?1;?j;?2;:::;?j?1;?j+1;:::;?m+1)U?12
=
?d
1 ?
? ?1 +?j ?d1
!
'diag(?2;:::;?j?1;?j+1;:::;?m+1):
We are going to show that
(d2;:::;dn) ` (?1 +?j ?d1;?2;:::;?j?1;?j+1;:::;?m+1): (2.1)
To proceed, notice that ?j?1 ? d1 ? d2,
d2 ? maxf?1 +?j ?d1;?2;:::;?j?1;?j+1;:::;?m+1g:
11
Moreover
kX
i=2
di ? (k ?1)d1 ?
kX
i=2
?i; k = 2;:::;j ?1;
kX
i=2
di =
kX
i=1
di ?d1 ?
kX
i=1
?i ?d1 = (?1 +?j ?d1)+
kX
i=2;i6=j
?i; k = j;:::;m;
m+1X
i=2
di =
m+1X
i=1
di ?d1 =
m+1X
i=1
?i ?d1 = (?1 +?j ?d1)+
m+1X
i=2;i6=j
?i:
Hence (2.1) is established. By the inductive hypothesis, there exists an m ? m
orthogonal matrix U3 such that
U3 diag(?1 +?j ?d1;?2;:::;?j?1;?j+1;:::;?m+1)U?13
has diagonal (d2;:::;dm+1). Then A := U4A1U?14 has diagonal d and eigenvalues
?, where U4 := 1'U3.
It is straight forward to show that the set of n?n doubly stochastic matrices
?n is a convex set in Rn?n. Birkhofi [16, p.527] showed that it is the convex hull
of the permutation matrices.
Theorem 2.6 (Birkhofi) A matrix D 2 Rn?n is a doubly stochastic matrix if
and only if it is a convex combination of permutation matrices, i.e., there are
permutation matrices P1;:::;Pm and nonnegative scalars fi1;:::;fim 2 R such
that fi1 +:::+fim = 1 and D = fi1P1 +:::+fimPm.
Combining and rewriting Theorem 2.4 and Theorem 2.5, we have the following
statement.
12
Theorem 2.7 (Schur and A.Horn) Let ? 2Rn. Then
diagfU(diag(?1;:::;?n))U?1 : U 2 O(n)g
= diagfU(diag(?1;:::;?n))U?1 : U 2 U(n)g
= fd 2Rn : d ` ?g
= convSn?;
where U(n) is the unitary group, O(n) is the orthogonal group, Sn is the full
symmetric group on f1;:::;ng and conv denotes the convex hull of the underlying
set in Rn.
Proof: Since O(n) ? U(n) and in view of Theorem 2.4,
diagfU(diag(?1;:::;?n))U?1 : U 2 O(n)g
? diagfU(diag(?1;:::;?n))U?1 : U 2 U(n)g
? fd 2Rn : d ` ?g:
By Theorem 2.5, fd 2Rn : d ` ?g ? diagfU(diag(?1;:::;?n))U?1 : U 2 O(n)g.
So we established the flrst two set equalities. Theorem 2.3 asserts that
fd 2Rn : d ` ?g = ?n?;
where ?n is the set of n?n doubly stochastic matrices. By Theorem 2.6
fd 2Rn : d ` ?g = convSn?
13
follows immediately.
Remark 2.8 We may replace O(n) by SO(n), the special orthogonal group and
U(n) by SU(n), respectively, in Theorem 2.7.
We will make use of Theorem 2.1 and Theorem 2.4 to prove the result of Ky
Fan, namely Theorem 1.2. Then we use Theorem 2.5 to prove Amir-Mo?ez-Horn-
Mirsky?s result, namely Theorem 1.3 which is the converse of Theorem 1.2. We
flrst combine Theorem 1.2 and Theorem 1.3 together in the following statement.
Theorem 2.9 (Ky Fan-Amir-Mo?ez-Horn-Mirsky) Let A 2Cn?n with eigenvalues
? = (?1;:::;?n)T and real singular values fi = (fi1;:::;fin)T. Then Re? ` fi.
Conversely, if ? 2 Cn, fi 2 Rn such that Re? ` fi, then there exists A 2 Cn?n
with eigenvalues ??s and real singular values fi?s.
Proof: (Ky Fan) Suppose that A has eigenvalues ?1;:::;?n. By Theorem 2.1
there exists a unitary matrix U 2 U(n) such that
Y := UAU?1 =
0
BB
B@
?1 ? ?
... ?
?n
1
CC
CA
is upper triangular. Now A = U?1YU and
A+A?
2 = U
?1(Y +Y
?
2 )U;
thus A and Y have the same eigenvalues and real singular values. Now 12(Y +Y ?)
is Hermitian and has diagonal entries Re??s and eigenvalues fi?s. By Theorem 2.4,
Re? ` fi.
14
(Amir-Mo?ez-Horn-Mirsky) Conversely, suppose ? 2Cn and fi 2Rn are given
such that Re? ` fi. By Theorem 2.5, there is a Hermitian matrix H = (hij) 2
Cn?n with eigenvalues fi?s and diagonal entries Re??s. The upper triangular matrix
A :=
0
BB
BB
BB
@
?1 2h12 ::: 2h1n
?2 ::: 2h2n
... ...
?n
1
CC
CC
CC
A
2 Cn?n
has eigenvalues ??s and real singular values fi?s since 12(A+A?) = H, of which the
eigenvalues are fi?s. This completes the proof.
The key point in the proof of Ky Fan?s result is to obtain an upper triangular
matrix Y which is similar to the original matrix A under unitary similarity. Since
eigenvalues and real singular values are invariant under unitary similarity, A and
Y have the same eigenvalues and real singular values. So the real part of the
eigenvalues of A are the real part of the diagonal of Y. Application of Schur?s
result on (Y+Y?)2 then flnishes the proof. The key element in the proof of Amir-
Mo?ez-Horn-Mirsky?s result is Theorem 2.5.
We may view taking the eigenvalues of A (the real part, respectively) as a
projection from Cn?n to Cn (Rn, respectively), after turning A into an upper
triangular matrix via Schur?s triangularization theorem. With this view in mind
we rewrite Theorem 2.9 in the following form.
15
Theorem 2.10 Given A 2Cn?n. Let Wn ?Cn?n be the space of upper triangular
matrices. Then
RediagffUAU?1 : U 2 U(n)g\Wng = fd 2Rn : d ` fig = convSnfi; (2.2)
where fi = (fi1;:::;fin)T is the vector of eigenvalues of 12(A+A?).
Proof: The last equality is in Theorem 2.7. The flrst equality is another form of
Theorem 2.9.
The above idea will be extended in Chapter 4 for the complex semisimple Lie
algebras.
16
Chapter 3
Preliminaries
In this chapter we introduce some notations and basic concepts of complex
semisimple Lie algebras. Most of the material can be found in any standard Lie
theory textbook such as [12, 18, 20]. In particular we will review the details of the
root space decomposition of classical semisimple Lie algebras.
In this chapter all Lie algebras are flnite dimensional overC, unless speciflcally
noted.
Deflnition 3.1 [20, p.2] A flnite dimensional vector space g over C is called a
complex Lie algebra if there is a product [X;Y] for X;Y 2 g that is linear in each
variable and satisfles
(a) [X;X] = 0 for all X 2 g (and hence [X;Y] = ?[Y;X]) and
(b) the Jacobi identity
[[X;Y];Z]+[[Y;Z];X]+[[Z;X];Y] = 0: (3.1)
The real Lie algebra is deflned analogously by changing the base fleld to R in the
above deflnition.
17
A Lie subalgebra h of g is a subspace satisfying [h;h] h. An ideal h in g is a
subspace satisfying [h;g] h. It is automatically a Lie subalgebra. The Lie algebra
g is said to be abelian if [g;g] = 0.
Deflnition 3.2 [20, p.8-9] Let g be a flnite dimensional Lie algebra. We deflne
recursively
g0 = g; g1 = [g;g]; gj+1 = [gj;gj];
g0 = g; g1 = [g;g]; gj+1 = [g;gj]:
The sequence of ideals g0 g1 g2??? is called the derived series of g and the
sequence g0 g1 g2??? is called the descending central series of g. We say that
g is solvable if gj = 0 for some j and g nilpotent if gj = 0 for some j.
Example 3.3 [20, p.3] Let g = gl(n;C) denote the associative algebra of all n?n
matrices with complex entries. We can turn g into a Lie algebra by introducing
the product
[X;Y] = XY ?YX; X;Y 2 g:
The set of all n?n upper triangular matrices u is a solvable subalgebra in g.
A Lie algebra g is simple if g is nonabelian and has no proper ideals. Hence
simple Lie algebras are at least 3-dimensional. A Lie algebra g is semisimple if g
has no nonzero solvable ideals. Every simple Lie algebra is semisimple.
For any complex Lie algebra g we get a linear map
ad : g ! Endg (adX)(Y) = [X;Y]; X;Y 2 g:
18
This linear map is called the adjoint representation of g, one of the most important
maps in the theory of Lie algebras. If X and Y are in g, then adX ? adY is a
linear transformation from g to itself, and it is meaningful to deflne
B(X;Y) = tr (adX ?adY):
Then B(?;?) is a symmetric bilinear form on g known as the Killing form of g.
Killing form is a very useful tool in the theory of Lie algebras. Cartan [20,
p.25] obtained two useful criteria for semisimplicity and solvability, respectively, of
a Lie algebra g by considering its Killing form.
Theorem 3.4 (Cartan?s Criterion for Semisimplicity) The complex Lie algebra g
is semisimple if and only if the Killing form for g is nondegenerate, i.e., if X 2 g
and B(X;Y) = 0 for all Y 2 g, then X = 0.
Theorem 3.5 (Cartan?s Criterion for Solvability) The complex Lie algebra g is
solvable if and only if its Killing form satisfles B(X;Y) = 0 for all X 2 g and
Y 2 [g;g].
Let us consider the root space decomposition of a complex Lie algebra. From
now on we assume that g is a complex semisimple Lie algebra unless specifled.
Deflnition 3.6 A Lie subalgebra h of g is called a Cartan subalgebra [12, p.130] if
h is maximal abelian and for each H 2 h, the endomorphism adgH is semisimple,
i.e., diagonalizable.
19
It is known that any flnite dimensional complex Lie algebra has a Cartan subal-
gebra. The importance of the Cartan subalgebras for unraveling the structure of
g lies in the following fundamental fact.
Theorem 3.7 [18, p.84] The Cartan subalgebras of a complex semisimple Lie
algebra g are all conjugate under the adjoint group Intg, where Intg ? GL(g) is
the analytic subgroup of GL(g) with Lie subalgebra adg ? End(g).
Since all the Cartan subalgebras of g are conjugate, there is no harm in choos-
ing one, say, h. Since h is abelian and the underlying fleldCis algebraically closed,
the adjoint representation ad : g ! End(g), restricted to h, splits g up as a direct
sum of one-dimensional subspaces. In other words, if fi 2 h? and if we set
gfi := fX 2 g : [H;X] = fi(H)X for all H 2 hg;
then g is a direct sum of the gfi. Since g is flnite dimensional, only flnitely many of
the gfi are nonzero. If fi 6= 0 and gfi 6= 0, then fi is called a root of g with respect to
h. Members of gfi are called root vectors for the root fi. Let ? denote the set of all
roots, a flnite subset of h?. We now have the well known root space decomposition
of g with respect to h.
Proposition 3.8 [20, p.88] Let g be a complex semisimple Lie algebra and let h
be a Cartan subalgebra of g. Then
g = g0 _+ _
X
fi2?g
fi: (3.2)
It satisfles
20
(a) h = g0,
(b) [gfi;gfl] = gfi+fl (with gfi+fl understood to be 0 if fi +fl is not a root),
(c) Bjh?h is nondegenerate; consequently there is an vector space isomorphism
? : h? ! h such that Hfi := ?(fi), fi 2 h? satisfles fi(H) = B(H;Hfi) for all
H 2 h,
(d) ? spans h?, the dual space of h,
(e) dimgfi = 1 for all fi 2 ?.
The real span V = Pfi2?Rfi ? h? is of dimension dimCh. Since the Killing
form B(?;?) remains nondegenerate in the restriction to h, hence deflnes an isomor-
phism fi 7! Hfi (as in Proposition 3.8(c) via Riesz?s representation theorem) of h?
onto h, and a bilinear form on the dual space h? of h by transporting via B(?;?) in
the following fashion:
h?;?i = B(H?;H?) = ?(H?) = ?(H?); ?;? 2 h?; (3.3)
where H? and H? are deflned in Proposition 3.8(c).
It turns out that the restriction of the bilinear form on V, h?;?ijV?V , is real
and positive deflnite, i.e., a real inner product so that V acquires the structure of
an Euclidean space.
Theorem 3.9 [20, p.101] Let ? be the root system of (g;h). Let V = Pfi2?Rfi ?
h? and h0 = Pfi2?RHfi ? h. Then
21
1. The real space V is an Euclidean space with the inner product h?;?ijV?V ,
the restriction of the bilinear form deflned by (3.3) to V ?V. Moreover, the
members of V are exactly those linear functionals in h? that are real on h0
and the restriction of the operation of those linear functionals from h to h0 is
an R isomorphism of V onto h?0. In particular Vjh0 = h?0.
2. h? = V 'iV.
3. h = h0 'ih0.
Theorem 3.10 [20, p.99-103] Let ? be the root system of (g;h). Then
1. ? spans h?0,
2. the orthogonal transformations sfi(?) := ? ? 2h?;fiihfi;fii fi, for all ? 2 V where
fi 2 ?, carry ? onto itself,
3. 2hfl;fiihfi;fii is an integer for any fi;fl 2 ?.
We now introduce a notion of positivity in V so that for any nonzero ? 2 V
so that
1. exactly one of ? and ?? is positive,
2. the sum of positive elements is positive and any positive multiple of a positive
element is positive.
