On the Derivation Algebras of Parabolic Lie Algebras
with Applications to Zero Product Determined Algebras
by
Daniel Brice
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
August 2, 2014
Keywords: Reductive Lie algebra, parabolic subalgebra, derivation, zero product
determined algebra, direct sum decomposition
Copyright 2014 by Daniel Brice
Approved by
Huajun Huang, Chair, Associate Professor of Mathematics
Randall R. Holmes, Professor of Mathematics
TinYau Tam, Lloyd and Sandra Nix Endowed Professor of Mathematics
Abstract
This dissertation builds upon and extends previous work completed by the author
and his advisor in [5]. A Lie algebra g is said to be zero product determined if for each
bilinear map j : g g !V that satisfies j(x, y) = 0 whenever [x, y] = 0 there is a linear
map f : [g,g] !V such that j(x, y) = f [x, y] for all x, y 2g. A derivation D on a Lie
algebra g is a linear map D : g !g satisfying D [x, y] = D(x), y + x, D(y) for all
x, y2g. Derg denotes the space of all derivations on the Lie algebra g, which itself forms
a Lie algebra. The study of derivations forms part of the classical theory of Lie algebras
and is well understood, though some work has been done recently that generalizes some
of the classical theory [9, 10, 14, 23, 24, 27, 30, 31, 34, 37]. In contrast, the theory of zero
product determined algebras is new, motivated by applications to analysis, and supports
a growing body of literature [1, 4, 5, 11, 33]. In this dissertation, we add to this body
of knowledge, studying the two concepts of derivations and of zero product determined
algebras individually and in relation to each other.
This dissertation contains two main results. Let K denote an algebraicallyclosed,
characteristiczero field. Let q be a parabolic subalgebra of a reductive Lie algebra g over
K or R. First we prove a direct sum decomposition of Derq. Derq decomposes as the
direct sum of ideals Derq = L adq, where L consists of all linear maps on q that map
into the center of g and map [q,q] to 0. Second, we apply the decomposition, along with
results of [5] and [33], to prove that q and Derq are zero product determined in the case
that g is a Lie algebra over K.
We conclude by discussing several possible directions for future research and by ap
plying the main results to providing tabular data for parabolic subalgebras of reductive
Lie algebras of types A5, G2, and F4.
ii
Acknowledgments
I?d like to thank the Department of Mathematics and Statistics at Auburn University
for supporting me in my studies and research for the past six years. My time at Auburn
has been wonderful and formative, and the people here have been nothing but welcoming
and friendly.
Chief among these is my advisor, Dr. Huajun Huang. Dr. Huang, thank you for
your patience with me when I often made the same mistake several times. Thank you for
making the effort to contact me when I would disappear for weeks at a time, swallowed
up in a miasma of depression. Thank you for having faith in me which I often lacked. I
certainly would not have had the emotional will to continue if not for your guidance.
I?d like to thank Dr. Holmes and Dr. Tam, whose work has influenced mine, who
have given me helpful input throughout the process of writing this dissertation, and from
whom I learned my craft. Dr. Holmes, I hope that I make you proud when I say that yours
is the primary influence on my mathematical writing style. Dr. Tam, you have made this
dissertation possible by giving me the necessary mathematical tools. Thank you both.
I?d also like to thank my undergraduate professors: Dr. Elliott who introduced me
to algebra, Dr. Grzegorczyk who introduced me to geometry, and Dr. Garcia who intro
duced me to mathematical reasoning and proof. It was at CSU  Channel Islands that,
under their expert tutelage, the elusive cognitive lubricant called mathematical maturity
first began to ferment inside me.
It would be impossible to name everyone who has made this dissertation possible.
I will close by thanking my family and my friends. I?ve had a great time these past six
years, thanks to all of you. Now, anybody up for a round of Spades?
iii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Background and Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Derivations and the adjoint map . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Real Lie algebras and complexification . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Root space decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Restricted root space decomposition . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Parabolic subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Langland?s decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.8 The center of a parabolic subalgebra . . . . . . . . . . . . . . . . . . . . . . . 35
3 Derivations of Parabolic Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1 The algebraicallyclosed, characteristiczero case . . . . . . . . . . . . . . . . 38
3.2 The real case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Zero Product Determined Derivation Algebras . . . . . . . . . . . . . . . . . . . 56
4.1 Zero product determined algebras . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Parabolic subalgebras of reductive Lie algebras . . . . . . . . . . . . . . . . . 59
4.3 Derivations of parabolic subalgebras . . . . . . . . . . . . . . . . . . . . . . . 64
5 Examples and Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
iv
5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.1.1 Type A5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.1.2 Type G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.3 Type F4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Directions for future research . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
v
List of Figures
1.1 Decomposition of qS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Block matrix form of derivations in L . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 A2 root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 sl(C3) root space decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Standard parabolic subalgebras of sl(C3) . . . . . . . . . . . . . . . . . . . . . . 29
2.4 F0 where D0 =fa2g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 The parabolic subalgebra q gl(C6) corresponding to D0 =fa1, a2, a4g. . . . . 34
2.6 The Levi factor decomposition of qS sl(C6) . . . . . . . . . . . . . . . . . . . . 35
2.7 A representative member of lZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Embedding of HomK(V1 ?+ V2, V1) in gl(V1 ?+ V2) . . . . . . . . . . . . . . . . . 41
4.1 Tensor definition of zero product determined . . . . . . . . . . . . . . . . . . . . . 59
4.2 G2 root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.1 Cartan matrix and transpose inverse for Type A5 . . . . . . . . . . . . . . . . . . 70
5.2 Cartan matrix and transpose inverse for Type G2 . . . . . . . . . . . . . . . . . . 70
5.3 Cartan matrix and transpose inverse for Type F4 . . . . . . . . . . . . . . . . . . 72
5.4 Logical relations among various types of derivationlike maps . . . . . . . . . . 76
vi
List of Tables
2.1 sl(C3) root spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Simple roots of sl(C6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Root spaces of sl(C6) relative to h . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.1 Partial multiplication table for sl(C6) in terms ofH . . . . . . . . . . . . . . . . 69
5.2 Partial multiplication table for sl(C6) in terms ofT . . . . . . . . . . . . . . . . 69
5.3 T in terms ofHand as matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4 Parabolic subalgebras of type A5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.5 Parabolic subalgebras of type G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.6 Parabolic subalgebras of type F4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
vii
Chapter 1
Introduction
Let K be an algebraicallyclosed field of characteristiczero. Let g be a reductive Lie
algebra overKorR. Let q be a parabolic subalgebra of g. The purpose of this dissertation
is, primarily, twofold: first, to construct a direct sum decomposition of the derivation
algebra Derq; second, to use this direct sum decomposition to extend work done in [5]
and [33] and show that Derq is zero product determined.
An algebraAwith multiplication is said to be zero product determined if for each
bilinear map j : A A ! V that satisfies j(x, y) = 0 whenever x y = 0 there is
a linear map f : A2 ! V such that j(x, y) = f x y for all x, y 2 A. This is a
relatively new concept, first appearing in [4] and expanded upon in [5, 11, 33] and others.
In the initial paper, published in 2009, Bre?ar, Gra?i?c, and S?nchez Ortega proved that
the full matrix algebra over a commutative ring is zero product determined when is
the usual matrix multiplication or when is the Jordan product x y = xy + yx, and
also that the general linear algebra gl over a field is zero product determined when is
the Lie bracket x y = [x, y] = xy yx [4]. Gra?i?c expanded the pool of Lie algebras
known to be zero product determined to include the classical algebras Bl, Cl, and Dl
over an arbitrary field of characteristic not 2 [11]. In 2011, Wang et al. proved that the
parabolic subalgebras of a simple Lie algebras over K are zero product determined for K
algebraicallyclosed and characteristiczero (in particular, the simple Lie algebras over K
are zero product determined) [33]. Our present research began as an attempt to extend
this result to parabolic subalgebras of reductive Lie algebras over K and over R and to
their derivation algebras.
1
A derivation D on a Lie algebra g is a linear map D : g !g satisfying D [x, y] =
D(x), y + x, D(y) for all x, y 2g. Denote by Derg the space of all derivations on the
Lie algebra g, which itself forms a Lie algebra. The study of derivations forms part of the
classical theory of Lie algebras. The classical theory of derivations can perhaps be said
to begin with the wellknown result that if g is a semisimple Lie algebra over a field of
characteristic not equal to two, then any derivation of g is an inner derivation [13, 26], so
in particular g = Derg in case g is semisimple charF6= 2.