One way to deflne positivity is by means of a lexicographic ordering. Fix a basis
f?1;:::;?ng of V, deflne positivity as follows: We say that ? > 0 if there exists
an index k such that h?;?ii = 0 for 1 ? i ? k ? 1, and h?;?ki > 0. We say
22
that a root fi is simple if fi > 0 and if fi does not decompose as fi = fl1 + fl2 with
fl1 and fl2 both positive roots. Then there are n linearly independent simple roots
fi1;:::;fin such that if a root fl is written as fl = x1fi1 +???+ xnfin, then all the
xj?s are integers with the same sign (if 0 is allowed to be positive or negative).
Denote by ?+ the set of of all positive roots which would uniquely determine a set
? = ffi1;:::;fing of simple roots and ? is called a simple system.
We know from Proposition 3.10 that sfi : h?0 ! h?0 deflned by
sfi(fl) = fl ? 2hfl;fiihfi;fii fi; fl 2 h?0;
is a re ection of the space h?0 that flxes ?. The set fsfi : fi 2 ?g generates a flnite
re ection group [19], denoted by W(g;h) or simply W, called the Weyl group of
(g;h). It is clearly a subgroup of O(h?0), the orthogonal group of h?0. The Weyl group
W can be generated by a smaller set of generators known as the simple re ections
fsfi : fi 2 ?g. Notice that the Weyl group W can be viewed as a subgroup of O(h0)
since we analogously have the re ections on h0:
sH(L) = L? B(H;L)B(H;H)H; H;L 2 h0;
and it is clear that
? ?sfi(fl) = s?(fi)(?(fl)); fi;fl 2 h?0:
In other words
? ?sfi ???1 = s?(fi);
23
and thus sfi is identifled with s?(fi), where ? is given in Proposition 3.8 (c).
Let fi 2 ? and P?fi be the hyperplane in h?0 deflned by
hfi;?i = 0; ? 2 h?0:
Let
C = h?0 ?
[
fi2?
P?fi;
i.e., C is the complement of the union of all P?fi (fi 2 ?). A component of C is
said to be a Weyl chamber of h?0 with respect to ?. The Weyl chamber
C0 = f? 2 h?0 : (fii;?) > 0; i = 1;:::;ng
is called the fundamental Weyl chamber. The choice of one among ?+, ? and C0
determines the others.
Let us consider some standard models of the classical simple Lie algebras an,
bn, cn and dn. We will look at their root space decompositions and Weyl groups.
Let Sn be the full symmetric group on f1;:::;ng and Eij be the square matrix of
appropriate size with (i;j)th entry 1 and 0 elsewhere.
Example 3.11 [12, p.186] [20, p.80] A model for the simple Lie algebra an (n ? 1)
is g := sl(n+1;C), the algebra of all (n+1)?(n+1) complex matrices with trace
0. Let h be the set of all diagonal matrices in g. Then h is a Cartan subalgebra of
24
g. Deflne ej 2 h? for j = 1;:::;n+1 by
ej
0
BB
B@
h1
...
hn+1
1
CC
CA = hj:
Direct matrix computation yields
(adH)Ejk = [H;Ejk] = (ej(H)?ek(H))Ejk for all H 2 h.
Thus Ejk is a simultaneous eigenvector for all adH, H 2 h. The root system of
(g;h) is
? = f?(ej ?ek) : 1 ? j < k ? ng:
The root space decomposition is
g = h _+ _
X
ej?ek2?g
ej?ek;
where
gej?ek =CEjk:
The simple roots are
? = fej ?ej+1 : j = 1;:::;ng:
Notice that ej ?ek can be written as
ej ?ek = (ej ?ej+1)+:::+(ek ?ek+1) if j < k,
25
and
ej ?ek = ?(ek ?ek+1)?:::?(ej?1 ?ej) if j > k.
The Killing form of (complex) g is [12, p.187]
B(X;Y) = 2(n+1)tr XY; X;Y 2 g:
For any fi = ej ?ek 2 ?, Hej?ek = 12(n+1)(Ejj ?Ekk) (see Proposition 3.8 (c)). So
h0 is the space of real diagonal matrices which is identifled with the hyperplane
f(h1;:::;hn+1)T 2Rn+1 :
n+1X
j=1
hj = 0g
in Rn+1 naturally:
h0 3 diag(h1;:::;hn+1) 7! (h1;:::;hn+1)T 2Rn+1:
If fi = ej ? ej+1 2 ?, the re ection sfi (identifled with sHfi) acts on H =
(h1;:::;hn+1)T 2 h0 by
sfi(H) = H ? 2B(Hfi;H)B(H
fi;Hfi)
Hfi
= H ?(hj ?hj+1)Hfi
= diag(h1;:::;hj?1;hj+1;hj;hj+2;:::;hn+1):
So the action of sej?ej+1 on H = diag(h1;:::;hn+1) is to switch the jth and
(j+1)st entries. Thus the Weyl group is the full symmetric group Sn+1 on the set
26
f1;:::;ng. With the identiflcation W acts on h0 by
(h1;:::;hn+1)T 7! (h (1);:::;h (n+1))T; 2 Sn+1:
Example 3.12 [20, p.83] A model of the simple Lie algebra bn (n ? 1) is g :=
so(2n+1;C), the set of all (2n+1)?(2n+1) complex skew symmetric matrices.
The subalgebra
h =
(
H =
? 0 ih
1
?ih1 0
!
'???'
? 0 ih
n
?ihn 0
!
'(0) : h1;:::;hn 2C
)
:
is a Cartan subalgebra of g. Let
ej(above H) = hj; 1 ? j ? n:
The root system of (g;h) is
? = f?ej ?ek : 1 ? j 6= k ? ng[f?ek : 1 ? k ? ng:
The root space decomposition is
g = h _+ _
X
fi2?g
fi; gfi =CEfi;
and with Efi as deflned below. To deflne Efi, flrst let j < k and let fi = ?ej ?ek.
Then Efi is 0 except in the sixteen entries corresponding to the jth and kth pairs
27
of indices, i.e.,
j k
Efi =
0
B@ 0 Xfi
?XTfi 0
1
CA j
k
with
Xej?ek =
0
B@ 1 i
?i 1
1
CA; X
ej+ek =
0
B@ 1 ?i
?i ?1
1
CA;
X?ej+ek =
0
B@ 1 ?i
i 1
1
CA; X
?ej?ek =
0
B@ 1 i
i ?1
1
CA:
To deflne Efi for fi = ?el, write
pair entry
l 2n+1
Efi =
0
B@ 0 Xfi
?XTfi 0
1
CA
with 0?s elsewhere and with
Xel =
0
B@ 1
?i
1
CA; X
?el =
0
B@ 1
i
1
CA:
The simple roots are
? = fej ?ej+1 : 1 ? j ? n?1g[feng:
28
The Killing form of (complex) g is [12, p.189]
B(X;Y) = (2n?1)tr XY; X;Y 2 g:
Notice that h0 = fH 2 h : h1;:::;hn 2Rg. If we identify h0 withRn in the natural
way,
h0 3
0
B@ 0 ih1
?ih1 0
1
CA'???'
0
B@ 0 ihn
?ihn 0
1
CA'(0) 7! (h
1;:::;hn)T 2Rn;
then
sej?ej+1(h1;:::;hn) = (h1;:::;hj?1;hj+1;hj;hj+2;:::;hn);
i.e., switching the j and the (j +1)st entries, and
sen(h1;:::;hn) = (h1;:::;hn?1;?hn):
Thus the Weyl group W of (g;h) acts on h0 by
(h1;:::;hn)T 7! (?h (1);:::;?h (n))T; 2 Sn:
Example 3.13 [20, p.85] The simple Lie algebra cn (n ? 1) may be realized as
g := sp(n;C) = sp(n)'isp(n), where is the set of 2n?2n complex matrices of the
following form:
sp(n;C) = fX 2 sl(2n;C) : XTJ +JX = 0g;
29
where J = Jn;n is the 2n?2n matrix
J =
0
B@ 0 In
?In 0
1
CA:
So [12, p.447]
sp(n;C) = f
?A
1 A2
A3 ?AT1
!
: A2;A3 2Cn?n complex symmetric; A1 2Cn?ng
and
sp(n) = f
?A ?B
B A
!
: A;B 2Cn?n;A? = ?A; BT = Bg:
Now
h := fdiag(h1;:::;hn;?h1;:::;?hn) : h1;:::;hn 2Cg
is a Cartan subalgebra of g. Let ej 2 h? be
ej(diag(h1;:::;hn;?h1;:::;?hn)) = hj; 1 ? j ? n:
Then the root system of (g;h) is
? = f?ej ?ek : 1 ? j 6= k ? ng[f?2ek : 1 ? k ? ng:
30
The corresponding root spaces are
gej?ek = C(Ej;k ?Ek+n;j+n); g2el = C(El;l+n);
gej+ek = C(Ej;k+n +Ek;j+n); g?2el = C(El+n;l);
g?ej?ek = C(Ej+n;k +Ek+n;j);
where 1 ? j 6= k ? n and 1 ? l ? n. The simple roots are
? = fej ?ej+1 : 1 ? j ? n?1g[f2eng:
The Killing form of (complex) g is [12, p.190]
B(X;Y) = (2n+2)tr XY; X;Y 2 g:
Notice that h0 = fH 2 h : h1;:::;hn 2Rg. If we identify h0 withRn in the natural
way,
h0 3 diag(h1;:::;hn;?h1;:::;?hn) 7! (h1;:::;hn)T 2Rn;
then Weyl group W of (g;h) acts on h0 by
(h1;:::;hn)T 7! (?h (1);:::;?h (n))T; 2 Sn:
Example 3.14 [20, p.85] The simple Lie algebra dn (n ? 3) may be realized as
g := so(2n;C) = so(2n)+iso(2n), the algebra of 2n?2n complex skew symmetric
31
matrices. The subalgebra
h =
8
><
>:H =
0
B@ 0 h1
?h1 0
1
CA'???'
0
B@ 0 hn
?hn 0
1
CA : h
1;:::;hn 2C
9
>=
>;;
is a Cartan subalgebra of g. The root system of (g;h) is
? = f?ej ?ek : 1 ? j < k ? ng;
where
ej(above H) = hj; 1 ? j ? n:
The corresponding root spaces g?ej?ek; ?ej ?ek 2 ? are similar to those deflned
for bn in Example 3.12. The simple roots are
? = fej ?ej+1 : 1 ? j ? n?1g[fen?1 +eng:
The Killing form of g is [12, p.188]
B(X;Y) = (2n?2)tr XY; X;Y 2 g:
Notice that h0 = fH 2 h : h1;:::;hn 2 Rg. If we identify h0 with Rn similar to
Example 3.12, i.e.,
h0 3
0
B@ 0 ih1
?ih1 0
1
CA'???'
0
B@ 0 ihn
?ihn 0
1
CA7! (h
1;:::;hn)T 2Rn;
32
then
sej?ej+1(h1;:::;hn) = (h1;:::;hj?1;hj+1;hj;hj+2;:::;hn);
i.e., switching the jth and the (j +1)st entries, and
sen?1+en(h1;:::;hn) = (h1;:::;?hn;?hn?1):
Thus the The Weyl group W of (g;h) acts on h0 by
(h1;:::;hn)T 7! (?h (1);:::;?h (n))T; 2 Sn;
where the number of negative signs is even.
33
Chapter 4
The complex semisimple case
In this chapter we assume that g is a complex semisimple Lie algebra and use
the notations in the previous chapter.
A complex Lie group is a Lie group G possessing a complex analytic structure
such that multiplication and inversion are holomorphic. For such a group the
complex structure induces a multiplication-by-i mapping in the Lie algebra g such
that g becomes a Lie algebra over C [20, p.55].
A semisimple Lie group G has a complexiflcation GC if GC is a complex con-
nected Lie group such that G is Lie subgroup of GC and the Lie algebra of GC is
the complexiflcation of the Lie algebra of G [20, p.404]. Not every semisimple Lie
group has a complexiflcation. Even if G has a complexiflcation, the complexiflca-
tion is not necessarily unique up to isomorphism. But if G is compact, GC exists
and is unique [20, p.375].
Let K be a real compact connected semisimple Lie group, G its complexiflca-
tion, and let k and g be their respective Lie algebras. Thus g = k ' ik. We flx a
maximal torus T of K and denote its Lie algebra by t. Then h = t'it is a Cartan
subalgebra of g (now it is h0 in Chapter 3). Let the root space decomposition of g
be
g = h'
M
fi2?
gfi;
34
where ? is the root system of (g;h) and gfi is the root space of the root fi 2 ?. Fix
a simple system ? for ?. The set of positive roots (with respect to ?) is denoted
by ?+. The Weyl group of (g;h) will be denoted by W. A subalgebra of g is called
a Borel subalgebra of g if it is a maximal solvable subalgebra of g. We introduce
the maximal nilpotent subalgebras n and n? of g:
n =
M
fi2?+
gfi; n? =
M
fi2?+
g?fi:
Then
b = h'n (4.1)
is a (standard) Borel subalgebra of g. Let B be the corresponding Borel subgroup
of G.
An automorphism of g is an invertible linear map L 2 GL(g) that respects
bracket
[L(X);L(Y)] = L[X;Y] for all X;Y 2 g:
Denote by Autg the group of automorphisms of g. The adjoint group Int(g) (see
the deflnition in Theorem 3.7) is a normal subgroup of Autg. The adjoint group
Int(g) is generated by eadX (= Ad(eX) [12, p.128]), where X 2 g. Its elements
are called inner automorphisms. Since the complexiflcation G of K is connected,
Ad(G) = Int(g) [12, p.129] and is the identity component of Aut(g) [12, p.132].
We have the following facts about the Borel subalgebras of a complex Lie
algebra g.
35
Theorem 4.1 [18, p.84] The Borel subalgebras of a complex semisimple Lie alge-
bra g are all conjugate under Intg.
The following is a recent generalization of the Schur triangularization theorem
(Theorem 2.1) in the context of complex semisimple Lie algebras by Djokovi?c and
Tam [6].
Proposition 4.2 (Djokovi?c and Tam) Let g be a complex semisimple Lie algebra.
1. The Borel subalgebras of g are all conjugate under AdK.
2. Let b be any Borel subalgebra of g. Then AdK(X) intersects b for each
X 2 g.
Proof: (1) Let b0 be any Borel subalgebra and let b the standard Borel algebra
given in (4.1). By Theorem 4.1 all Borel subalgebras are conjugate under Intg =
AdG. So there is g 2 G such that b0 = Ad(g)b. The global Iwasawa decomposition
[12, p.275] states that G = KAN (G is viewed as a real group), where K, A, and
N denote the analytic subgroups of G with Lie algebras k, it, and n, respectively.
Thus G = KB, where B AN is the analytic subgroup of G with Lie algebra b.
Therefore there exist k 2 K and b 2 B such that g = kb. So
b0 = Ad(g)b = Ad(k)Ad(b)b = Ad(k)b:
(2) Let b be any Borel subalgebra of g. For any X 2 g, X is contained in some
Borel subalgebra b0 of g. Thus Ad(k)X 2 Ad(k)b0 = b by the flrst part for some
k 2 K.
36
Let be the Cartan involution of g = k'ik, i.e., is identity on k and negative
identity on ik. Then is semilinear, i.e., (?X + ?Y) = ?X + ?Y, X;Y 2 g,
respects the bracket [X;Y] = [ X; Y] and is an involution, i.e., 2 = 1. Moreover
B (X;Y) := 2ReB(X; Y) is an inner product on g and k and ik are orthogonal
with respect to B (?;?).
Example 4.3 Consider g := sl(n;C). The Hermitian decomposition sl(n;C) =
su(n) + isu(n) suggests that k = su(n) and the corresponding Cartan involution
is (X) = ?X?; X 2 g. Moreover (Eij) = ?Eji; i 6= j and B (X;Y) =
?B(X; (Y)) = 2(n+1)tr XY ?.
Proposition 4.4 (h) = h and (gfi) = g?fi for all fi 2 ?.
Proof: Clearly (it) = it and t = t so that h = h. Let X 2 gfi and H :=
iH1 + H2 2 t _+it, H1;H2 2 it. Then [H; X] = [ H;X] = [ (iH1 + H2);X] =
[ iH1;X]+ [ H2;X] = [iH1;X]+ [?H2;X] = ifi(H1)X?fi(H2) X = ?fi(H) X
since fi takes real values on it.
We will use the following result of Kostant [22] to prove Theorem 4.6.
Theorem 4.5 (Kostant) Use the same notations of K, t and W as above. Let
? : k ! t be the orthogonal projection with respect to the Killing form. If Z 2 t,
then ?(AdK(Z)) = convWZ,
For any complex semisimple Lie algebra g whose connected Lie group is G, it
is known that [20, p.302] that g always has a compact real form k, i.e., g = k 'ik.
Let K be a connected subgroup of G corresponding to k.
37
The following is an extension of Ky Fan-Amir-Mo?ez-Horn-Mirsky?s result in
the context of complex semisimple Lie algebras.
Theorem 4.6 Let g = k'ik be a complex semisimple Lie algebra, where k is the
Lie algebra of a semisimple Lie group K. Let t be a Cartan subalgebra of k. Let
? : g ! it be the orthogonal projection with respect to the Killing form of the
realiflcation gR of g. If Z 2 it, then
[X2k ?(AdK(X +Z)\b) = convWZ; (4.2)
where b is given in (4.1) and W is the Weyl group of (g;h). Equivalently
?((k +AdK(Z))\b) = convWZ: (4.3)
In particular, for each U 2 g, ?(AdK(U)\b) ? convWZ, where Z 2 AdK(12(U ?
U))\it.
Proof: Notice that
[X2k ?(AdK(X +Z)\b)
= ?([X2k AdK(X +Z)\b)
= ?([[X2k [k2K (Adk(X)+Adk(Z))]\b)
= ?([[k2K [X2k (Adk(X)+Adk(Z))]\b))
= ?([k2K (k +Adk(Z))\b)) since AdG(k)jk = AdK(k) 2 Aut(k)
= ?((k +AdK(Z))\b)
38
and Proposition 4.2 ensures that ?((k +AdK(Z))\b) is nonempty.
We now give a proof of (4.2). Let Y 2 k + AdK(Z). Then Y = X + Adk(Z)
for some X 2 k and k 2 K. Since k ? it under B (?;?),
?(Y) = ?(X +Adk(Z)) = ?(Adk(Z)):
By Theorem 4.5 ?(AdK(Z)) = convWZ. Thus ?(Y) 2 convWZ and
?((k +AdK(Z))\b) ? ?(k +AdK(Z)) ? convWZ: (4.4)
Conversely, let fl 2 convWZ. By Theorem 4.5 again, there exists Y 2
AdK(Z) such that ?(Y) = fl. Recall the root space decomposition from Proposi-
tion 3.8
g = h _+ _
X
fi2?+(g
fi 'g?fi):
The direct sum gfi 'g?fi is not orthogonal. Write
Y = Y0 +
X
fi2?+
(Yfi +Y?fi);
where Y0 2 h, Yfi 2 gfi and Y?fi 2 g?fi. Since Y 2 ik, ik is the ?1 eigenspace of ,
we have
?Y0 +
X
fi2?+
(?Yfi ?Y?fi) = ?Y = Y = Y0 +
X
fi2?+
( Yfi + Y?fi):
39
Since the sums are direct and gfi = g?fi (fi 2 ?), it follows that Y0 2 h \ik, i.e.,
Y0 2 it. Moreover Y?fi = ? Yfi for all fi 2 ?. Then
Y = Y0 +
X
fi2?+(Yfi ? Yfi);
and Y0 = ?(Y) = fl. Set
X :=
X
fi2?+(Yfi + Yfi) 2 k:
Then
X +Y = Y0 +2
X
fi2?+Yfi 2 (X +AdK(Z))\b:
Clearly ?(X +Y) = ?(Y) = fl. This proves
?((k +AdK(Z))\b) convWZ: (4.5)
Combining (4.4) and (4.5) we conclude
?((k +AdK(Z))\b) = convWZ:
For U 2 g, AdK(U) \ b is nonempty by Proposition 4.2. We decompose
U = 12(U + U)+ 12(U ? U). Clearly
?(AdK(U)\b) = ?(AdK(12(U + U)+ 12(U ? U))\b)
? [X2k?(AdK(X + 12(U ? U))\b)
40
= convWZ;
where Z 2 AdK(12(U ? U))\it.
Remark 4.7 When g = sl(n;C), the theorem is simply Ky Fan-Amir-Mo?ez-Horn-
Mirsky?s result with an appropriate translation.
Remark 4.8 Let N(T) denote the normalizer of t in K with respect to the adjoint
action of K, i.e., N(T) = fk 2 K : Ad(k)t = tg. Consider the group homomor-
phism ? : N(T) ! O(t), n 7! (Adn)jt, from N(T) into O(t) which denotes the
group of orthogonal linear transformations of the real space t. The kernel of ?
is T and W is the image of ?. So ? deflnes a group isomorphism between the
group N(T)=T onto the Weyl group W of (g;h) if we identify ih0 and t. Knapp
calls N(T)=T the analytically deflned Weyl group and W the algebraically deflned
Weyl group [20, p.207].
Remark 4.9 The statement of Theorem 4.6 remains true when the Cartan sub-
space ik is replaced by k. But we need to change the projection ? to ?1 : g ! t and
assume that Z 2 t. This becomes the generalization of the Ky Fan-Amir-Mo?ez-
Horn-Mirsky?s result about the imaginary part of the eigenvalues and imaginary
singular values.
Remark 4.10 Notice that g is an inner product space equipped with the natural
inner product hX;Yi := B (X;Y) = ?B(X; Y), and k and ik are orthogonal since
[k;ik] ? ik, [k;k] ? k and [ik;ik] ? k. Thus
kXk2 = k12(X + X)k2 +k12(X ? X)k2:
41
When g = sl(n;C), it simply asserts that the square of the Frobenius norm of X
is the sum of squares of the real and imaginary singular values [1, Theorem 5].
The complex classical Lie algebras an (n ? 1), bn (n ? 1), cn (n ? 1) and dn
(n ? 2 and d1 is not semisimple) are semisimple. Indeed they are simple except
d2. So Theorem 4.6 holds for them. The following
a1 ? b1 ? c1; b2 ? c2;; a3 ? d3; d2 ? a1 'a1:
are the only isomorphisms [12, p.465] which hold between the complex classical Lie
algebras. All occur among low dimensional algebras. We now draw the pictures of
the convex hull convWfl for fl 2 it for several low dimensional cases.
Example 4.11 (1) a1: For g = sl(2;C), h =
n?h1 0
0 ?h1
!
: h1 2C
o
and the
only simple root is fi1 = e1 ? e2 2 h? (See the notations in Example 3.11),
where
fi1
?h
1 0
0 ?h1
!
= 2h1:
TheWeylgroupof(g;h)isW = f1;sfi1g, wheresfi1(fi1) = ?fi1 andsfi1(?fi1) =
fi1. Thus W acts on h0 = it =
n?h1 0
0 ?h1
!
: h1 2R
o
:
?h
1 0
0 ?h1
!
7!
?h
1 0
0 ?h1
!
; or
??h
1 0
0 h1
!
:
So if fl = diag(fl1;?fl1) 2 h0, then
convWfl =
n?x 0
0 ?x
!
: ?fl1 ? x ? fl1
o
;
42
which is identifled with the line segment [?fl1;fl1].
(2) a2 : For g = sl(3;C), the simple roots are fi1 = e1 ?e2, fi2 = e2 ?e3 (See the
notations in Example 3.11). The Weyl group is generated by the re ections
fsfi1;sfi2g with respective to the hyperplanes perpendicular to fi1 and fi2,
respectively. If we identify h0 = it with the hyperplane H0 := fx 2 R3 :
x1 +x2 +x3 = 0g in R3. The Weyl group W of (g;h) acts on h0 by
(h1;h2;h3)T 7! (h (1);h (2);h (3))T; 2 S3;
where S3 is the full symmetric group on the set f1;2;3g. For any fl 2 it,
Wfl consists of six points on the hyperplane H0, hence convWfl is a hexagon
including its interior. The shaded region is the intersection of convWfl and
the (closed) fundamental Weyl chamber.
?1=e1?e2
?2=e2?e3
?
Figure 4.1: The Convex Hull convWfl For a2
43
(3) b2 : For g = so(5;C), the simple roots are fi1 = e1?e2, fi2 = e2 (See Example
3.12). The Weyl group is generated by the re ections fsfi1;sfi2g and if we
identify h0 = it with R2 in the natural way, then the Weyl group acts on it
by
(h1;h2)T 7! (?h (1);?h (2))T; 2 S2:
For any fl 2 it, the convex hull convWfl is an octagon including its in-
terior. The shaded region is the intersection of convWfl and the (closed)
fundamental Weyl chamber.
?1=e1?e2
?2=e2 ?
Figure 4.2: The Convex Hull convWfl For b2
(4) c2 : Similar to so(5;C), the simple roots of g = sp(2;C) are fi1 = e1 ? e2,
fi2 = 2e2 (See Example 3.13). The graph of the convex hull is identical with
the one of g = so(5;C) since they have the same Weyl group.
(5) d2 : For g = so(4;C), the simple roots are fi1 = e1 ? e2, fi2 = e1 + e2 (See
Example 3.14). The Weyl group is generated by the re ections fsfi1;sfi2g
44
?1=e1?e2
?2=2e2
?
Figure 4.3: The Convex Hull convWfl For c2
and acts on h0 = it by
(h1;h2)T 7!?(h (1);h (2))T; 2 S2;
if we identify it with R2. The graph of convWfl for fl 2 it is a rectangle
including its interior. The shaded region is the intersection of convWfl and
the (closed) fundamental Weyl chamber.
45
?1=e1?e2
?2=e1+e2
?
Figure 4.4: The Convex Hull convWfl For d2
46
Chapter 5
The inequalities associated with an and cn
In Chapter 4 we obtained the extension of Ky Fan-Amir-Mo?ez-Horn-Mirsky?s
result in the context of complex semisimple Lie algebras. In this chapter and the
next we will analyze the result for the classical Lie algebras an, bn, cn and dn, which
are realized in matrix models. We obtain some interesting inequalities which are
similar to majorization. In this chapter, we use the same notations as we did in the
previous chapters. Let K be a real semisimple compact connected Lie group whose
complexiflcation is G. Let g and k be the Lie algebras of G and K, respectively.
Let h = t 'it be the Cartan subalgebra of g, where t is the Cartan subalgebra of
k. Let b be the (standard) Borel subalgebra of g:
b = h'
X
fi2?+
gfi;
where ?+ is the set of positive roots in the root system ? of (g;h). Denote by
? = ffij; j = 1;:::;ng the set of simple roots and set V = Pni=1Rfii. Let
? : g ! h and ? : g ! it be the orthogonal projections with respect to the Killing
form. In this chapter, we use (?;?) to denote the Killing form B(?;?) of g restricted
to h0 = it, and identify (it)? with it under the identiflcation deflned by Proposition
3.8 (c) (? : fi 7! Hfi). Under this identiflcation, the (closed) fundamental Weyl
47
chamber of g is
(it)+ = fH 2 it : fij(H) ? 0; j = 1;:::;ng:
The vectors 2fii(fii;fii), i = 1;:::;n again form a basis of V. Let f?1;:::;?ng be the
dual basis:
2(?i;fij)
(fij;fij) = ?ij; i;j = 1;:::;n:
They are called the fundamental dominant weights [18, p.67].
Proposition 5.1 Let g be a complex semisimple Lie algebra. The (closed) fun-
damental Weyl chamber (it)+ for g is the cone C generated by the fundamental
dominant weights ?j; j = 1;:::;n, i.e., C = fPnj=1 aj?j : aj ? 0;j = 1;:::;ng.
Proof: Denote by
h?; i := 2(?; )( ; ) :
Let fi 2 (it)+. Thus (fi;fij) ? 0 for j = 1;:::;n. We have
fi =
nX
j=1
hfi;fiji?j;
hence fi 2 C. This proves (it)+ ? C. Conversely, let fi 2 C. Then fi = Pnj=1 aj?j
with aj ? 0; j = 1;:::;n. Thus hfi;fiji = aj ? 0 and hence (fi;fij) ? 0 for
j = 1;:::;n. This proves fi 2 (it)+. So C ? (it)+. Therefore (it)+ = C.