In 1955, Jacobson proved that a Lie algebra g is nilpotent if it has a derivation f 2
Derg that is nonsingular [14]. Dixmier and Lister provided an example in 1957 show
ing that the converse was not possible [9]. They constructed a particular Lie algebra L,
showed that L is nilpotent, characterized the derivation algebra Der L through explicit
computation, and showed that every derivation of L had a nontrivial kernel. Of note is
that Dixmier and Lister explicitly decompose Der L in this particular case and arrive at
results similar to our results for general parabolic subalgebras of reductive Lie algebras.
T?g?, in 1961, proved a partial converse to our result in the special case when q = g
[27]. By 1972, Leger and Luks ? working over an arbitrary field of characteristic not equal
to two ? were able to show that all derivations of a Borel subalgebra b of a semisimple
Lie algebra g are inner derivations, analogous to the known result for the semisimple
algebra g itself [19]. More generally, their result applies to the class of Lie algebras g
that can be expressed as the semidirect product g = aog0 where the subalgebra g0 is
nilpotent and the ideal a is abelian and acts diagonally on g0 [19]. This wider class of Lie
algebras includes Borel subalgebras but does not include parabolic subalgebras. Working
independently, Tolpygo extended the result of Leger and Luks to apply to any parabolic
subalgebra q between b and g, but only in case the scalar field is the complex numbers C
[29]. The balance of the work done in this era provides characterizations of derivations of
special classes of Lie algebras [10, 24, 28].
2
The recent direction that work on derivations has taken has been to relax the defini
tion of Lie algebra to include consideration of Lie algebras that draw scalars from com
mutative rings rather than from fields and to attempt to reproduce as much of the clas
sical theory as can be , characterizing derivations of specific classes of such Lie algebras
[23, 30, 31]. Zhang in 2008 takes a different approach, defining a new class of solvable Lie
algebras over C and characterizing their derivation algebras [37].
Other work has been in the direction of considering certain maps that are similar to
but may fail to be derivations [6, 8, 34]. Unrelated to zero product determined algebras,
except perhaps inspired by the concept, Wang et al. recently defined a product zero deriva
tion of a Lie algebra g as a linear map f : g ! g satisfying [f(x), y] + [x, f(y)] = 0
whenever [x, y] = 0 [34]. In the aforementioned paper, the authors go on to characterize
the product zero derivations of parabolic subalgebras q of simple Lie algebras over an
algebraicallyclosed, characteristiczero field, ultimately showing all product zero deriva
tions of q to be sums of inner derivations and scalar multiplication maps [34]. In papers
appearing in 2011 and 2012, Chen et al. consider nonlinear maps satisfying derivability
and nonlinear Lie triple derivations respectively [6, 8]. The authors characterize all such
maps on parabolic subalgebras of a semisimple Lie algebra over C as the sums of inner
derivations and certain maps called quasiderivations that may fail to be linear [6, 8].
This dissertation serves two purposes: to expand on the results characterizing deriva
tions of classes of Lie algebras and to apply these results to further the study of zero prod
uct determined algebras. We prove, among other results, the following two theorems.
Theorem. Let q be a parabolic subalgebra of a reductive Lie algebra g over an algebraicallyclosed,
characteristiczero field or over R. The derivation Lie algebra Derq decomposes as the direct sum
of ideals
Derq = L adq
where L consists of all linear transformation mapping q into its center and mapping [q,q] to 0.
3
Theorem. Let q be a parabolic subalgebra of a reductive Lie algebra g over an algebraicallyclosed,
characteristiczero field. q and Derq are zero product determined.
We began this research with the goal of proving the latter theorem, as it is a natural
extension of results found in [33]. As a tool to this end, we required an understanding of
the structure of Derq, especially as it relates to the structure of q. Considering the vast
body of literature on derivations, we assumed that results on the structure of Derq for
such a q would be readily available; however, a review of the literature found only partial
results, as we summarized above. Without the necessary tools to proceed, developing
those tools quickly overtook our work on zero product determined algebras and became
a central part of our research in and of itself. The results are complete and satisfying, and
enable us to pursue our original goals.
The method of proof of the former theorem relies on utilization of the grading on q
afforded by the root system F. In order to motivate the methods employed, we offer the
following example. The reader is encouraged to keep this example in mind during the
general treatment in chapter 3.
Example 1. We consider the parabolic subalgebra q of g = gl(C6) consisting of block
uppertriangular matrices in block sizes 3, 2, 1 (see figure 1.1). We write gl(C6) = gZ gS,
where gZ = CI and gS = sl(C6). We decompose q similarly: q = gZ qS, where qS =
q\gS.
coroot contained in t
coroot contained in c
Figure 1.1: Decomposition of qS
4
gS has root space decomposition gS = h ?+ i6=j Cei,j where h consists of traceless
diagonal 6 6 matrices. It is well known that the coroots hi = eii ei+1,i+1 form a basis
of h. We further decompose h into t ?+c, where t = Spanfh1, h2, h4gand c = Spanfh3, h5g
(see figure 1.1). It follows that t = h\[q,q] and that q has the vector space direct sum
decomposition
q = gZ ?+c ?+[q,q].
In light of this decomposition, a linear transformation that sends q to gZ and sends [q,q]
to 0 has the block matrix form illustrated by figure 1.2.
0
@
gZ c [q,q]
gZ 0
c 0 0 0
[q,q] 0 0 0
1
A
Figure 1.2: Block matrix form of derivations in L
The claims of the two theorems ? that Derq = L adq and that Derq is zero product
determined ? may then be explicitly verified via computation in this special case. The
proofs of the theorems in general will rely on carrying out the same decomposition of q
and the accompanying computations in abstract.
A brief outline of this dissertation: Chapter 2 provides the necessary background def
initions and tools needed to understand the results in the sequel. Except where noted in
section 2.8, this chapter does not contain original research and may be skimmed or even
skipped by an experienced Lie algebraist. Chapter 3 proves the former theorem and sev
eral ancillary results, as mentioned, primarily relying on the root space decomposition.
Chapter 4 formally introduces the notion of a zero product determined algebra, summa
rizes some of the known results, and extends those results. Chapter 5 contains a short
discussion on possible directions in which to generalize the results of this dissertation for
future research.
5
Before we begin, we shall make note of some conventions of terminology and nota
tion. For the convenience of the reader, we shall use the term proposition for any result
that is not original to this dissertation. The terms lemma, theorem, and corollary are used
for results appearing for the first time in this dissertation. R and C will denote the field
of real numbers and the field of complex numbers, respectively. If F is a field, we will
say that F is complexlike to mean that F is algebraicallyclosed and of characteristiczero.
Typically, K will be used to denote complexlike fields and F will be used to denote more
general fields.
If V is a vector space, we denote the identity map on V by idV. If V1 and V2 are
vector spaces over the same field F, we denote by HomF(V1, V2) the set of Flinear maps
from V1 to V2, which is itself a vector spaces over F. If V1 and V2 are both subspaces of
a vector space V, intersect trivially, and together span V, we write V = V1 ?+ V2. If V is
a vectorspace, W a subspace, and f a linear map f : V ! V, we say f stabilizes W,
or W is f invariant, to mean f(W) W. We write fjW to denote the restriction of f to
W; namely, the linear map W ! V defined by fjW(v) = f(v) for each v 2 W. If V is
a vector space whose scalar field is ambiguous or nonstandard in some way, we write
V = VF to note that F is the scalar field. For example, the notation C3 will denote the
set of 3entry column vectors with entries in C viewed as a 3dimensional complex vector
space, and the notation C3R will denote the same set viewed as a 6dimensional vector
space with real number scalars.
If M is a matrix, Mt denotes the transpose of M. If M is a matrix with complex entries,
M denotes the conjugate transpose of M, ie M = M t. The notation Tr M denotes the
trace of a matrix or linear transformation M (ie, the sum of the diagonal entries of M or
a matrix representing M, respectively). In case Tr M = 0 we may say M is traceless for
brevity. The notation ei,j is used to denote the matrix with 1 in the ith row, jth column
entry and zeros elsewhere. The notation In (or simply I if n is understood) denotes the
n n identity matrix with 1s along the main diagonal and zeros elsewhere.