48
The dual cone dualit(it)+ ? it of the fundamental Weyl chamber (it)+ is
deflned as
dualit(it)+ := fX 2 it : (X;Y) ? 0; for all Y 2 (it)+g;
which may be written as
fX 2 it : (X;?j) ? 0; j = 1;:::;ng:
Kostant [22, Lemma 3.3] proved the following result which is very useful to derive
inequalities that completely describe convWZ.
Lemma 5.2 (Kostant)
1. Let Z 2 (it)+. For any w 2 W, Z ?wZ 2 dualit(it)+.
2. Let Y;Z 2 (it)+, then Y 2 convWZ if and only if Z?Y 2 dualit(it)+, where
W is the Weyl group of (g;h).
Proof: (1) If Z?w0Z 62 dualit(it)+ for some id 6= w0 2 W, then there would exist
X 2 (it)+ such that (Z ? w0Z;X) < 0. Let us flx this X. Because W is flnite,
there exist a id 6= w 2 W so that (Z ? wZ;X) = (Z;X) ? (wZ;X) is minimal,
or equivalently (wZ;X) is maximal. Since W acts simply transitively on the Weyl
chambers, wZ =2 (it)+ and there exists a simple root fi 2 ? such that fi(wZ) < 0.
Let Hfi be the be the corresponding element of fi in it. Then we have
(sfi(wZ);X) = (wZ ? fi(wZ)jfij2 Hfi;X)
49
= (wZ;X)?( fi(wZ)(H
fi;Hfi)
Hfi;X)
= (wZ;X)? fi(wZ)(H
fi;Hfi)
fi(X)
> (wZ;X)
since fi(X) > 0 and fi(wZ) < 0. This contradicts the maximality of (wZ;X).
(2) If Y;Z 2 (it)+ and Y 2 convWZ, then there exist aw ? 0, w 2 W such
that
Y =
X
w2W
awwZ;
X
w2W
aw = 1:
Thus
Z ?Y =
X
w2W
awZ ?
X
w2W
awwZ
=
X
w2W
aw(Z ?wZ):
Since Z ?wZ 2 dualit(it)+ by the flrst part and aw ? 0 for all w 2 W, Z ?Y 2
dualit(it)+.
Conversely, let Y;Z 2 (it)+ and suppose that Y =2 convWZ Then Y and
convWZ lie on difierent sides of some hyperplane in it. Then there exists X 2 it
such that (Y;X) > (wZ;X) for all w 2 W. Since W 2 O(it), (Z;w?1X) =
(wZ;X) < (Y;X). Choose w 2 W such that w?1X 2 (it)+. Since Z;w?1X 2 (it)+,
we have (Z ?w?1Z;w?1X) ? 0 by the flrst part, or equivalently,
(Z;w?1X) ? (w?1Z;w?1X) = (Z;X):
50
Thus (Z ?Y;X) < 0, i.e., Z ?Y 62 dualit(it)+.
The following picture illustrate the geometric meaning of Lemma 5.2 for
sl(3;C) in which C := dualit(it)+ is the dual cone generated by fi1 and fi2.
Z
Y
C
?1=e1?e2
?2=e2?e3
Figure 5.1: Z ?Y in C := dualit(it)+ for sl(3;C)
The shaded region is the intersection of the fundamental Weyl chamber (it)+
and the backward cone, ?C +Z, centered at Z.
Lemma 5.3 Given Y;Z 2 it, Y 2 convWZ if and only if Y+ ?Z+ 2 dualit(it)+,
where W is the Weyl group of (g;h), where Y+ denotes the element in the singleton
set WY \(it)+.
Proof: The Weyl group W acts transitively on Weyl chambers, i.e., WY \(it)+
is nonempty [18, p.51] for each Y 2 (it)+, which is a singleton set since W is a
group.
Clearly each ! 2 W flxes convWZ, i.e., !(convWZ) = convWZ and hence
WY ? convWZ if and only if Y 2 convWZ. Thus Y 2 convWZ if and only
51
if Y+ ? convWZ+. By Lemma 5.2, Y 2 convWZ if and only if Y+ ? Z+ 2
dualit(it)+.
Example 5.4 ReferringtoExample3.11concerningthesimpleLiealgebra an (n ?
1), we still use the model g = sl(n+1;C). Let K = SU(n+1). Then k = su(n+1)
and (X) = ?X?, X 2 g. Denote h by the set of all diagonal matrices in g, which
is a Cartan subalgebra. The root space decomposition of g with respect to h is
g = h _+ _
X
fi2?+g
fi 'g?fi;
where ? = f?(ej ?ek) : 1 ? j < k ? n + 1g and gej?ek = CEjk. Identify h0 = it
with the hyperplane fx 2Rn+1 : x1 +:::+xn+1 = 0g in Rn+1:
diag(h1;:::;hn+1) 7! (h1;:::;hn+1)T:
The Weyl group W of (g;h) acts on it by
(h1;:::;hn+1)T 7! (h (1);:::;h (n+1))T; 2 Sn+1:
The positive roots in ? are
ej ?ek; 1 ? j < k ? n+1;
and the simple roots are
fij = ej ?ej+1; j = 1;:::;n:
52
The fundamental dominant weights are [18, p.69] [20, p.289]
?k = n?k +1n+1
kX
j=1
ej ? kn+1
n+1X
j=k+1
ej; k = 1;:::;n:
The (closed) fundamental Weyl chamber it+ is
f(h1;:::;hn+1)T 2 it : h1 ? ::: ? hn+1g:
The dual cone of (it)+ in it is
dualitit+ = f(x1;:::;xn+1)T 2 it :
kX
j=1
xj ? 0;k = 1;:::;n; and
n+1X
j=1
xj = 0g:
Since W = Sn+1, fi+ = (fi[1];:::;fi[n+1]) 2 (it)+, i.e., rearrangement of fi 2 it in
nonincreasing order. By Lemma 5.3, given fi;fl 2 it, fi 2 convWfl if and only if
fl+ ?fi+ 2 dualit(it)+, i.e.,
kX
j=1
fi[j] ?
kX
j=1
fl[j]; k = 1;:::;n; (5.1)
n+1X
j=1
fi[j] =
n+1X
j=1
fl[j] = 0; (5.2)
In other words, fi ` fl, i.e., majorization.
Theorem 4.6 holds for the simple g = sl(n+1;C), n ? 1. Let us consider the
inequalities associated with g = sl(n + 1;C). It is easy to see that the projections
? : b ! h and ? : b ! it amount to taking the eigenvalues and the real part of the
eigenvalues, respectively, of the matrices in b. We know that AdK(Y)\b ? it for
53
any Y 2 ik, hence AdK(Y)\ b are the eigenvalues of Y 2 ik (essentially spectral
theorem on Hermitian matrices). For any X 2 sl(n+1;C), if fi = (fi1;:::;fin+1)T 2
?(AdK(X)\b), and fl = (fl1;:::;fln+1)T is the eigenvalues of 12(X +X?) = 12(X ?
(X))(2 ik), then by Theorem 4.6 we know that fi 2 Wfl. By (5.1) and (5.2) we
have
kX
j=1
fi[j] ?
kX
j=1
fl[j]; k = 1;:::;n;
n+1X
j=1
fi[j] =
n+1X
j=1
fl[j] = 0;
where (fi[1];:::;fi[n+1])T;(fl[1];:::;fl[n+1])T 2 (it)+ are the rearrangements of the
entries of fi and fl, respectively, in nonincreasing order. Notice that the rearrange-
ment of fi, say, is simply to map fi to its representative in the fundamental Weyl
(it)+ via the Weyl group action. The result is essentially Ky Fan. If we change
the projection ? to ?1 : g ! t, by similar argument, we get the result of Ky Fan?s
result about the imaginary part of eigenvalues and imaginary singular values. Con-
versely, if fi ` fl, where fi;fl 2 Rn+1, i.e., (5.1) and (5.2) hold for fi and fl, then
Theorem 4.6 on g = sl(n + 1;C) asserts that there exists X 2 g such that the
real part of the eigenvalues of X is majorized by the real singular values of X,
i.e., the result of Amir-Moe?z-Horn and Mirsky up to a translation. Therefore Ky
Fan-Amir-Moe?z-Horn-Mirsky?s result is a special case of Theorem 4.6.
54
Example 5.5 [20, p.85] Consider the simple complex Lie algebra cn, which is
realized as g = sp(n;C) = sp(n)'isp(n) and (X) = ?X?. Recall that
sp(n;C) = fX 2 sl(2n;C) : XTJ +JX = 0g;
where J = Jn;n is the 2n?2n matrix
J =
0
B@ 0 In
?In 0
1
CA:
Recall
sp(n;C) = f
?A
1 A2
A3 ?AT1
!
: A2;A3 2Cn?n complex symmetric; A1 2Cn?ng:
If ? 2C is an eigenvalue of A =
?A
1 A2
A3 ?AT1
!
2 sp(n;C) with eigenvector
x =
?u
v
!
2C2n;
i.e., Ax = ?x, then ?
?v
u
!
2C2n
is also an eigenvector of A corresponding to the eigenvalue ??. Thus the eigen-
values of A 2 sp(n;C) occur in pair but opposite in sign. So do the real singular
values of A. We will see that again when we discuss the projection ? : b ! h.
55
The symplectic group K = Sp(n) consists of the matrices of the form
0
B@ U ?V
V U
1
CA2 U(2n);
and k = sp(n). Thus (X) := ?X? 2 g.
The group G = Sp(n;C) = fg 2 GL(2n;C) : gTJg = Jg, i.e., the group of
matrices g that preserves the bilinear form
hx;yi := x1yn+1 +x2yn+2 +???+xny2n; x;y 2Cn:
Matrices in Sp(n;C) are called symplectic matrices. Indeed Sp(n;C) ? SL(2n;C)
by considering the Pfa?an (See Remark 6.6). As we did in Example 3.13, let
h = fdiag(h1;:::;hn;?h1;:::;?hn) : h1;:::;hn 2Cg:
Let us identify h0 = it with Rn in the natural way:
diag(h1;:::;hn;?h1;:::;?hn) 7! (h1;:::;hn)T:
The positive roots are
fej ?ek : 1 ? j < k ? ng[f2el : 1 ? l ? ng:
56
The Weyl group W of (g;h) acts on it:
(h1;:::;hn)T 7! (?h (1);:::;?h (n))T; 2 Sn:
The simple roots are
fij = ej ?ej+1; j = 1;:::;n?1; fin = 2en;
and the fundamental dominant weights [18, p.67] [20, p.289] are
?k =
kX
j=1
ej; k = 1;:::;n:
The (closed) fundamental Weyl chamber (it)+ is
(it)+ = f(h1;:::;hn)T : h1 ????? hn ? 0g:
The dual cone of (it)+ in it is
dualit(it)+ = f(h1;:::;hn)T 2 it :
kX
j=1
hj ? 0; k = 1;:::;ng:
The condition that fl ?fi 2 dualit(it)+ is equivalent to
kX
j=1
fij ?
kX
j=1
flj; k = 1;:::;n: (5.3)
57
Because of the Weyl group action, if fi = (fi1;:::;fin) 2 it, then
fi+ = (jfij[1];:::;jfij[n]) 2 (it)+;
where jfij = (jfi1j;:::;jfinj). In other words, fi+ is the rearrangement of the ab-
solute values of fi?s in nonincreasing order. By Lemma 5.3, if fi;fl 2 it, then
fi 2 convWfl if and only if
kX
j=1
jfij[j] ?
kX
j=1
jflj[j]; k = 1;:::;n:
We remark that the orthogonal projection ? : b ! h with respective to the Killing
form of g amounts to taking eigenvalues and ? : b ! h is equivalent to taking the
real part of the eigenvalues, since
b =
n
0
B@ A1 A2
0 ?AT1
1
CA; A
1 is upper triangular;A2T = A2
o
:
and h consists of diagonal matrices.
Deflnition 5.6 Let a;b 2 Rn. We say that a is weakly majorized by b, denoted
by a `w b, if
kX
i=1
a[i] ?
kX
i=1
b[i]; k = 1;:::;n;
where a[1] ? a[2] ? ??? ? a[n] and b[1] ? b[2] ? ??? ? b[n] are the rearrangements of
the entries of a and b, respectively, in nonincreasing order.
58
Thus we have the following result which basically asserts that majorization
plays the same role sp(n;C) as in sl(n;C).
Proposition 5.7 The n largest nonnegative real parts of the eigenvalues of an
A 2 sp(n;C) are weakly majorized by the n largest nonnegative real singular
values of A. Conversely given two nonnegative n-tuples fi = (fi1;:::;fin) and fl =
(fl1;:::;fln), if fi `w fl, then there exists an A 2 sp(n;C) such that ?fi1;:::;?fin
are the real parts of the eigenvalues of A and ?fl1;:::;?fln are the real singular
values of A.
59
Chapter 6
The inequalities associated with bn and dn
As we already noticed in Chapter 1, eigenvalues of a skew symmetric matrix
occur in pair, opposite in sign, since A and AT = ?A have the same characteristic
polynomial. We now proceed to investigate the relation between the real parts of
eigenvalues of A and the real singular values of A.
Example 6.1 [18, p.3] In Example 3.12, we used the model so(2n + 1;C) for bn.
In the model g = so(2n+1;C), G = SO(2n+1;C), K = SO(2n+1), k = so(2n+1),
(X) = ?X?, X 2 g. Unlike sl(n;C) and sp(n;C), the form of the Borel subalgebra
b does not make it transparent that ?(Ad(K)X \ b) and ?(Ad(K)X \ b) amount
taking the eigenvalues and the real parts of the eigenvalues of X, respectively. In
order to see that we switch to another model.