6
Chapter 2
Background and Setting
This chapter reviews the basic facts of the classical theory of Lie algebras which are
required for an understanding of the subsequent discussion. Proofs are included as space
permits. Where a particular definition, theorem, or proof is not cited in the line of the text,
the reader is referred to any of the standard texts on the subject, eg, [2, 13, 15, 17, 21, 25].
2.1 Lie algebras
Definition 2.1. A Lie algebra is a vector space g over a field F together with a binary
operation [ , ] : g g !g satisfying:
1. [ , ] is Fbilinear, ie,
8x, y, z2g,8a2F, [x + ay, z] = [x, z]+ a[y, z] and [x, y + az] = [x, y]+ a[x, z];
2. [ , ] is alternating, ie,
8x2g, [x, x] = 0; and
3. [ , ] satisfies the Jacobi identity, ie,
8x, y, z2g, x,[y, z] + y,[z, x] + z,[x, y] = 0.
Proposition 2.1. For all x, y2g, [x, y] = [y, x].
Proof. Consider [x + y, x + y]. By condition 2, [x + y, x + y] = 0 and by conditions 1 and
2, [x + y, x + y] = [x, x] + [x, y] + [y, x] + [y, y] = [x, y] + [y, x], so 0 = [x, y] + [y, x], or
rather [x, y] = [y, x].
7
Example 2. Real threespace R3, together with the familiar cross product defined by
0
BB
BB
@
x1
y1
z1
1
CC
CC
A
0
BB
BB
@
x2
y2
z2
1
CC
CC
A
=
0
BB
BB
@
y1z2 y2z1
x2z1 x1z2
x1y2 x2y1
1
CC
CC
A
is a Lie algebra. It is straightforward to verify that satisfies the three conditions required
of [ , ] in definition 2.1.
Example 3. Let n be some positive integer. A space of n n matricesMmay be endowed
with a bracket product by defining
8M, N 2M, [M, N] = MN NM.
IfMis closed under taking linear combinations of bracket products of its members, then
Mis a Lie algebra.
Example 4. Denote by so(n) the space of all n n matrices with entries in R satisfying
M = Mt (ie. M is skewsymmetric). so(n) is closed under taking linear combinations
of matrix bracket products MN NM, so so(n) is a Lie algebra under [M, N] defined in
example 3.
Example 5. A matrix M with complex entries is called Hermitian if M = M . M is called
skewHermitian is M = M . If M is Hermitian (res. skewHermitian), complex scalar
multiple of M may fail to be Hermitian (res. skewHermitian). In fact, for Hermitian
M, the scalar multiple iM is skewHermitian, and vice versa. Because of this, the vector
space consisting of Hermitian (res. skewHermitian) matrices are vector spaces over R,
despite Hermitian (res. skewHermitian) matrices admitting complex entries.
Denote by su(n) the space of all n n skewHermitian traceless matrices. su(n) is
closed under the bracket [M, N] defined in examples 3, and as such forms a Lie algebra
8
over R. The space of Hermitian matrices is not closed under the bracket. In fact, if M and
N are Hermitian, [M, N] is skewHermitian.
Example 6. Let V be a vector space over a fieldF. Denote by gl(V) the space of allFlinear
maps from V to V. Denote by sl(V) the subspace of gl(V) consisting of traceless linear
maps. Define the bracket product [ , ] by
8f , g2gl(V), [f , g] = f g g f .
Then, gl(V) is a Lie algebra, and sl(V) is a Lie algebra under the bracket restricted to
sl(V) sl(V). If the dimension of V is n, and if a basis for V is chosen, then gl(V) and
sl(V) are concretely realized as the space of all n n matrices with entries in F and the
space of all n n traceless matrices with entries inF, respectively, and the bracket defined
here agrees with the bracket defined in example 3.
Example 7. The vector space Cn n consisting of n n matrices with complex entries can
be considered a Lie algebra over C, as gl(Cn) with dimension n2, or as a Lie algebra over
R, as gl(CnR) with dimension 2n2.
Definition 2.1 does not exclude the possibility of considering infinitedimensional Lie
algebras or Lie algebras over primecharacteristic fields, as is the case with ? for exam
ple ? [16] and [26], respectively. In addition, recent work in the study of Lie algebras has
relaxed the definition to include the consideration of Lie algebras with scalars from a com
mutative ring rather than a field, such as in [2, 4, 11, 23, 30, 31]. This dissertation follows
none of the aforementioned directions. Instead, all of the Lie algebras considered in this
dissertation are finitedimensional vector spaces, drawing scalars from a characteristic
zero field such as R or C.
If g is a Lie algebra and if A, B are subsets of g (written A, B g) we use the notation
[A, B] to denote the Flinear span of members of g of the form [a, b] where a 2 A and
9
b2B. Taking the Flinear span is essential here, as the setf[a, b]ja2 A, b2Bgoften fails
to be a linear subspace of g.
Definition 2.2. h is a subalgebra of g (written h g) means that h is a linear subspace of g
and that [h,h] h. a is an ideal of g (written a g) means that a is a subalgebra of g and
that [a,g] a.
In light of proposition 2.1, [a,g] = [g,a], so the condition that [a,g] a is equivalent
to the condition that [g,a] a.
Example 8. Let g be any Lie algebra. g g trivially. Moreover, the bracket product of two
ideals is an idea, so [g,g] is an ideal and is called the derived algebra of g.
Example 9. Consider gl(R3). Notice that so(3) gl(R3), and sl(R3) = [gl(R3),gl(R3)]
gl(R3).
A subalgebra a g induces an equivalence relation on g by partitioning g into cosets
x +a =fx + aja2agfor each x 2g. The set of all such cosets has a natural structure as
a Lie algebra if, and only if, a is an ideal.
Definition 2.3. Let g be a Lie algebra and let a g. The quotient algebra g/a is the Lie
algebra consisting of the set
g/a =fx +ajx2gg
together with the bracket
[x +a, y +a] = [x, y]+a.
We omit the verification that the bracket on g/a is welldefined, while we note that
the proof relies on the fact that a is an ideal.
Definition 2.4. A Lie algebra g is abelian when [g,g] = 0.
The name is not accidental or arbitrary. In fact, ifMis a space of commuting matri
ces, then for any M, N 2Mwe have [M, N] = MN NM = MN MN = 0, so that
10
the term abelian used here agrees with the familiar usage of the term from group theory
when used to mean commutative.
Definition 2.5. The center of g (written gZ) is the set of all z2g such that [z,g] = 0.
The notation [x, S] is undefined in the case we consider, where S g and x 2g, but
we now fix this notation to mean [fxg, S], ie, the linear span of the bracket products [x, s]
for fixed x2g and for all s2S.
Proposition 2.2. gZ is abelian and gZ g.
Proof. We omit the verification that gZ is a linear subspace of g. We have left to show that
[gZ,gZ] = 0 and that [gZ,g] gZ. Both follow from the observation that [gZ,g] = 0.
Definition 2.6. A Lie algebra g is simple when g is nonabelian and the only ideals of g are
0 and g itself.
Example 10. Let dim V 2. gl(V) is not simple because (gl(V))Z = FidV (where idV is
the identity map on V). sl(V), however, is simple. It is somewhat nontrivial to prove
this, though a proof may be found in any of the standard texts.
The simple Lie algebras were completely classified and enumerated, in case F is
complexlike, by Killing and Cartan as early as the 1890?s [13]. The classification of all
simple Lie algebras over R appears in [17].
Definition 2.7. Let g1 and g2 be Lie algebras over F. An Flinear map f : g1 ! g2 is
called a homomorphism (or endomorphism in case g1 = g2) if
8x, y2g1, f [x, y] = f(x), f(y) .
If in addition f is injective (ie, onetoone) and surjective (ie, onto), then f is called an
isomorphism (or automorphism in case g1 = g2) and g1 and g2 are said to be isomorphic,
written g1 = g2.
11
Example 11. Let V be a vector space, and suppose that T : V ! V is a change of basis,
represented my an invertible matrix A in the sense that T(v) = Av. Let g gl(V). There
is a change of basis T0 : g !g corresponding to T, represented by matrix conjugation in
that
T0(x) = A 1xA.
Then, the map T0 is an automorphism of g, since
T0 [x, y] = A 1(xy yx)A
= A 1xyA A 1yxA
= (A 1xA)(A 1yA) (A 1yA)(A 1xA)
= T0(x), T0(y) .