Notice that
G = SO(2n+1;C) := fg 2 SL(2n+1;C) : gTg = I2n+1g;
is the group of matrices preserving the symmetric bilinear form [20, p.70]
hx;yi := x1y1 +???+x2n+1y2n+1; x 2C2n+1:
60
If we change the quadratic form to
hx;yi := x1y1 +x2yn+1 +???+xny2n+1; x 2C2n+1;
then the group becomes
~G := fg 2 SL(2n+1;C) : gTJg = Jg;
where
J :=
0
BB
B@
1 0 0
0 0 In
0 In 0
1
CC
CA:
The two groups G and ~G are isomorphic via the isomorphisms
iS(S) : ~G ! G; iS(g) := SgS?1; (6.1)
where
STS = J: (6.2)
SuchS 2 GL(2n+1;C)existsbyTakagi?sfactorization[16, p.204-205], forexample,
S = (1)' e
?i?=4
p2
?iI
n In
In iIn
!
: (6.3)
It may be view as the restriction of the automorphism iS 2 GL(2n+1;C) deflned
by iS(g) := SgS?1, g 2 GL(2n + 1;C). Obviously S 2 GL(2n + 1;C) but not in
G := SO(2n + 1) nor ~G. By matrix difierentiation, the Lie algebra ~g is the set
61
of matrices A such that JA = ?ATJ. Direct computation [18, p.3] leads to the
explicit form of ~g:
~g =
n
0
BB
BB
B@
0 ?bT ?aT
a A1 A2
b A3 ?AT1
1
CC
CC
CA : A1;A2;A3 2Cn?n; A2 = ?A
T
2 ;A3 = ?A
T
3 ;a;b 2C
n
o
;
which can also be deduced from the Lie algebra isomorphism of ~g onto g:
Ad(S) : ~g ! g; (6.4)
in which Ad(S) (abuse of notation) is identifled with the restriction of Ad(S) :
gl(2n+1;C) ! gl(2n+1;C) onto ~g. We will see that the change of model enables
us to see that the projection ~? : ~b !~h amounts to taking eigenvalues, where
~h := fdiag(0;h1;:::;hn;?h1;:::;?hn) : h1;:::;hn 2Cg;
a Cartan subalgebra of ~g. To describe ~b, consider the root space decomposition of
~g with respect to ~h:
~g = ~h _+ _
X
fi2?+~g
fi '~g?fi;
where ~? is the set of all roots of (~g;~h). Notice that
~b := ~h _+ _X
fi2~?+~g
fi
62
is a Borel subalgebra, where the positive roots in ~?+ are
f~ej ? ~ek : 1 ? j < k ? ng[fel : 1 ? l ? ng;
where
~ej(diag(0;h1;:::;hn;?h1;:::;?hn)) = hj; j = 1;2;:::
The root spaces are [34, p.7]
~g~ej?~ek =C
0
BB
B@
0
Ejk
?Ekj
1
CC
CA; ~g?~ej+~ek =C
0
BB
B@
0
Ekj
?Ejk
1
CC
CA;
~g~ej+~ek =C
0
BB
B@
0
0 Ejk ?Ekj
0
1
CC
CA; ~g?~ej?~ek =C
0
BB
B@
0
0
?Ejk +Ekj 0
1
CC
CA;
where 1 ? j < k ? n, and
~g~el =C
0
BB
B@
0 0 ?lT
??l 0 0
0 0 0
1
CC
CA; ~g?~el =C
0
BB
B@
0 ??Tl 0
0 0 0
?l 0 0
1
CC
CA;
where f?l : 1 ? l ? ng is the standard basis of Rn. So
~b =
n
0
BB
B@
0 0 ?uT
u A1 A2
0 0 ?AT1
1
CC
CA : A1;A2 2Cn?n; A1 upper triangular; A2 = ?AT2 ; u 2Cn
o
:
63
Clearly the diagonal elements of each matrix in ~b are its eigenvalues in which the
nonzero ones appear in pair but of opposite signs. Thus the projection ~? : ~b ! ~h
amounts to taking the eigenvalues of the elements in ~b. Decompose
~h =~t'i~t;
where
i~t = fdiag(0;h1;:::;hn;?h1;:::;?hn) : h1;:::;hn 2Rg
and we identify i~t with Rn in the natural way that
diag(0;h1;:::;hn;?h1;:::;?hn) 7! (h1;:::;hn)T;
then the Weyl group W of (~g;~h) acts on i~t by
(h1;:::;hn)T 7! (?h (1);:::;?h (n))T; 2 Sn:
The simple roots are
fij = ej ?ej+1; j = 1;:::;n?1; fin = en:
The fundamental dominant weights [18, p.69] [20, p.289] are
?k =
kX
j=1
ej; k = 1;:::;n?1; ?n = 12
nX
j=1
ej:
64
The (closed) fundamental Weyl Chamber (i~t)+ is identifled as
(i~t)+ = f(h1;:::;hn)T 2 i~t : h1 ? ::: ? hn ? 0g:
The dual cone of (i~t)+ in i~t is identifled as
duali~t(i~t)+ = f(h1;:::;hn)T 2 i~t :
jX
k=1
hk ? 0; j = 1;:::;ng:
By Lemma 5.2, given fi;fl 2 (i~t)+, i.e., fi and fl are identifled as nonnegative
vectors inRn, the condition fi 2 convWfl is equivalent to fl?fi 2 duali~t(i~t)+, i.e.,
kX
j=1
fij ?
kX
j=1
flj; k = 1;:::;n; (6.5)
i.e., fi `w fl.
Lemma 6.2 Let : g1 ! g2 be a Lie algebra isomorphism of g1 onto g2. If g1 is
semisimple, then g2 is semisimple. Moreover,
1. if h1 is a Cartan subalgebra in g1, then (h1) is a Cartan subalgebra in g2,
2. if b1 is a Borel subalgebra in g1, then (b1) is a Borel subalgebra in g2.
Proof: Since adg2( X) = ?adg1X ? ?1, X 2 g1 and tr AB = tr BA, we have
[12, p.131]
Bg2( (X); (Y)) = Bg1(X;Y); X;Y 2 g2:
By Cartan?s criterion of semisimplicity (Theorem 3.4), g2 is semisimple.
65
(1) Clearly (h1) is a subalgebra of g2. We need to prove that (h1) is maximal
abelian and adh2 diagonalizable. First h2 is abelian since isomorphisms respect
bracket:
[ (h1); (h1)]g2 = [h1;h1]g1 = 0:
If H2 2 g2 such that [H2; (h1)]g2 = 0, then
0 = [H2; (h1)]g2 = [ ?1(H2);h1]g1:
Since h1 is maximal abelian, ?1(H2) 2 h1, i.e., H2 2 (h1). So (h1) is maximal
abelian. Since adh1 is diagonalizable, : g1 ! g2 is an isomorphism of g1 onto g2,
and
ad( (h1)) = ?adh1 ? ?1;
ad( (h1)) is diagonalizable. Therefore h2 = (h1) is a Cartan subalgebra of g2.
(2) Let b2 := (b1). Since b1 is a Borel subalgebra, the kth derived subalgebra
b1k = 0 for some positive integer k. By induction the jth derived algebra of (b1)
is
[ (b1)]j = (bj1); j = 1;2;:::
Thus bk2 = [ (b1)]k = (bk1) = 0, i.e., b2 = (b1) is solvable. If b02 b2 is a maximal
solvable subalgebra of g2, then ?1(b02) b1 is solvable subalgebra of g1. Thus
?1(b02) = b1. Hence b02 = (b1) = b2, i.e., b2 = (b1) is a Borel subalgebra of g2.
Lemma 6.3 Let g, b, h be given in Example 3.12. There exists a matrix similarity
: ~g ! g such that
66
1. (~h) = h and (~b) = b,
2. (~h?) = h?,
3. ?? = ? ~?, where ? : g ! h and ~? : ~g !~h are the projections,
4. ?jb : b ! h amounts to take eigenvalues of X 2 b.
Proof: (1)Recallfrom(6.4)thatAd(S) : ~g ! g deflnedbyAd(S)X = SXS?1; X 2
~g. By Lemma 6.2 Ad(S)~h is a Cartan subalgebra of g and Ad(S)~b is a Borel sub-
algebra of g containing Ad(S)~h. By Theorem 4.1, there exists ? 2 Ad(g) such that
?(Ad(S)~b) = b. Now ?(Ad(S)~h) and h are both Cartan subalgebras of the solvable
algebra b, so[18, Theorem16.2]thereexists?0 2 Int(b)forwhich?0??(Ad(S)~h) = h.
But ?0 is the restriction to b of some ? 2 Intg [18, p.84] so that ? ??(Ad(S)~h) = h.
Then := ? ?? ?Ad(S) : ~g ! g is a matrix similarity (thus a Lie algebra isomor-
phism from ~g onto g) satisfying
(~g) = g; (~h) = h; (~b) = b: (6.6)
(2) Since adg( ~X) = ?ad ~gX ? ?1 and tr AB = tr BA, we have [12, p.131]
Bg( ( ~X); (~Y)) = B~g( ~X; ~Y);
where Bg(?;?) and B~g(?;?) are the respective Killing forms of g and ~g. Let ~X 2~h?.
For each Y 2 h, there exists ~Y 2~h such that (~Y) = Y. Thus
Bg( ( ~X);Y) = Bg( ( ~X); (~Y)) = B~g( ~X; ~Y) = 0:
67
So (X1) 2 h?. This proves that (~h?) ? h?. For dimension reason, (~h?) = h?.
(3) Each ~X 2 ~g can be decomposed as ~X = ~X~h + ~X~h?, where ~X~h 2 ~h and
~X~
h? 2~h
?. So
?? ( ~X) = ?? ( ~X~h + ~X~h?) = ?( ( ~X~h)) = ( ~X~h) = (~?( ~X~h + ~X~h?)) = ? ~?( ~X):
Thus ?? = ? ~?.
(4) We have the following commuting diagram.
~b //
~?
??
b
?
??~
h // h
The projection ~?j~b : ~b ! ~h amounts to taking eigenvalues of ~X 2 ~g and is a
matrix similarity which of course preserves eigenvalues. For each X 2 b, there
exist ~X 2 b0 such that ( ~X) = X. So ?(X) = ?? ( ~X) = ?~?( ~X). Since ?(X) 2 h
and ~?( ~X) 2~h is a diagonal matrix having eigenvalues as the diagonal elements. So
? : b ! h amounts to taking eigenvalues because of the form of h.
Proposition 6.4 With the notations in Example 3.12 and given X 2 so(2n+1;C),
?(Ad(K)X \ b) and ?(Ad(K)X \ b) amount taking the eigenvalues and the real
parts of the eigenvalues of X, respectively.
Proof: Let : ~g ! g be in Lemma 6.3. Let g 2 GL(2n + 1;C) such that
( ~X) = g ~Xg?1 = Ad(g) ~X, X 2 ~g since the groups under discussion are matrix
groups. Let ~K := g?1Kg = ig?1(K). For each X 2 g, set ~X = Ad(g?1)X 2 ~g.
68
Then
h 3 ?(Ad(K)X \b) = ? ~?? ?1(Ad(g ~Kg?1)Ad(g) ~X \Ad(g)~b)
= Ad(g)? ~??Ad(g?1)(Ad(g)Ad( ~K) ~X \Ad(g)~b)
= g~?(Ad( ~K) ~X \~b)g?1
Now Ad( ~K) ~X \~b yields the eigenvalues of ~X = g?1Xg, i.e., the eigenvalues of X.
We conclude that ?(Ad(K)X \b) yields the eigenvalues of X because of the form
of h. The argument for ? is similar.
So we have the following result for so(2n+1;C) by using Theorem 4.6.
Proposition 6.5 The n nonnegative real parts of the eigenvalues of a (2n +
1) ? (2n + 1) complex skew symmetric matrix A are weakly majorized by the
n nonnegative real singular values of A. Conversely given two nonnegative n-
tuples fi = (fi1;:::;fin) and fl = (fl1;:::;fln), if fi `w fl, then there exists a
(2n + 1)?(2n + 1) skew symmetric matrix A such that ?fi1;:::;?fin;0 are the
real parts of the eigenvalues of A and ?fl1;:::;?fln;0 are the real singular values
of A.
Remark 6.6 To facilitate further discussion we introduce a notion known as Pfaf-
flan [10, Appendix D] of a complex skew symmetric matrix. Let X = (xij) be a
complex skew-symmetric matrix, i.e., XT = ?X. If X 2 so(2n+1;C), then
det(X) = det(?XT) = (?1)ndetX = 0:
69
On the other hand, if X 2 so(2n;C), then its determinant is a perfect square:
detX = Pf(X)2;
where
Pf(X) :=
X
2S2n
sgn( )x (1) (2) ?:::?x (2n?1) (2n)
such that (2r ? 1) < (2r) for 1 ? r ? n, and (2r ? 1) ? (2r + 1) for
1 ? r ? n ? 1. There are (2n ? 1) ? (2n ? 3) ? ::: ? 3 ? 1 terms in this sum.
Equivalently,
Pf(X) = 12nn!
X
sgn( )x (1) (2) ?:::?x (2n?1) (2n):
Example 6.7
Pf
0
B@ 0 x12
?x12 0
1
CA = x
12;
Pf
0
BB
BB
BB
BB
@
0 x12 x13 x14
?x12 0 x23 x24
?x13 ?x23 0 x34
?x14 ?x24 ?x34 0
1
CC
CC
CC
CC
A
= x12x34 ?x13x24 +x14x23;
and
Pf
"? 0 x
1
?x1 0
!
'???'
? 0 x
n
?xn 0
!#
= x1???xn:
The following are some properties of Pfa?an.
Proposition 6.8 For any A 2 so(2n;C).
70
1. Pf(A)2 = det(A).
2. Pf(BABT) = det(B)Pf(A), B 2C2n?2n.
3. Pf(?A) = ?nPf(A), ? 2C.
4. Pf(AT) = (?1)nPf(A)
5. Pf(A1 'A2) = Pf(A1)Pf(A2).
6. For any M 2Cn?n,
Pf
? 0 M
?MT 0
!
= (?1)n(n?1)=2detM:
It follows from Proposition 6.8 (2) that the determinant of any symplectic matrix
A, i.e., ATJA = J, is 1:
Pf(J) = Pf(ATJA) = det(A)Pf(J):
If X 2 so(2n;C) and A 2C2n?2n, then AXAT 2 so(2n;C) and
Pf(AXAT) = (detA)?Pf(X):
For any Y 2 g = so(2n;C) and any k 2 SO(2n), then k?1 = kT, detk = 1 and
hence Adk(Y) = kYk?1 = kYkT, we have
Pf(Y) = Pf(Adk(Y)): (6.7)
71
So the Pfa?an is an invariant polynomial of a skew-symmetric matrix and is
invariant under under special orthogonal similarity. It is important in the theory
of characteristic classes. In particular, it can be used to deflne the Euler class of a
Riemannian manifold which is used in the generalized Gauss-Bonnet theorem.
Example 6.9 [20, p.85] Similar to the Lie algebra ~g in Example 6.1, we choose
another model ~g for Lie algebra dn, which is equivalent to the model g = so(2n;C)
in Example 3.14. Let
~g =
n
0
B@ A1 A2
A3 ?AT1
1
CA : A
1;A2;A3 2Cn?n; AT2 = ?A2;AT3 = ?A3
o
:
Then
~h = fdiag(h1;:::;hn;?h1;:::;?hn) : hj 2C;j = 1;:::;ng
is a Cartan subalgebra. The positive roots are
fej ?ek : 1 ? j < k ? ng:
The root spaces are
~gej?ek =
0
B@ Ejk 0
0 ?Ekj
1
CA; ~g?ej+ek =
0
B@ Ekj 0
0 ?Ejk
1
CA;
~gej+ek =
0
B@ 0 Ejk ?Ekj
0 0
1
CA; ~g?ej?ek =
0
B@ 0 0
?Ejk +Ekj 0
1
CA;
72
where 1 ? j < k ? n. Identify ~h0 = i~t with Rn in the natural way. The Weyl
group W of (~g;~h) acts on ~h0:
(h1;:::;hn)T 7! (?h (1);:::;?h (n))T; 2 Sn;
where the number of negative signs is even. The simple roots are
fij = ej ?ej+1; j = 1;:::;n?1; fin = en?1 +en;
and the fundamental dominant weights [18, p.69] [20, p.289] are
?j =
kX
j=1
ej; k = 1;:::;n?2; ?n?1 = 12(
n?1X
j=1
ej ?en); ?n = 12
nX
j=1
ej:
The (closed) fundamental Weyl chamber (i~t)+ is
(i~t)+ = f(h1;:::;hn)T : h1 ? ::: ? hn?1 ?jhnjg:
The dual cone of (i~t)+ in i~t is
duali~t(i~t)+ = f(h1;:::;hn)T 2 i~t :
kX
j=1
hj ? 0; k = 1;:::;n?1;
n?1X
j=1
hj ?hn ? 0g:
The condition that fl ?fi 2 duali~t(i~t)+ amounts to the following inequalities.
kX
j=1
fij ?
kX
j=1
flj; k = 1;:::;n?2;
73
n?1X
j=1
fij ?fin ?
n?1X
j=1
flj ?fln
nX
j=1
fij ?
nX
j=1
flj
The inequalities are equivalent to
kX
j=1
fij ?
kX
j=1
flj; k = 1;:::;n; (6.8)
n?1X
j=1
fij ?fin ?
n?1X
j=1
flj ?fln: (6.9)
When fi;fl 2 (it)+, i.e., fi1 ? ??? ? fin?1 ? jfinj and fl1 ? ??? ? fln?1 ? jflnj, (6.8)
and (6.9) are equivalent to
fi `w fl; i:e:;
kX
j=1
fij ?
kX
j=1
flj; k = 1;:::;n;
n?1X
j=1
fij ?fin ?
n?1X
j=1
flj ?fln
The above condition is clearly stronger than weakly majorization.
Similar to the simple Lie algebra bn, there is an matrix similarity between
the two models ~g in Example 6.9 and so(2n;C) of dn which preserves eigenvalues.
Moreover we have the following commuting diagram.
~b //
~?
??
b
?
??~
h // h
74
Proposition 6.10 With the notations in Example 3.14 and given X 2 so(2n;C),
?(Ad(K)X \ b) and ?(Ad(K)X \ b) amount taking the eigenvalues and the real
parts of the eigenvalues of X, respectively.
Let Y 2 ik = isp(n) have eigenvalues ?fl1;:::;?fln such that fl1;:::;fln are
nonnegative. There exists k 2 SO(2n) [16, p.107] such that
Adk(Y) =
? 0 ifl
1
?ifl1 0
!
'???'
? 0 ifl
n?1
?ifln?1 0
!
'
? 0 ?ifl
n
??ifln 0
!
2 it;
where ? = ?1. So by (6.7) and Example 6.7
Pf(Y) = ?infl1???fln?1fln;
and ? = sign[(?i)nPf(Y)] is uniquely determined by Y.
Proposition 6.11 Let A 2 so(2n;C) and let ?fl1;:::;?fln be the real singular
values of A with flj ? 0;j = 1;:::;n. Suppose
? 0 ifi
1
?ifi1 0
!
'???'
? 0 ifi
n?1
?ifin?1 0
!
'
? 0 ifi
n
?ifin 0
!
2 ?(AdK(A)\b);
sothat?fi1;:::;?fin aretherealpartoftheeigenvaluesofA. Setjfij = (jfi1j;:::;jfinj).
Then
kX
j=1
jfij[j] ?
kX
j=1
fl[j]; k = 1;:::;n?2;
n?1X
j=1
jfij[j] +sign(fi1???fin)jfij[n] ?
n?1X
j=1
fl[j] +[signf(?i)nPf(12(A+A?))g]fl[n];
75
n?1X
j=1
jfij[j] ?sign(fi1???fin)jfij[n] ?
n?1X
j=1
fl[j] ?[signf(?i)nPf(12(A+A?))g]fl[n];
(thesignofzeromaybetaken1or?1). Conversely, suppose(fi1;:::;fin)T;(fl1;:::;fln)T 2
Rn satisfying the inequalities,
kX
j=1
jfij[j] ?
kX
j=1
jflj[j]; k = 1;:::;n?2;
n?1X
j=1
jfij[j] +sign(fi1???fin)jfij[n] ?
n?1X
j=1
jflj[j] +sign(fl1???fln)jflj[n];
n?1X
j=1
jfij[j] ?sign(fi1???fin)jfij[n] ?
n?1X
j=1
jflj[j] ?sign(fl1???fln)jflj[n];
or equivalently
kX
j=1
jfij[j] ?
kX
j=1
jflj[j]; k = 1;:::;n?2;
n?1X
j=1
jfij[j] +jfij[n] ?
n?1X
j=1
jflj[j] +sign(?nj=1(fijflj))jflj[n];
n?1X
j=1
jfij[j] ?jfij[n] ?
n?1X
j=1
jflj[j] ?sign(?nj=1(fijflj))jflj[n];
where jfij[j] and jflj[j]; j = 1;:::;n; are the rearrangements of the entries of jfij and
jflj, respectively, in nonincreasing order. Then we can flnd A 2 so(2n;C) such that
?fi?s are the real part of the eigenvalues of A, ?fl?s are the real singular values of
A and
sign[(?i)nPf(12(A+A?))] = sign(fl1???fln):
76
Proof: Since
Y :=
? 0 ifi
1
?ifi1 0
!
'???
? 0 ifi
n?1
?ifin?1 0
!
'
? 0 ifi
n
?ifin 0
!
2 ?(AdK(A)\b);
the element Y+ 2 (it)+ \WY is of the form
Y+ =
? 0 ijfij
[1]
?ijfij[1] 0
!
'???'
? 0 ijfij
[n?1]
?ijfij[n?1] 0
!
'
? 0 ?ijfij
[n]
??ijfij[n] 0
!
;
where ? = sign(fi1???fin). Similarly, there exists Z+ 2 AdK(12(A + A?) \ (it)+
such that
Z+ =
? 0 ijflj
[1]
?ijflj[1] 0
!
'???'
? 0 ijflj
[n?1]
?ijflj[n?1] 0
!
'
? 0 ?0ijflj
[n]
??0ijflj[n] 0
!
;
where ?0 = signfPf((?i)n12(A+A?))g. Now Y+ 2 convWZ+ by Theorem 4.6 and
Lemma 5.3. So Z+ ? Y+ 2 dualit(it)+ by Lemma 5.2. Under the identiflcation
(jfij[1];:::;jfij[n?1];?jfij[n]) and (jflj[1];:::;jflj[n?1];?0jflj[n]) satisfy (6.8) and (6.9),
i.e.,
(jfij[1];:::;jfij[n?1];?jfij[n])T `w (jflj[1];:::;jflj[n?1];?0fl[n])T;
n?1X
j=1
jfij[j] ??jfij[n] ?
n?1X
j=1
fl[j] ??0jflj[n];
which are equivalent to the flrst set of inequalities.
Conversely, if fi;fl 2Rn satisfy the second set of inequalities, then
(jfij[1];:::;jfij[n?1];sign(fi1???fin)jfij[n])T `w (jflj[1];:::;jflj[n?1];sign(fl1???fln)fl[n])T;
77
n?1X
j=1
jfij[j] ??jfij[n] ?
n?1X
j=1
fl[j] ??0jflj[n]:
Thus
(jfij[1];:::;jfij[n?1];sign(fi1???fin)jfij[n])T 2 convW(jflj[1];:::;jflj[n?1];sign(fl1???fln)fl[n])T:
Then by Theorem 4.6 and Proposition 6.12 there exists A 2 so(2n;C) such that
the real parts of the eigenvalues are ?fi1;:::;?fin and the real singular values
are ?fl1;:::;?fln, and by the invariance of the Pfa?an under adjoint action of
K = SO(2n) we have
sign[(?i)nPf(12(A+A?))] = sign(fl1???fln):
Notice that sign(ab) = sign(a)sign(b) for any real numbers a and b. If ?1 = 1,
the second set of inequalities is exactly the third set of inequalities. If ?1 = ?1,
the last two inequalities are identical to the last two inequalities in the second set.
Thus the last set of three inequalities is equivalent to the second set. Hence if the
last three equations are true, then there exists A 2 so(2n) satisfying the required
conditions.
Proposition 6.12 Let A 2 so(2n;C) and let fi1;:::;fin be the largest n nonnega-
tive real part of the eigenvalues of A and let fl1;:::;fln be the largest n nonnegative
real singular values of A. Then either
1. (fi1;:::;fin?1;fin)T 2 convW(fl1;:::;fln)T, or
78
2. (fi1;:::;fin?1;?fin)T 2 convW(fl1;:::;fln)T.
Conversely if fi;fl 2Rn have the above relationship, then there exists A 2 so(2n;C)
such that?fi1;:::;?fin are the real parts of the eigenvalues of A and?fl1;:::;?fln
are the real singular values of A.
Proof: Since fi1;:::;fin are the largest n nonnegative real part of the eigenvalues
of A and fl1;:::;fln are the largest n real singular values of A, by considering the
action of the Weyl group on A, either (fi1;:::;fin)T or (fi1;:::;fin?1;?fin)T (but
usually not both) is in ?(AdK(A)\b) under the identiflcation, where K = SO(2n).
Similarly either (fl1;:::;fln)T or (fl1;:::;fln?1;?fln)T is in AdK(12(A + A?)) \ it.
By Theorem 4.6 we have the following four possibilities
(fi1;:::;fin?1;?1fin)T 2 convW(fl1;:::;fln?1;?2fln)T;
where ?1 = ?1 and ?2 = ?1. But
(fi1;:::;fin?1;?1fin)T 2 convW(fl1;:::;fln?1;?fln)T
is the same as
(fi1;:::;fin?1;??1fin)T 2 convW(fl1;:::;fln?1;fln)T:
Thus either
1. (fi1;:::;fin?1;fin)T 2 convW(fl1;:::;fln)T, or
2. (fi1;:::;fin?1;?fin)T 2 convW(fl1;:::;fln)T.
79
Conversely, if either of the conditions is true, then Theorem 4.6 and Proposition
6.12 guarantee the existence of the required A.
Remark 6.13 Proposition 6.5 is no longer true for dn. We can see this clearly
from the following example for n = 2. Let fi = (1=2;0)T and fl = (1;1)T. Ob-
viously fi `w fl. By Proposition 6.12, there is no A 2 so(2;C) with the real
part of the eigenvalues ?fi?s and real singular values ?fl?s. See the following flg-
ure: L1 = convW(1;1)T, L2 = convW(1;?1)T, and the square with vertices
(1;1);(1;?1);(?1;1);(?1;?1) (the shaded area) is the set of all the vectors, in
particular fi = (1=2;0)T, weakly majorized by fl = (1;1)T. But the union of L1
and L2 is a \cross" which is clearly not convex.
?=(1,1)
(1,?1)
?=(1/2,0)
L1
L2
x
Figure 6.1: The Union of the Convex Hulls: L1 [L2
80
Chapter 7
The real semisimple case
The proof [1] given by Amir-Mo?ez and Horn for the converse of Ky Fan?s result
also works for sl(n;R), a normal real form of sl(n;C). However the study would
be intricate for real semisimple Lie algebras. Theorem 4.6 concerns a complex
semisimple Lie algebra g, a Cartan subalgebra h and a Borel subalgebra b. In the
complex semisimple case g, all maximal solvable subalgebras in g are conjugate via
theadjointgroup AdGof g [18, Section 16.4] (itis also true for Cartan subalgebras).
The Borel subalgebra b in Section 2 and 3 is the \standard" one with respect to
the chosen Cartan subalgebra h = t'it and the basis ? for the root system ?.
One may consider the real semisimple case. From now on g denotes a real
semisimple Lie algebra with Killing form B(?;?). A subalgebra h is called a Car-
tan subalgebra of g if the complexiflcation hC of h is a Cartan subalgebra of the
complexiflcation gC of g [20, p.318]. It is well known [21] [31, p.397] that Cartan
subalgebras of a real semisimple Lie algebra are not conjugate in general (un-
less g is compact). But there are only flnitely many conjugacy classes of Cartan
subalgebras [31, p.395].
Example 7.1 Let g = sl(2;R) and let
a =R
?1 0
0 ?1
!
; b =R
? 0 1
?1 0
!
:
81
Then a and b are Cartan subalgebras of g. They cannot be conjugate in g since
ea = fdiag((e?;e??) : ? 2Rg is not compact, but eb = SO(2) is compact.