Example 12. R3, is isomorphic to so(3) by the isomorphism r : R3 !so(3) defined by
r
0
BB
BB
@
x
y
z
1
CC
CC
A
=
0
BB
BB
@
0 z y
z 0 x
y x 0
1
CC
CC
A
.
Definition 2.8. Let f : g1 ! g2 be a homomorphism. The kernel of f (written Ker f ) is
the set
Ker f =fx2g1jf(x) = 0g,
and the image of f (written Im f ) is the set
Im f =ff(x)2g2jx2g1g.
Proposition 2.3. Let f : g1 !g2 be a homomorphism. Then the kernel of f is an ideal of g1,
and the image of f is a subalgebra of g2.
12
The underlying vector space of a Lie algebra g may be decomposed as a direct sum
of subspaces. For example g = h ?+ k if h and k are subspaces of g, if h\k = 0, and if
Span(h[k) = g, but an otherwise unqualified decomposition does not reflect any of the
Lie algebra structure of g, meaning that the vector space decomposition is not necessarily
compatible in any meaningful way the with bracket. We define two notions of direct sums
of Lie algebras that are, to various degrees, compatible with the bracket.
Definition 2.9. Let g = a ?+b. g is said to be the Lie algebra direct sum of a and b, written
g = a b if both a and b are ideals of g.
The requirement that a and b be ideals of g guarantees that the bracket acts diagonally
on either summand. In other words, [a,b] = 0 since [a,b] a\b = 0. In this way, g is
thought of as taking two distinct Lie algebras, a and b, and combining them together with
the natural componentwise bracket rule
[a1 + b1, a2 + b2] = [a1, a2] {z }
2a
+[b1, b2] {z }
2b
for a1, a2 2a, b1, b2 2b.
Definition 2.10. Let g be a Lie algebra. g is called semisimple if g is the direct sum of simple
ideals. g is called reductive if g = gZ gS for some semisimple ideal gS.
Semisimple and reductive Lie algebras are natural generalization of simple Lie al
gebras, in the sense that much of the theory of simple Lie algebras can be extended to
semisimple and reductive Lie algebras. We state without proof that the semisimple ideal
gS of a reductive g is maximal and unique up to isomorphism and that the simple sum
mands of a semisimple g are unique up to isomorphism.
Example 13. sl(Cn) is semisimple because it is a sum of one simple ideal. gl(Cn) = CIn
sl(Cn) is neither simple nor semisimple but is reductive, with center consisting of scalar
13
matrices CIn and maximal semisimple ideal sl(Cn) (Recall In denotes the n n identity
matrix).
Definition 2.11. Let g = a ?+ b. g is said to be the Lie algebra semidirect sum of a and b,
written g = aob, if a is an ideal of g and b is a subalgebra of g.
The notation g = aob is chosen to remind us that a g. Since a is an ideal, [b,a] a.
Explicitly, for each x 2 b, the map x : a ! a defined by x a = [x, a] is a linear
transformation on a. Because of this, we say that b acts on a and that g is an extension
of b by a. Then the bracket rule on g may be described in terms of the brackets on the
individual summands and the action of b on a as
[a1 + b1, a2 + b2] = [a1, a2]+ b1 a2 b2 a1 {z }
2a
+[b1, b2] {z }
2b
.
2.2 Derivations and the adjoint map
Proposition 2.4. Let g = aob. Define the map r : b !gl(a) by r(x) = x (ie, r(x)(a) =
[x, a]) for x2b. The map r is a homomorphism of Lie algebras.
Proof. We must show r [x, y] = r(x), r(y) for all x, y2b. Let a2a.
r [x, y] (a) = [[x, y], a]
= [x,[y, a]] [y,[x, a]] by definition 2.1
= r(x) r(y) r(y) r(x) (a)
= [r(x), r(y)] (a).
Since a was arbitrary, r [x, y] = r(x), r(y) for all x, y2b.
The map r is said to be a representation of b, and a is said to be a bmodule.
14
Definition 2.12. Let g be a Lie algebra. A representation of g is a homomorphism r : g !
gl(V) for some (finitedimensional) vector space V. V, in this case, is said to be a gmodule.
When r is injective it is said to be a faithful representation.
Representations are a tool by which an abstract Lie algebra g may be studied more
concretely by considering Lie algebras consisting of linear transformations. If a basis for V
is chosen, the linear transformations themselves are then represented by matrices, further
simplifying the study of g.
Proposition 2.5 (Ado?s Theorem). Let F be a characteristiczero field. Let g be a (finite
dimensional) Lie algebra over F. Then, g admits a faithful representation. Explicitly, g is iso
morphic to a space of matrices with entries in F and bracket [M, N] = MN NM [2, Ch. I,
?7.3].
Example 14. The map r of example 12 is a faithful representation of R3, onto so(3).
Since r is an isomorphism, it has an inverse r 1 : so(3) ! R3; however, r 1 is not a
representation because it does not map into a subspace of some gl(V).
Definition 2.13. Let g be a Lie algebra. A linear map D : g !g is called a derivation if
8x, y2g, D [x, y] = D(x), y + x, D(y) .
The definition of derivation is motivated by the familiar product rule of differenti
ation. In fact, the differential operator ddx is a derivation in a suitable context. We will
not spend time developing this idea, other than to mention it. The interested reader is
referred to [3] for an algebraic treatment or to [12] or [36] for a geometric point of view.
Definition 2.14. For a Lie algebra g, Derg denotes the set of all derivations on g.
We note that if D1 and D2 are derivations, D1 D2 need not necessarily be a deriva
tion; however, [D1, D2] = D1 D2 D2 D1 is a derivation. In light of this, we have
Derg gl(g).
15
Definition 2.15. Let g be a Lie algebra. For x2g the adjoint of x is the map ad x : g !g
defined by ad x(y) = [x, y] for all y2g. The adjoint map is the map ad : g !gl(V).
Proposition 2.6. ad : g ! gl(g) is a representation (in particular, a Lie algebra homomor
phism). Moreover, for each x2g, ad x is a derivation on g.
The proposition follows from definition 2.1. In a sense, the definition of a Lie algebra
g is intended to ensure that the action of multiplication by an element x (ie, the map ad x)
is a derivation on g for all x 2g. It is for this reason that derivations take a primary role
in the study of Lie algebras.
Proof of proposition 2.6. We omit the proofs that ad and ad x are linear maps. Let x, y, z2g
and consider ad x. We must to show that ad x([y, z]) = [ad x(y), z]+[y, ad x(z)].
ad x [y, z] = x,[y, z]
= z,[x, y] y,[z, x] by condition 3
= [x, y], z + y,[x, z] by condition 2
= ad x(y), z + y, ad x(z) ,
so ad x is a derivation. Next, we must show ad[x, y] = [ad x, ad y].
ad[x, y](z) = [x, y], z
= [y, z], x [z, x], y by condition 3
= x,[y, z] y,[x, z] by condition 2
= ad x ad y(z) ad y ad x(z)
= [ad x, ad y](z)
so ad[x, y] = [ad x, ad y], that is, ad is a homomorphism.
We write adg = Im ad and note that adg Derg. Furthermore, notice Ker ad = gZ.
16
Example 15. Consider the adjoint representation of gl(Cn). The kernel isCIn and the image
is isomorphic to gl(Cn)/CIn = sl(Cn).
In the example above, the traceless matrices sl(Cn) act as derivations on the space
of matrices gl(Cn). A natural question that we will return to often in this dissertation is
whether there are other derivations on gl(Cn), and if so, how we may characterize them.
Definition 2.16. Let g be a Lie algebra. An inner derivation of g is a member of adg. Any
member of Derg not in adg is called an outer derivation.
Proposition 2.7. An inner derivation maps g into [g,g] and stabilizes ideals.
Proof. Let D be an inner derivation, so D = ad x for some x 2 g. Let y 2 g be arbitrary
and notice D(y) = ad x(y) = [x, y] 2 [g,g], verifying the first assertion. Next, let a g.
D(a) = ad x(a) = [x,a] a by the definition of ideal.
In general, for a Lie algebra g we observe the chain of subspaces
adg Derg gl(g)
consisting of inner derivations, all derivations, and all linear maps respectively. The
next theorem, a classical result in the theory of Lie algebras, completely characterizes
the derivations of semisimple Lie algebras. Our work in chapter 3 of this dissertation can
be understood as a generalization and extension of this classical result.