Borel subalgebras in a complex Lie algebra are (complex) maximal solvable
algebras and there is only one conjugacy class since all Borel subalgebras are con-
jugate. However, in the real case, there are difierent conjugacy classes of maximal
solvable subalgebras [25, 26, 27]. The conjugacy classes of maximal solvable sub-
algebras may be obtained via the non-conjugate Cartan subalgebras [25] and the
procedure will be outlined.
A decomposition
g = k'p
into a direct sum is called a Cartan decomposition if
1. the map : X + Y 7! X ?Y (X 2 k;Y 2 p) is an automorphism of g, i.e.,
2 Aut(g).
2. The bilinear form B (X;Y) = ?B(X; Y) is positive deflnite on g.
Since 2 = 1, B is a symmetric bilinear form. The flrst condition amounts to the
following:
[k;k] ? k; [k;p] ? p; [p;p] ? k:
So k and p are orthogonal to each other under B(?;?) and under B (?;?). Since
B (?;?) is positive deflnite, B(?;?) is negative deflnite on k and positive deflnite on
p. The subspace p ? g is called a Cartan subspace of g and k is a subalgebra of g.
An involutory automorphism 2 Aut(g) such that the symmetric bilinear
form B (X;Y) = ?B(X; Y) is positive deflnite is called a Cartan involution. A
82
Cartan involution determines a Cartan decomposition of g and vice versa. The
importance of the Cartan decomposition is that it is unique up to conjugacy, i.e.,
if g = k0 ' p0 is another Cartan decomposition, there exists ? 2 Int(g) such that
k0 = ?(k) and p0 = ?(p) [20, p.301].
For a complex semisimple Lie algebra u, the only Cartan involutions of its
realiflcation uR are the conjugations with respect to the compact real forms of u
[20, p.302].
Let g = k _+p be the Cartan decomposition associated with the Cartan involu-
tion , that is, k is the +1 eigenspace of and p is the ?1 eigenspace of . Since
the adjoint of adX is [20, p.304]
(adX)? = ?ad X; X 2 g;
adX is represented by a symmetric matrix if X 2 p, and by a skew symmetric
matrix if X 2 k. Fix a maximal abelian subspace ap in p. For any H 2 ap ? p,
(adH)? = adH;
and hence adap is a commuting family of self-adjoint transformations of g. So [16,
p.103] g is the orthogonal direct sum of the subspaces
gfi = fX 2 g : (adH)X = fi(H)X for all H 2 apg; fi 2 a?p
If fi 6= 0 and gfi 6= 0, we call fi a restricted root of (g;ap). The set of all restricted
roots is denoted by ?. Any nonzero gfi is called a restricted root space, and each
83
member of gfi is called a restricted root vector for the restricted root fi. The
decomposition of g,
g = g0 _+ _
X
fi2?+(g
fi 'g?fi); (7.1)
is called the restricted root space decomposition of g relative to ap [20, p.313], where
?+ is the set of restricted positive roots (with respect to a flxed base ? of ?) of
the root system ? of (g;ap). Unlike the root system for a complex semisimple
Lie algebra, the restricted root system of (g;ap) need not be reduced, i.e., it may
happen that fi 2 ? and 2fi 2 ?. Moreover dimgfi may be larger than 1 and usually
g0 is bigger than ap. We also have the orthogonal sum [20, p.313]
g0 = ap _+m;
where m = Zk(ap), the centralizer of ap in k. Hence
k\g0 = m; p\g0 = ap:
The root system ? does not of itself determine g up to isomorphism. The Weyl
group of (g;ap) which may be deflned as the normalizer of ap in K modulo the
centralizer of ap in K, will be denoted by W.
Since Ad(K) preserves the Killing form and k, Ad(K)ap ? p. The following
is the original Kostant?s convexity theorem [22] and Theorem 4.5 is a particular
case.
Theorem 7.2 (Kostant) Let Z 2 ap, then ?(Ad(K)Z) = convWZ where ? : p !
ap is the orthogonal projection with respect to the Killing form.
84
Since is?1on p andthuson ap, and[H; X] = [ H;X] = ? [H;X] = ?fi(H) X
for all X 2 gfi, H 2 ap, we have [20, 313]
Proposition 7.3 (ap) = ap and (gfi) = g?fi for all fi 2 ?.
Let
n =
X
fi2?+
gfi
which is a nilpotent subalgebra of g. Let a be a maximal abelian subalgebra of g
containing ap. Then a is a Cartan subalgebra. We have ap = a \ p and if we put
ak := a\k, then a = ak 'ap. Let
b := a _+n = a _+ _
X
fi2?+g
fi
which is a maximal solvable subalgebra of g.
Theorem 7.4 Let g = k _+p be a Cartan decomposition associated with the Cartan
involution . Let ap be a maximal abelian subspace of p. Let g = g0 _+ _Pfi2?+(gfi '
g?fi) be the restricted root space decomposition of the real semisimple Lie algebra
g with respect to ap and set b = a _+ _Pfi2?+gfi. Let ? : g ! ap be the orthogonal
projection with respect to B (?;?). Then for each fl 2 ap,
?((k +AdK(fl))\b) = convWfl:
Proof: Notice that
?((k +AdK(fl))\b)) ? ?(k +AdK(fl)) = ?(AdK(fl)) = convWfl
85
by Theorem 7.2.
Suppose 2 convWfl, where ;fl 2 ap. By Theorem 7.2 again, there exists
Y 2 AdK(fl) such that ?(Y) = . Let Y = Y0 + Pfi2?+(Yfi + Y?fi), where
Y0 2 ap = g0 \ p, Yfi 2 gfi, Y?fi 2 g?fi for fi 2 ?+. Since Y 2 p, p is the ?1
eigenspace of , and gfi = g?fi by Proposition 7.3, we have
?Y0 +
X
fi2?+
(?Yfi ?Y?fi) = ?Y = (Y) = Y0 +
X
fi2?+
( Yfi + Y?fi):
Since the sums are direct, Y?fi = ? Yfi. Then Y = Y0 +Pfi2?+(Yfi ? Yfi), and
Y0 = ?(Y) = . Similar to the proof of Theorem 4.6, set X := Y0+2Pfi2?+Yfi 2 b.
Clearly ?(X) = Y0 = , 12(X ? X) = Y, and fl 2 AdK(Y)\ap.
Remark 7.5 Theorem 7.4 provides Amir-Mo?ez-Horn-Mirsky?s type result for the
real semisimple Lie algebras. The algebra b := a _+ _Pfi2?+gfi will be called the
standard maximal solvable subalgebra of g containing ap. For example, when
g = sl(n;R), a = ap may be chosen as the space of real diagonal matrices and
ak = 0.
Unlike the complex case, given X 2 g, the adjoint orbit AdK(X) may not
intersect b; in this case Ky-Fan?s type result would be trivial. For example, consider
g = sl(2;R), b the algebra of upper triangular matrices, X =
? a b
?b a
!
(b 6= 0)
whose eigenvalues are in complex conjugate pair. In general, in sl(n;R) one would
encounter the same problem and the following result in matrix theory is well known
[17, p.82].
86
Proposition 7.6 For any X 2 sl(n;R), there exists k 2 SO(n) such that kXk?1
is of block upper triangular form where the (main diagonal) blocks are either 1?1
or 2?2: 0
BB
BB
BB
BB
@
A1 ?
A2
...
As
1
CC
CC
CC
CC
A
;
with zero trace, where Ak =
? a
k bk
?bk ak
!
, or Ak = (ck), ak;bk;ck 2R, k = 1;:::;s.
IndeedtheaboveformsinProposition7.6areassociatedwiththesocalledstandard
maximal solvable subalgebras of sl(n;R). Since each conjugacy class is uniquely
determined by (degA1;degA2;:::;degAs) where Ak = 1 or 2, k = 1;:::;s. There
are
Nn = 1p5
"?
1+p5
2
!n
?
?
1?p5
2
!n#
(the Fibonacci number deflned by Nn = Nn?1 +Nn?2, N1 = 1, N2 = 2) conjugacy
classes of maximal solvable subalgebras [26, p.1032] of sl(n;R).
Motivated by Proposition 4.2 and the case sl(n;R) in Proposition 7.6, we now
ask whether for any element X 2 g, AdK(X) intersects some standard maximal
solvable subalgebra s.
To be speciflc, we introduce some notions. Fix a Cartan decomposition of the
real semisimple Lie algebra g = k _+p and let be the associated Cartan involution.
Fix a maximal abelian subspace ap in p .
87
A Cartan subalgebra c is called a standard Cartan subalgebra (relative to
and ap) [28, p.405]. If c = ck _+cp, where ak ? ck := c\k and cp := c\p ? ap. In this
case ck is called the toral part, and cp is called the vector part [31, p.379].
Sugiura [31, Theorem 2, Theorem 3] proved that
Theorem 7.7 (Sugiura) Let g be a real semisimple Lie algebra.
1. Every Cartan subalgebra of g is conjugate to a standard Cartan subalgebra
via Int(g).
2. Two standard Cartan subalgebras are conjugate via Int(g) if and only if their
vector parts are conjugate under the Weyl group W of (g;ap).
Therefore it is su?cient to consider standard Cartan subalgebras in order to
flnd the conjugacy classes of Cartan subalgebras. Rothschild [28, p.405] showed
that two standard Cartan subalgebras are conjugate if and only if their toral parts
are conjugate.
The idea of (1) in Theorem 7.7 is as follow: If c is any Cartan subalgebra of g,
there exists a conjugate of c which is -stable, i.e., c = ck _+cp such that ck = c\k and
cp = c\p. The vector part cp is an abelian subalgebra of p and hence is contained
in some maximal abelian subalgebra in p. By conjugating c via K, we may arrange
that cp ? ap, and then by conjugating again, leaving cp flxed, we can also arrange
that ak ? ck. One has [12, p.259]
Proposition 7.8 (Sugiura) Any maximal abelian subalgebra a of g that contains
ap is a Cartan subalgebra of g and ap = a\p and ak = a\k.
88
A result of Mostow [25, Theorem 4.1] asserts that each maximal solvable
subalgebra s of g contains a Cartan subalgebra c, for example, compact Cartan
subalgebra of g is maximal solvable [25, Lemma 4.1]. If c is a standard Cartan
subalgebra, such s is called a standard maximal solvable subalgebra (with respect
to ap and ). Each G-conjugate of s is still a maximal solvable subalgebra, due to
Cartan?s criterion of solvability [20, Proposition 1.43], i.e., Theorem 3.5 remains
true for real case, and the adjoint action of G respects the bracket and preserves the
Killing form. Thus each conjugacy class of maximal solvable subalgebras under the
adjoint action of G contains a standard maximal solvable subalgebra s by Theorem
7.7.
Let c = ck _+cp be a standard Cartan subalgebra of a real semisimple Lie algebra
g. Since cp ? ap, adcp is a family of commuting self adjoint linear transformations
of g. So g has a root space decomposition with respect to cp:
g = g0 _+
X
fi2R
gfi;
where gfi := fX 2 g : [H;X] = fi(H)X for all H 2 cpg and R ? cp? is the
set of roots which do not vanish identically on cp. An element H 2 cp is called
cp-singular if there exists fi 2 R such that fi(H) = 0, otherwise H is called cp-
general. A connected component in the set of cp-general elements of cp is called a
cp-chamber. For any H 2 cp, fi 2 ?, set
gfi(H) := fX 2 g : (adH ?fi(H))X = 0g:
89
Let C be a cp-chamber in cp. Then gfi(H) is independent of the choice of H in C
and thus may be written as gfi(C). Set
g+(H) :=
X
fi>0
gfi(H) =
X
fi>0
gfi(C)
which is also independent of the choice of H in C so it may be denoted by g+(C).
The following is Mostow?s result [23, Theorem 4.1].
Theorem 7.9 (Mostow) Let g be a real semisimple Lie algebra.
1. Any maximal solvable subalgebra of g contains a Cartan subalgebra. Hence
it is conjugate via Int(g) to a standard maximal solvable subalgebra.
2. Any maximal solvable subalgebra of g containing a standard Cartan subal-
gebra c = ck _+cp is of the form c + g+(C) for some cp-chamber C.
Let N := fg 2 G : Ad(g)cp = cpg and NK := fk 2 K : Ad(k)c = cg. Then the
cp-restrictions of N and NK coincide [23, p.515]. Notice that g+(C1) and g+(C2)
are conjugate for two difierent cp-chambers C1 and C2 if and only if the chambers
are conjugate under the cp-restriction of NK.
Example 7.10 For the special case cp = ap, the cp-chambers are the Weyl cham-
bers which are permuted transitively by the Weyl group. So all standard maximal
solvable subalgebras containing the standard Cartan subalgebra a = ak _+ap are
conjugate. One may pick b = (ak _+ap) _+Pfi>0 gfi in Theorem 7.4.
90
To list all non-conjugate standard maximal solvable subalgebras of g, we flrst flnd
the standard Cartan subalgebras c = ck _+cp. Then flnd the non-conjugate cp-
chambers of c, which yields all the standard maximal solvable subalgebras of g
containing c via Theorem 7.9.
Example 7.11 When g = sl(3;R), there are two conjugacy classes of Cartan
subalgebras [20] represented by the standard Cartan subalgebras:
c1 =
n
0
BB
B@
a 0 0
0 b 0
0 0 ?a?b
1
CC
CA : a;b 2R
o
; c2 =
n
0
BB
B@
a b 0
?b a 0
0 0 ?2a
1
CC
CA : a;b 2R
o
:
However there are three conjugacy classes of maximal solvable subalgebras [26,
p.1032] represented by the standard maximal solvable subalgebras:
s1 :=
n
0
BB
B@
a c e
0 b d
0 0 ?a?b
1
CC
CA : a;b;c;d;e 2R
o
;
s2 :=
n
0
BB
B@
a b c
?b a d
0 0 ?2a
1
CC
CA : a;b;c;d 2R
o
;
s3 :=
n
0
BB
B@
?2a c d
0 a b
0 ?b a
1
CC
CA : a;b;c;d 2R
o
:
91
Notice that s2 and s3 contain conjugate standard Cartan subalgebras corresponding
to c2.