Proposition 2.8. Let g be a semisimple Lie algebra over a field F of characteristic not equal to
two. The only derivations of g are inner derivations [13, 26].
The proposition states that Derg = adg in case g is semisimple. A large portion of
our work is to characterize outer derivations when q is a parabolic subalgebra (cf. section
2.6) of a reductive g.
17
Let g be semisimple and write as the sum of its simple ideals g = g1 ... gk. Each
simple gi is of course semisimple, so Dergi = adgi = gi/(gi)Z = gi by proposition 2.8
and since each (gi)Z = 0. Applying proposition 2.8 to the semisimple g gives
Derg = adg = g/gZ = g = g1 ... gk
so that we have
Der(g1 ... gk) = Der(g1) ... Der(gk)
in case each gi is simple. In this fashion, the direct sum structure of g is carried over to the
derivation algebra Derg when g is semisimple. This is not universally applicable to all
direct sums of Lie algebras ? it is true for semisimple Lie algebras because of propositions
2.7 and 2.8. An arbitrary derivation of a general Lie algebra does not necessarily stabilize
ideals, and direct sum decomposition is not necessarily preserved. There are two ideals,
however, that are stabilized by every derivation, related in the following proposition.
Proposition 2.9. Let D be a derivation on an arbitrary Lie algebra g. D stabilizes [g,g] and gZ.
Proof. Let x, y2g.
D([x, y]) = [D(x), y]+[x, D(y)]2[g,g],
so D stabilizes [g,g] as desired. Next, let z 2gZ. We need to show D(z)2gZ. Let x 2g
and consider D([z, x]).
0 = D([z, x]{z}
=0
) = [D(z), x]+[z, D(x)] {z }
=0
= [D(z), x],
so [D(z), x] = 0 for all x2g, meaning D(z)2gZ as desired.
18
2.3 Real Lie algebras and complexification
Let i denote the imaginary unit. Let g be a Lie algebra over R. g is an Rvector space,
but it is possible that vectors in g admit complex entries (cf. example 5). We would like
to define ?g as the vector space g + ig, but we are concerned about possible notational
collisions between i and entries of members of g. However, in light of Ado?s Theorem
(proposition 2.5), g is isomorphic to a real Lie algebra consisting of matrices with real
entries, and we think of g in this way as we proceed in order to avoid this issue.
The vector space
?g = g ?+ ig =fx + iyjx, y2gg
of dimension 2 dimg over R may be thought of as a Cvector space of dimension dimg.
We may define a bracket on ?g by
[x + iy, u + iv] = [x, u] [y, v] {z }
2g
+ i([x, v]+[y, u]) {z }
2ig
.
We may verify that ?g, together with the bracket defined above, satisfies definition 2.1 with
F = C, making ?g a Lie algebra over C.
Definition 2.17. The complex Lie algebra ?g is called the complexification of the real Lie
algebra g. g is called a real form of ?g.
It is possible for nonisomorphic real Lie algebras g1 and g2 to have isomorphic com
plexifications bg1 = bg2 = g. In that case, both g1 and g2 are real forms of the complex Lie
algebra g.
Example 16. Since sl(Rn) ?+ isl(Rn) = sl(CnR) = su(n) ?+ isu(n), both sl(Rn) and su(n) are
real forms of sl(Cn).
Proposition 2.10. Let g be real, let ?g = g ?+ ig be the complexification of g. Then the center of ?g
is the complexification of the center of g, namely ?gZ = cgZ = gZ ?+ igZ.
19
Proof. Let z 2 ?gZ. Write z = x + iy with x, y 2g. Now, for arbitrary w = u + iv 2 ?g with
u, v2g we have
0 = [z, w] = [x + iy, u + iv]
= [x, u] [y, v]+ i([x, v]+[y, u])
and by direct sum decomposition [x, u] = [y, v] and [x, v] = [y, u]. Adding these equa
tions gives
8u, v2g, [x, u + v] = [y, v u] (2.1)
Setting v = u in equation 2.1 produces [x, 2u] = 0 for all u 2g, so x 2gZ. Similarly,
setting u = v in equation 2.1 produces 0 = [y, 2v] for all v2g, so y2gZ.
Proposition 2.11. Let g be a semisimple (res. reductive) Lie algebra over R. The complexification
?g of g is semisimple (res. reductive) [17, Ch. VI, ?9].
Proposition 2.12. Let D be a derivation of the real Lie algebra g. Then ?D defined by ?D(x + iy) =
D(x)+ iD(y) is a derivation of ?g. Moreover, ?D stabilizes g.
Proof. Let z = x + iy, w = u + iv be arbitrary elements of ?g.
?D([z, w]) = ?D [x + iy, u + iv]
= ?D
[x, u] [y, v]+ i [x, v]+[y, u]
= D [x, u] [y, v] + iD [x, v]+[y, u]
= D([x, u]) D([y, v])+ i D([x, v])+ D([y, u])
= [D(x), u]+[x, D(u)] [D(y), v] [y, D(v)]
+ i [D(x), v]+[x, D(v)]+[D(y), u]+[y, D(u)]
20
= [D(x), u + iv]+[x + iy, D(u)]+ i[x + iy, D(v)]+ i[D(y), u + iv]
= [D(x), w]+ i[D(y), w]+[z, D(u)]+ i[z, D(v)]
= [D(x)+ iD(y), w]+[z, D(u)+ iD(v)]
= ?D(z), w + z, ?D(w)
So ?D is a derivation on ?g.
?D stabilizes g by definition. Indeed, if x2g, then ?D(x) = ?D(x + i0) = D(x)2g.
2.4 Root space decomposition
LetKdenote a complexlike field. Fix g as denoting a semisimple Lie algebra overK.
We will show how to decompose g into a (vector space) direct sum of certain subspaces
that have desirable interactions with the bracket [ , ], made explicit below. Our approach
largely follows Humphreys, and the reader is referred to [13, Ch. II, ?8] for proofs.
Definition 2.18. A toral subalgebra of g is a subalgebra h g that is abelian and for each
x 2 h, the map ad x : g ! g is diagonalizable. If h is a maximal toral subalgebra, it is
called a Cartan subalgebra of g.
For a general Lie algebra g, a Cartan subalgebra is typically defined to be a self
normalizing nilpotent subalgebra, but this is more generality than we require. Within the
class of semisimple Lie algebras, the general notion coincides with the simplertostate
definition 2.18. Cartan subalgebras are unique in the sense made precise below.
Proposition 2.13. The Cartan subalgebras of g are conjugate to one another, in the sense that any
one may be transformed into any other by an appropriate automorphism of g.
As a result of proposition 2.13, each Cartan subalgebra of g has the same dimension.
We call this number the rank of g. Now, fix h as denoting a specific Cartan subalgebra of
g. Consider the dual vector space h consisting of linear functionals a : h !C.
21
Definition 2.19. For a nonzero a2h , define the subspace ga of g by
ga =fx2gj8h2h,[h, x] = a(h)xg.
In case ga 6= 0, a is called a root and ga is called a root space. Denote by F the set of roots.
F is called a root system. The rank of F is the dimension of h .
Some basic facts about the root spaces ga for a2F:
Proposition 2.14. Let a, b2F. [ga,gb] ga+b if a + b2F and [ga,gb] = 0 if a + b /2F.
Moreover, each ga is onedimensional.
Let Z+ denote the set of positive integers and Z denote the set of negative integers.
Follows are some basic facts about the geometric properties of the root system F.
Proposition 2.15. Let r denote the rank of F. A set D of r roots (refered to as a base of F) may
be selected so that each root b2F can be written uniquely as either a Z+linear combination or
a Z linear combination of roots in D. Moreover, F = F and if b2F and b = ki=1 ai with
each (notnecessarily nonrepeating) ai 2D, we may rearrange the terms of the sum so the partial
sums ji=1 as(i) lie in F for each j k each (where s denotes an appropriate permutation).
In light of propositions 2.14 and 2.15 we fix the following notation and terminology:
a D as in proposition 2.15 is called a base of F, and roots in D are called simple roots; roots
in F generated by positive integer combinations of simple roots are called positive roots,
and F+ denotes the set of positive roots; and similarly for negative roots and F . From
each gb for b2F we arbitrarily choose an xb 2gb so that gb = Kxb by proposition 2.14.
Proposition 2.16 (Root space decomposition). The semisimple Lie algebra g decomposes as the
vector space direct sum
g = h ?+
b2F
Kxb.
22
This decomposition is called the root space decomposition of g relative of h, and is essentially unique
in that the root space decompositions of g relative to two Cartan subalgebras h1 and h2 differ only
by an automorphism of g.
We may extend these notions to reductive Lie algebras over K. When g = gZ gS
is reductive, then a Cartan subalgebra of g takes the form gZ h, where h is a Cartan
subalgebra of gS. In this case, by root system F of the reductive g we mean the root system
of the semisimple part gS, and by root space decomposition of g we mean the vector space
direct sum decomposition
g = gZ ?+h ?+
b2F
Kxb.
Example 17. Consider sl(C3), concretely the Lie algebra of 3 3 traceless matrices with
complex entries. One choice of Cartan subalgebra is the subalgebra h consisting of di
agonal matrices. We take as basis for h the matrices h1 = e1,1 e2,2 and h2 = e2,2 e3,3.
(Recall, ei,j denotes the matrix with a 1 in the i, j position and zeros elsewhere.)
Our next task is to find all b 2 h such that gb 6= 0. Write b = b1h 1 + b2h 2, where
h i (hj) = di,j is the dual functional to the vector hi. h 1 and h 2 are not roots, as gh 1 = gh 2 = 0.
However, since h 1, h 2 together span h , any root b will be of the form b = b1h 1 + b2h 2
with b1, b2 2 C. In particular, straightforward computation show that g2h 1 h 2 6= 0 and
g h 1+2h 2 6= 0, among others. We may write a1 = 2h 1 h 2, a2 = h 1 + 2h 2 and as we
shall see below, D =fa1, a2gis a base of the root system F. We record the individual root
spaces, our choice base D, and our choice of basis vector for each root space in table 2.1.
Root b b in terms of D Root Space gb Choice of xb
2h 1 h 2 a1 Ce1,2 xa1 = e1,2
h 1 + 2h 2 a2 Ce2,3 xa2 = e2,3
h 1 + h 2 a1 +a2 Ce1,3 xa1+a2 = e1,3
2h 1 h 2 a1 Ce2,1 x a1 = e2,1
h 1 + 2h 2 a2 Ce3,2 x a2 = e3,2
h 1 + h 2 a1 a2 Ce3,1 x a1 a2 = e3,1
Table 2.1: sl(C3) root spaces
23
Because the root system F may be written F =fa1, a2, a1 +a2, a1, a2, a1 a2g,
we see that D = fa1, a2g was a suitable choice of base. Since dimh = 2, F lies in a
twodimensional plane. Figure 2.1 provides an illustration of F.

A
A
A
A
AAU
A
A
A
A
AAK
a1
a2 a1 +a2
a1
a2 a1 a2
Figure 2.1: A2 root system
Finally, sl(C3) decomposes as the vector space direct sum of root spaces:
sl(C3) = h ?+
b2F
Cxb.
Refer to figure 2.2 for a graphical representation of the root space decomposition of sl(C3).
2
4
h ga1 ga1+a2
g a1 h ga2
g a1 a2 g a2 h
3
5
Figure 2.2: sl(C3) root space decomposition
The root space decomposition of sl(C3) ? and more generally of sl(Cn) ? is, in
a sense, a triviality, since it merely write sl(C3) in terms of the standard unit matrices
ei,j. Partly this is because we chose a wellbehaved Cartan subalgebra, but partly this
24
situation is intentional. The root space decomposition of sl(Cn) relative to the Cartan
subalgebra consisting of diagonal matrices can be thought of as the motivating example
for root space decomposition in general. In this sense, the root space decomposition of a
general reductive g provides an slesque basis for general g [13], allowing one to reduce
questions in Lie algebra theory to questions in linear algebra and matrix theory. Root
space decomposition also allows for induction on the height of roots as a proof technique
(where, for example, the roots a2 and a1 a2 have respective heights 1 and 2).
2.5 Restricted root space decomposition
In this section, let g denote a semisimple Lie algebra over R. We will define the re
stricted root space decomposition of g, a coarser decomposition than the analogous root
space decomposition for Lie algebras over complexlike fields. Except for some nota
tional changes made for internal consistency, the development of these ideas here follows
Knapp?s, and the reader is referred to [17, Ch. VI, ?24] for proofs.
Definition 2.20. Let q : g !g be an automorphism. q is called a Cartan involution of g if
q q = idg and if the bilinear map
Bq(x, y) = Tr
ad x ad q(y)
(2.2)
is positive definite.
For x, y 2 g, notice that ad x, ad q(y) are linear transformations g ! g, which we
may think of as a matrices. Observing this, we see that the Bq, defined by equation 2.2, is
well defined. The interested reader may note that, in general, the bilinear map B(x, y) =
Tr(ad x)(ad y) on g g is called the Killing form, after Wilhelm Killing, and has many
applications in the structure theory of Lie algebras [2, 13, 17]. We will not make use of
the Killing form in the sequel, though Cartan involutions are necessary to arrive at the
Cartan decomposition of g, described below.
25
Example 18. Recall sl(Rn), su(n), and sl(CnR) are Lie algebras over R. The map q(M) =
Mt on sl(Rn) is a Cartan involution. The identity map on su(n) is a Cartan involution.
Complex conjugation q(M) = M is a Cartan involution on sl(CnR), as well as is the map
q(M) = M .
Proposition 2.17. Every real Lie algebra g has a Cartan involution q. The Cartan involution
q of g is essentially unique, in the sense that if q0 is another Cartan involution of g, then q0 =
j q j 1 for some automorphism j of g.
Given a Cartain involution q we may consider the eigenvalues of q. Since q q = idg
the eigenvalues of q are 1 and 1. We write k for the eigenspace corresponding to 1 and p
for the eigenspace corresponding to 1, and observe the following results:
Proposition 2.18. For q a Cartan involution of g we have g = k ?+ p, with k and p as above.
Furthermore, [k,k] k, [k,p] p, and [p,p] k.
Definition 2.21. Notation as above, the decomposition g = k ?+ p is called the Cartan
decomposition of g.
Starting from a Cartan decomposition, we select a maximal abelian subspace a p
and we set m =fx2kj[x,a] = 0g k. For each l2a we write
gl =fx2gj8h2a,[h, x] = l(h)xg
analogously to the complexlike case.
Definition 2.22. Let l2a be nonzero. In case gl 6= 0, we call l a restricted root of g and
we call gl the restricted root space associated to l. F = fl2a jl is a restricted rootg is
called the restricted root system of g relative to a.
Proposition 2.19 (Restricted root space decomposition). g decomposes as the vector space
direct sum
g = a ?+m ?+
l2F
gl.
26
Furthermore, the restricted root spaces ga,gb satisfy [ga,gb] ga+b.
In contrast to the complexlike case, the restricted root space gl need not be one
dimensional. The restricted root system exhibits some of the same geometric properties
exhibited by the root system of a complexlike Lie algebra, which we discuss in the next
two propositions.
Proposition 2.20. The restricted root system F satisfies F = F. Furthermore, a set of positive
restricted roots F+ may be selected with the properties that for each l 2F exactly one of l or
l lies in F+ and for each a, b2F+, if a+ b2F then a+ b2F+.
With F and F+ fixed, by a simple restricted root we mean a positive restricted root
that does not decompose as the sum of two or more other positive restricted roots. Write
D for the set of simple restricted roots. We call D a base of F.
Proposition 2.21. There are dima simple restricted roots. D is linearly independent and spans
F. F+ SpanZ+(D) and F SpanZ (D). If a, b2D, then a b /2F.
In case g = gZ gS is reductive, then when we refer to the restrict root space de
composition or restricted root system of g, we mean respectively the restricted root space
decomposition and restricted root system of gS.
Example 19. Let g = sl(R3). With respect to the Cartan involution q(M) = Mt, the Car
tan decomposition g = k+p is given by k = so(3) and p = M2sl(R3) M is symmetric .
a p may be chosen as consisting of diagonal traceless matrices, and then m = 0 since
skewsymmetric matrices have zeros along the diagonal. For each pair (i, j) with i, j 3
and i6= j we have the restricted root li,j corresponding to the onedimensional root space
gli,j = Rei,j giving the restricted root space decomposition
sl(R3) = a ?+
i,j 3
i6=j
Rei,j
similar to the root space decomposition of sl(C3) given in example 17.