We now address the question whether for any X 2 g, AdK(X) \ s 6= ` for
some standard maximal solvable subalgebra s of g.
Proposition 7.12 If g is a compact semisimple Lie algebra, then for any X 2 g,
AdK(X)\s 6= ` for some standard maximal solvable subalgebra s of g.
Proof: When g is compact we have g = k, p = 0, ap = 0 and ak = t which is a
Cartan subalgebra. By [25, Lemma 4.1] t is a maximal solvable subalgebra of k.
Since maximal tori are conjugate in compact Lie group [20, p.202], AdK(X)\t 6= `
for any X 2 k.
In this case Theorem 7.4 is reduced to Theorem 5.2.
Proposition 7.13 In general it is not true that given arbitrary X 2 g, AdK(X)
intersects s for some standard maximal solvable subalgebra s of g.
Proof: Consider the real simple Lie algebra g = su1;1. We consider the group:
SU(1;1) =
n?fi fl
fl fi
!
: jfij2 ?jflj2 = 1
o
;
whose Lie algebra is a real form of sl(2;C):
su1;1 =
n?ia c
c ?ia
!
: a 2R;c 2C
o
;
K =
n
diag(ei ;e?i ) : 2R
o
;
92
k =
n?ia 0
0 ?ia
!
: a 2R
o
;
p =
n?0 c
c 0
!
: c 2C
o
;
ap =
n?0 b
b 0
!
: b 2R
o
:
There are two non-conjugate standard Cartan subalgebras: k and ap [31, p.401].
They are also the two [27, p.518] standard maximal solvable subalgebras of su1;1.
Since k is a compact Cartan subalgebra of su1;1, it is maximal solvable [25, Lemma
4.1]. We know that all maximal solvable subalgebras containing ap are conjugate.
Let us consider the root space decomposition of g = su(1;1) with respect to ap.
The roots are R = ffi;?fig [20, p.314], where
fi
0
B@ 0 b
b 0
1
CA = 2b; b 2R; gfi =R
0
B@ ?i i
?i i
1
CA; g?fi =R
0
B@ ?i ?i
i i
1
CA:
Let
H =
0
B@ 0 1
1 0
1
CA
(indeed any cp-general element H in cp works). By Theorem 7.9,
s = ap + gfi(H) =R
0
B@ ?ia ia+b
?ia+b ia
1
CA; a;b 2R:
93
is the only standard maximal solvable subalgebra containing ap. Let
X =
0
B@ ?i ?
? i
1
CA; 0 < ? < 1:
Clearly k\AdK(X) = `. Now s\AdK(X) = `, otherwise let
Ad
0
B@ ei 0
0 e?i
1
CAX =
0
B@ ?i ?e?i2
?ei2 i
1
CA
=
0
B@ ?ia ia+b
?ia+b ia
1
CA2 s; for some a;b; 2R:
Thus a = 1, and ?e?i2 = ia+b = i+b. Then ? = ji+bj? 1, a contradiction.
Proposition 7.14 Let c = ck _+cp be a standard Cartan subalgebra of g. Let
?cp : g ! cp be the orthogonal projection onto cp ? ap. Then for any fl 2 cp,
convWfl \cp = ?cp(AdK(fl)) = ?cp(convWfl);
Proof: ThesecondequalityfollowsimmediatelyfromTheorem7.2. If 2 convWfl\
cp, by Theorem 7.2 there is k 2 K such that = ?ap(Ad(k)fl). Now 2 cp implies
= ?cp( ) = ?cp(?ap(Ad(k)fl)) = ?cp(Ad(k)fl). So we have
convWfl \cp ? ?cp(AdK(fl)):
94
To establish the converse inclusion, we will show that convWfl is symmetric
with respect to cp, i.e., if 2 convWfl such that = 1 + 2, where 1 2 cp and
2 2 c?p \ap, then 1 ? 2 2 convWfl.
Let a be the maximal abelian subalgebra containing ap. By Proposition 7.8
a = ak + ap is a Cartan subalgebra of g, where ak = a \ k and ap = a \ p. Let
h := aC and gC be the complexiflcations of a and g respectively. The root space
decomposition of gC with respect to h is
gC = h +
X
fi2?
(gC)fi;
where ? ? h?0 is the root system of (gC;h) and h0 := ap + iak is spanned by Hfi,
fi 2 ? (see Proposition 3.8). Let ? : h?0 ! a?p be the restriction to ap. Then for
each fi 2 ?, either ?(fi) is zero or is an element of ? (that is why ? is called the
restricted root system of (g;ap)) [12, p.260-263].
Since cp is the vector part of some standard Cartan subalgebra c, by [31,
Theorem 5], there exist ? roots fi1;:::;fi? 2 ? such that
1. fii ?fij 62 ?, 1 ? i;j ? ? and fii ?fij 6= 0 if i 6= j.
2. c?p \ap = P?i=1RHfii.
In particular Hfii 2 c?p \ap ? ap and thus ?(fi1);:::;?(fi?) are in ?. By [12, p.457]
fi1;:::;fi? are orthogonal.
Extend fHfi1;:::;Hfi?g to an orthogonal basis fHfi1;:::;Hfi?;X1;:::;Xmg of
ap. To show that convWfl is symmetric about cp, it su?ces to show the symmetry
for Wfl. Let ? = ?1 + ?2 2 Wfl, where ?1 2 cp and ?2 2 c?p \ ap. Then ?1 =
95
Pm
i=1 ?iXi and?2 =
P?
i=1 ?iHfii. Thenapplythere ections := s?(fi1)???s?(fi?) 2 W
on ? 2 convWfl ? ap:
s? = s?(fi1)???s?(fi?)?1 +s?(fi1)???s?(fi?)?2 = sfi1 ???sfi??1 ?sfi1 ???sfi??2 = ?1 ??2;
to have the desired symmetry. Now if 2 convWfl with = 1 + 2, 1 2 cp
and 2 2 c?p \ ap, then ?cp( ) = 1. But 1 ? 2 2 convWfl by the symmetry of
convWfl with respect to cp. So 1 2 convWfl and thus 1 2 convWfl \cp.
Suppose s is a standard maximal solvable subalgebra containing the standard
Cartan subalgebra c. Since cp ? ap ? p, cp ? k so that
?cp((k +AdK(fl))\s) ? ?cp(k +AdK(fl)) = ?cp(AdK(fl)) = convWfl \cp:
Due to Theorem 7.4 and Proposition 7.12, one may ask whether the set equality
holds. The following example shows that the set inclusion is strict, even for some
normal real Lie algebras. (A real semisimple Lie algebra g is called normal [31,
p.392] if g has a Cartan subalgebra whose toral part is zero).
Example 7.15 Consider g = sl(3;R) in Example 7.15. Choose the Cartan decom-
position g = k _+p, where k is the set of all real skew symmetric matrices and p is
the set of all real symmetric matrices in g. Let ap = fdiag(a;b;?a?b) : a;b 2Rg.
Consider the Cartan subalgebra c = ck _+cp of g corresponding to
s2 =
n
0
BB
B@
a b c
?b a d
0 0 ?2a
1
CC
CA : a;b;c;d 2R
o
;
96
such that
cp = fdiag(a;a;?2a) : a 2Rg:
Let fl = diag(1;1;?2) 2 cp. Then Wfl = S3fl = fH1;H2;H3g where
H1 = diag(1;1;?2); H2 = diag(1;?2;1); H3 = diag(?2;1;1):
Then
H = 12(H2 +H3) = diag(?12;?12;1) 2 convWfl \cp:
We now claim H =2 ?cpf(k +AdK(fl))\s2g. Otherwise, let H = ?cp(Y +Adk(fl))
for some Y 2 k and k 2 K and Y + Adk(fl) 2 s2. Since diagY = 0, the diagonal
of Adk(fl) must be
diag(Adk(fl)) = diag(?12;?12;1):
That is to say, there exists
k =
0
BB
BB
B@
u1 u2 u3
v1 v2 v3
w1 w2 w3
1
CC
CC
CA2 SO(3)
such that
diag(k(diagfl)kT) = diag(u21 +u22 ?2u23;v21 +v22 ?2v23;w21 +w22 ?2w23)
= (?12;?12;1):
97
So
u21 +u22 ?2u23 = ?1=2;
v21 +v22 ?2v23 = ?1=2;
w21 +w22 ?2w23 = 1:
Because k 2 SO(3), we have
u21 +u22 +u23 = 1;
v21 +v22 +v23 = 1;
w21 +w22 +w23 = 1:
So
u3 = ?
p2
2 ; v3 = ?
p2
2 ; w3 = 0:
Let w1 = cos , w2 = sin . Because u and v are perpendicular to w, we can
assume that
u = (u1;u2;u3) = (?1
p2
2 sin ;??1
p2
2 cos ;?2
p2
2 );
where ?1; ?2 = ?1. Similarly, we can assume that
v = (v1;v2;v3) = (?1
p2
2 sin ;??1
p2
2 cos ;?2
p2
2 );
where ?1; ?2 = ?1. By direct computation, the (1;2) and (2;1) entries of Adk(fl)
are equal to 12?1?1??2?2 6= 0. Thus no X 2 k (skew symmetric matrices) can make
98
X + Adk(fl) an element of s2. Therefore we cannot flnd k 2 K and X 2 k such
that X +Adk(fl) 2 s2 and
?cp(X +Adk(fl)) = diag(?12;?12;1) 2 convWfl \cp:
One can get the same conclusion on s2 similarly, but not on s1 due to Theorem
7.4.
Remark 7.16 In Example 7.15, sl(3;R) is the simplest nontrivial example. Con-
sider g := sl(2;R) = k+p where k is the set of all real skew symmetric matrices and
p is the set of all real symmetric matrices in g. Let ap = fdiag(a;?a) : a 2 Rg.
Now g has two standard Cartan subalgebras
c1 := ap =
n?a 0
0 ?a
!
: a 2R
o
; c2 := so(2) =
n? 0 b
?b 0
!
: b 2R
o
;
and the corresponding standard maximal solvable subalgebras are
s1 :=
n?a b
0 ?a
!
: a;b 2R
o
; s2 := so(2) =
n? 0 b
?b 0
!
: b 2R
o
;
respectively. Now Theorem 7.4 implies
?cp((k +AdK(fl))\s1) = convWfl \(c1)p; fl 2 (c1)p:
Since (c2)p = 0, the statement
?cp((k +AdK(fl))\s2) = convWfl \(c2)p; fl 2 (c2)p
99
is then trivial.
We conclude this chapter by asking whether Ad(K)X \s 6= ` for some stan-
dard maximal solvable subalgebra s of g, if g is a normal real semisimple Lie algebra.
The question is motivated by Proposition 7.6.
100
Chapter 8
The eigenvalues and the real and imaginary singular values for
sl(2;C) and sl(2;R)
Ky Fan-Amir-Mo?ez-Horn-Mirsky?s result asserts that the real part of the
eigenvalues of a matrix is majorized by the real singular values, and conversely
if there exist ? 2 C, fl 2 Rn such that fl ` Re?, then there is a matrix with
eigenvalues ??s and real singular values fl?s. A similar result for the imaginary part
of the eigenvalues and the imaginary singular values is also given. Then we may
ask the following question: what is the necessary and su?cient condition on the
given vectors ? 2C, fi;fl 2Rn so that a matrix A 2Cn?n exists with eigenvalues
??s, real singular values fi?s and imaginary singular values fl?s? Condition stronger
than majorization is expected. The following is the simplest case and it shows that
the norm condition in Remark 4.10 is not su?cient.
Proposition 8.1 Let fi;fl 2 R and a + ib 2 C. Then there exists A 2 sl(2;C)
whose eigenvalues, real singular values, and imaginary singular values are?(a+ib),
?fi, and ?fl, respectively, if and only if (?a;a) ` (?fi;fi), (?b;b) ` (?fl;fl), and
fl2 ?b2 = fi2 ?a2.
Proof: Let A 2 sl(2;C) whose eigenvalues, real singular values, and imaginary
singular values are ?(a + ib), ?fi, and ?fl, respectively. After an appropriate
101
unitary similarity, we may assume that A is in upper triangular form:
A =
?a+ib c
0 ?a?ib
!
:
Then
(A+A?)=2 =
? a c=2
c=2 ?a
!
; (A?A?)=2i =
? b c=2i
?c=2i ?b
!
:
The eigenvalues of the matrices are ?(a + ib), ?fi = ?(a2 + 14jcj2)1=2, and ?fl =
?(b2 + 14jcj2)1=2. So (?a;a) ` (?fi;fi), (?b;b) ` (?fl;fl) and fl2 ?b2 = fi2 ?a2 =
1
4jcj
2. Conversely, if the conditions are satisfled, the above triangular matrix A
(thus not unique) is the required one with fi2 ?a2 = 14jcj2,
The following is the corresponding result for the real case.
Proposition 8.2 Let fi;fl 2 R and a + ib 2 C. Then there exists A 2 sl(2;R)
whose eigenvalues, real singular values, and imaginary singular values are?(a+ib),
?fi, and ?fl, respectively, if and only if (1) b = 0, (?a;a) ` (?fi;fi), and fl2 =
fi2 ?a2, or (2) a = fi = 0, b = ?fl.
Proof: If the eigenvalues of A 2 sl(2;R) are complex, they must be conjugate to
each other, that is, ?ib, b 2R. Otherwise, they must be of the form ?a, a 2R. By
Proposition 8.1, or by observing each A 2 sl(2;R) is (special) orthogonally similar
to one of the forms:
(a)
?a c
0 ?a
!
; a;c 2R; (b)
? 0 b
?b 0
!
; b 2R;
102
accordingly, we have the necessary conditions. Conversely, (1) let A be in the form
(a) whose eigenvalues, real singular values, and imaginary singular values are ?a,
?fi = ?(a2 + 14c2)1=2, and ?fl = ?(14c2)1=2. Thus set c = ?2jflj; (2) let A be in
the form (b) and it is obvious.
But for the case n ? 3, the problem becomes much more complicated. The
problem is similar to the recently settled Horn?s problem on the eigenvalues of sum
of Hermitian matrices [8, 9]. Further research is needed for a clear understanding.
103
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