27
Example 20. Consider g = sl(C3R) as a Lie algebra over R. The map q defined by q(M) =
M is a Cartan involution on g with corresponding Cartan decomposition sl(C3R) =
su(3) + isu(3) with k = su(3) and p = isu(3) (cf. example 5). Chose a p to consist
of the diagonal matrices in p; namely, a consists of traceless diagonal matrices with pure
imaginary entries. Then m consists of the diagonal matrices in k ? traceless diagonal
matrices with real entries. For each pair (i, j) with i, j 3 and i 6= j we have a restricted
root li,j where the corresponding restricted root space gli,j is 2dimensional and takes the
form
gli,j = a1ei,j + a2iei,j a1, a2 2R .
The restricted root space decomposition is then given by the vector space direct sum
sl(C3R) = a ?+m ?+
i,j 3
i6=j
gli,j.
2.6 Parabolic subalgebras
In this section, we will define Borel subalgebras and parabolic subalgebras of re
ductive Lie algebras over a complexlike field K or over R. We first consider parabolic
subalgebras in the case that g is semisimple over K and close by extending the definition
to the larger class of Lie algebras.
Definition 2.23. Let K be complexlike. Let g be semisimple over K. A Borel subalgebra
b of g is a subalgebra of the form
b = h ?+
b2F+
gb
where h is a Cartan subalgebra of g and F is the root system of g relative to h.
Example 21. In case g = sl(C3) (as a complex Lie algebra) and h consists of diagonal
matrices, b is the subalgebra of g consisting of upper triangular matrices.
28
0
@
1
A
0
@
1
A
0
@
1
A
0
@
1
A
q(?) = b q(fa1g) q(fa2g) q(fa1, a2g) = g
Figure 2.3: Standard parabolic subalgebras of sl(C3)
Definition 2.24. Let K be complexlike. Let g be semisimple over K. With notation as
above, a standard parabolic subalgebra of g relative to the Cartan subalgebra h is a subalge
bra q of g satisfying b q g. A parabolic subalgebra of g is a subalgebra q g that is a
standard parabolic subalgebra for some appropriate choice of h.
Example 22. Where g = sl(C3) and b is as above, any standard parabolic subalgebras of g
consists of block upper triangular matrices. Each parabolic subalgebra of g is simply one
of the standard parabolic subalgebras conjugated by a change of basis of C3.
Proposition 2.22. The standard parabolic subalgebras of g relative to b are in onetoone cor
respondence with subsets of D. Explicitly, for a subset D0 D, the corresponding parabolic
subalgebra q = q(D0) is
q = h ?+
b2F0
gb
where
F0 = F+[ F\SpanD0 ,
ie, F0 consists of all positive roots and the negative roots spanned by D0.
Example 23. Since D = fa1, a2g, g = sl(C3) has four standard parabolic subalgebras,
illustrated in figure 2.3. Additionally, figure 2.4 gives an illustration of F0 when D0 =
fa2g. That a particular root b is included in F0 is indicated by placing at the head of b,
while b /2F0 is indicated with .
29
a1  a1
a1 +a2
a
1 a2
A
A
A
A
AAU
a
2
A
A
A
A
AAK
a2
Figure 2.4: F0 where D0 =fa2g
Since a parabolic subalgebra differs from a standard parabolic subalgebra only by
an automorphism of g, proofs valid for any parabolic subalgebra q g need only con
sider the case where q is a standard parabolic subalgebra. We will often make use of this
principle in the sequel.
In case g = gZ gS is reductive over K, a Borel subalgebra b of g is of the form
b = gZ bS where bS = b\gS is a Borel subalgebra of gS. A parabolic subalgebra q of g
is of the form q = gZ qS where qS = q\gS is a parabolic subalgebra of gS.
Having defined parabolic subalgbras in the complexlike semisimple and reductive
cases, we now extend the definition to include real semisimple and reductive Lie algebras.
Definition 2.25. Let g be a reductive Lie algebra over R. A parabolic subalgebra q of g is a
subalgebra such that the complexification ?q is a parabolic subalgebra of ?g.
Notice that because (?g)Z = d(gZ), a parabolic subalgebra q of a real reductive g = gZ
gS has the form q = gZ qS, where qS is a parabolic subalgebra of g. In addition, parabolic
subalgebras of a real Lie algebra exhibit a structure theory relating to the restricted root
30
space decomposition analogous to the structure theory of parabolic subalgebras of Lie
algebras over a complexlike field.
Proposition 2.23. Let g be reductive over R. Let q be a parabolic subalgebra of g. We may choose
a restricted root space decomposition
g = a ?+m ?+
l2F
gl
with restricted root system F and set of simple restricted roots D so that q has the form
q = a ?+m ?+
l2F0
gl
where
F0 = F+[ F\SpanD0 ,
for an appropriate subset D0 of D.
2.7 Langland?s decomposition
What follows is perhaps not part of the classical theory of Lie algebras, but can be
found in chapter V, section 7 of [17] in case g is complexlike and in chapter VII, section 7
of [17] in case g is real.
Let g be semisimple over a complexlike fieldKor overRand let q g be a parabolic
subalgebra. Without loss of generality, we may assume that q arrises from a (restricted)
root space decomposition of g.
In the complexlike case, we have the following situation:
g = h ?+ b2Fgb, where
h is a Cartan subalgebra of g,
F is the root system of g relative to h,
31
D is a base of F,
D0 D is the subset of D corresponding to q, and
F0 = F+[(F\SpanD0).
Then q = h ?+ b2F0gb.
Considering the case where g is real, we have the analogous situation:
g = a ?+m ?+ l2Fgl where,
F is the restricted root system of g relative to a,
D is a set of simple restricted roots of F,
D0 D is the subset of D corresponding to q, and
F0 = F+[(F\SpanD0),
so that q = a ?+m ?+ l2F0gl.
F0 may be partitioned into two subsets, F0\ F and F0n F. This partition of F
results in a vector space direct sum decomposition of q as
q = l ?+n
where
l = h ?+
b2F0\ F
gb
and
n =
b2F0n F
gb.
Proposition 2.24. Notation as above, n is an ideal of q, l is a subalgebra of q, and l is reductive.[17,
Ch. V, ?7; Ch. VII, ?7]
32
Definition 2.26. Notation as above, l is called the Levi factor of q and n is called the nilrad
ical of q. The decomposition q = l ?+n is called Langland?s decomposition. (Notice that since
n is an ideal, q is the Lie algebra semidirect sum of the subalgebra l and the ideal n, and
we may write q = lnn.)
Finally, we extend this terminology and these notions to the case there g = gZ gS is
reductive and q = gZ qS, by simply writing
q = gZ (l ?+n)
where l is the Levi factor and n is the nilpotent radical of qS. In such case, we say l (res. n)
is the Levi factor (res. nilradical) of q and of qS interchangeably.
Example 24. Let g = gl(C6) = CI6 sl(C6), let h consist of traceless diagonal matrices.
Then h has dimension 5. The root system F will embed into a fivedimensional eucliedan
space h , and so we need a base D consisting of five simple roots.
Write hi = ei,i ei+1,i+1 for 1 i 5. Then fhig spans h and the dual functionals
fh igspan h . By computing [hi, ej,j+1] for each pair (i, j)2f1, ..., 5g2 we find five simple
roots D =fa1, ..., a5g, recorded in table 2.2.
Root ai ai in terms offh ig Root space gai
a1 (2, 1, 0, 0, 0) Ce1,2
a2 ( 1, 2, 1, 0, 0) Ce2,3
a3 (0, 1, 2, 1, 0) Ce3,4
a4 (0, 0, 1, 2, 1) Ce4,5
a5 (0, 0, 0, 1, 2) Ce5,6
Table 2.2: Simple roots of sl(C6)
The root spaces of sl(C6) are listed in table 2.3. We enumerate each (positive) root
b 2 F as a vector with respect to the basis D = fa1, ..., a5gand also with respect to the
basisfh 1, ..., h 2g.
33
b in terms of D Root space gb Root space g b b in terms offh ig
(1, 0, 0, 0, 0) Ce1,2 Ce2,1 (2, 1, 0, 0, 0)
(0, 1, 0, 0, 0) Ce2,3 Ce3,2 ( 1, 2, 1, 0, 0)
(0, 0, 1, 0, 0) Ce3,4 Ce4,3 (0, 1, 2, 1, 0)
(0, 0, 0, 1, 0) Ce4,5 Ce5,4 (0, 0, 1, 2, 1)
(0, 0, 0, 0, 1) Ce5,6 Ce6,5 (0, 0, 0, 1, 2)
(1, 1, 0, 0, 0) Ce1,3 Ce3,1 (1, 1, 1, 0, 0)
(0, 1, 1, 0, 0) Ce2,4 Ce4,2 ( 1, 1, 1, 1, 0)
(0, 0, 1, 1, 0) Ce3,5 Ce5,3 (0, 1, 1, 1, 1)
(0, 0, 0, 1, 1) Ce4,6 Ce6,4 (0, 0, 1, 1, 1)
(1, 1, 1, 0, 0) Ce1,4 Ce4,1 (1, 0, 1, 1, 0)
(0, 1, 1, 1, 0) Ce2,5 Ce5,2 ( 1, 1, 0, 1, 1)
(0, 0, 1, 1, 1) Ce3,6 Ce6,3 (0, 1, 1, 0, 1)
(1, 1, 1, 1, 0) Ce1,5 Ce1,5 (1, 0, 0, 1, 1)
(0, 1, 1, 1, 1) Ce2,6 Ce2,6 ( 1, 1, 0, 0, 1)
(1, 1, 1, 1, 1) Ce1,6 Ce6,1 (1, 0, 0, 0, 1)
Table 2.3: Root spaces of sl(C6) relative to h
Take D0 = fa1, a2, a4g. The standard parabolic subalgebra q corresponding to D0
consists of block upper triangular matrices corresponding to the partition 3 + 2 + 1 of 6,
as illustrated in figure 2.5.
q =
0
BB
BB
BB
@
1
CC
CC
CC
A
Figure 2.5: The parabolic subalgebra q gl(C6) corresponding to D0 =fa1, a2, a4g
Consideration of the root systemFshows thatF\ F =f a1, a2, (a1 +a2), a4g.
Fn F consists of the remaining positive roots.
l (respectively n) consists of the block diagonal matrices (block strictly upper trian
gular matrices) in q that preserve the existing block structure, illustrated in figure 2.6.
34
l =
0
BB
BB
BB
@
1
CC
CC
CC
A
, n =
0
BB
BB
BB
@
1
CC
CC
CC
A
Figure 2.6: The Levi factor decomposition of qS sl(C6)
As a reductive Lie algebra, l decomposes as l = lZ lS. Write hi = ei,i ei+1,i+1 for
1 i 5. Direct computation shows that center lZ is two dimensional:
lZ =fa(h1 + 2h2 + 3h3 + 3h4)+ b(h4 + 2h5)ja, b2Cg.
Perhaps more naturally, we may describe lZ in terms of matrices, illustrated in figure 2.7.
l =
0
BB
BB
BB
@
a
a
a
b
b
3a 2b
1
CC
CC
CC
A
Figure 2.7: A representative member of lZ
A Cartan subalgebra of l is spanned by h1, h2, and h4, and lS is the Lie algebra direct
sum of simple ideals isomorphic to sl(C3) and sl(C2) whereby l = C2 sl(C3) sl(C2).
2.8 The center of a parabolic subalgebra
We conclude this chapter with a lemma ? characterizing the center of parabolic sub
algebras ? that will be required later. We did not find the following result in the standard
texts, but it is elementary. As such, we presume it is already wellknown, and we include
it here rather than in chapter 3.
35
Lemma 2.25. Let q = gZ qS be a parabolic subalgebra of the reductive Lie algebra g = gZ gS
over a complexlike field K or over R. The center of q is gZ.
Proof. We consider first the special case where g = h ?+ b2F is semisimple over K. We
assume without loss of generality that q is a standard parabolic subalgebra and write
q = h ?+ b2F0gb.
Let z2qZ so z = zh + b2F0 zgb. Then for any x2q we have
0 = [z, x] = [zh, x]+
b2F0
[zgb, x]. (2.3)
Specifically, for 06= x2ga with a2F0, equation 2.3 becomes
0 = a(zh)x {z }
2ga
+
b2F0
[zgb, x] {z }
2ga+b
and by direct sum decomposition 0 = a(zh) for all a 2F0. Since F0 spans h it must be
the case that zh = 0, so z = b2F0 zgb.
Now, let 06= h2h and apply equation 2.3. We see
0 =
b2F0
[zgb, h] =
b2F0
b(h)zgb
which by direct sum decomposition yields 0 = b(h)zgb for all b2F0. Since h is arbitrary
in h, zgb = 0 for all b, so z = 0.
Having established that qZ = 0 when g is semisimple over K, that qZ = gZ when g is
reductive over K follows from the Lie algebra direct sum decomposition q = gZ qS. We
now consider the case where q is a parabolic subalgebra of a real reductive g. We have ?q
is a parabolic subalgebra of ?g by definition. Then
gZ + igZ = d(gZ) = (?g)Z = (?q)Z {z }
by above case
= d(qZ) = qZ + iqZ. (2.4)
36
Finally, by Ado?s Theorem (proposition 2.5), we may assume that g consists of real matri
ces, so that we may seperate the real and imaginary part in equation 2.4, giving gZ = qZ,
as desired.
37
Chapter 3
Derivations of Parabolic Lie Algebras
We begin this chapter by stating the main result of this dissertation:
Theorem. Let q be a parabolic subalgebra of a reductive Lie algebra g over R or over a complex
like field. Let L be the set of all linear transformations mapping q into qZ that send [q,q] to 0.
Then L is an ideal of Derq and Derq decomposes as the direct sum of ideals
Derq = L adq.
Our main result is valid for R or for any complexlike field (such as C). However, the
method of proof in the two cases is quiet different. The proof of the complexlike case is
highly technical: given a derivation D on q, we explicitly construct a linear map L and an
element x2q such that D = L +ad x, after which we prove that our construction satisfies
the stated properties. The real case, in contrast, is highlevel and abstract, appealing to
the complex case of the central theorem as applied to ?q, the complexification of the real
parabolic subalgebra q.
Lie algebras over complexlike fields support a more regular structural decompo
sition then real Lie algebras affords. In particular, Langland?s decomposition ? while
possible in the former case ? is completely unnecessary in order to prove theorem 3.1.
Because of the less regular structure of real Lie algebras, Langland?s decomposition be
comes an important tool for the proof of theorem 3.4
3.1 The algebraicallyclosed, characteristiczero case
Throughout this section we use the following notational conventions:
38
K denotes a complexlike field;
g = gZ gS denotes a reductive Lie algebra over K, where
gZ is the center of g, and
gS is the maximal semisimple ideal of g;
q = gZ qS is a given parabolic subalgebra of g, where
qS = q\gS is a parabolic subalgebra of gS.
We choose a Cartan subalgebra h, a root system F, and a base D compatible with qS in the
sense that qS is a standard parabolic subalgebra of gS relative to (h,F,D) and corresponds
to a subset D0 D. Then
qS = h ?+
a2F0
Kxa,
where
F0 = F+[ F\SpanD0
and where each xa is chosen arbitrarily from the onedimensional root space it spans.
Define t and c by
t = h\[q,q] and
c = Spanf[xa, x a]ja2DnD0g.
Claim. h decomposes as h = c ?+t.
Proof. Notice that h = Spanf[xa, x a]ja2Dg and that t = Spanf[xa, x a]ja2D0g.
From these observations, we see that c\t = 0 and that Span(c[t) = h.
39
Noting that [q,q] = t ?+ a2F0Kxa, we arrive at the desired vector space directsum
decompositions of q:
q = gZ ?+h ?+
a2F0
Kxa
= gZ ?+
hz}{
c ?+t ?+
a2F0
Kxa
 {z }
[q,q]
= gZ ?+c ?+[q,q].
We take a moment to note that alternatively c may have been chosen so that it coin
cides with the center of l in Langland?s decomposition q = l ?+n. This approach is not
required in order to prove the complexlike case, but it is taken in order to simplify the
proof of theorem 3.4 in the real case.
For the remainder of the section, we assume all of the notational conventions de
scribed above without further mention, starting with a restatement of the central theorem
in terms of the adopted notation.
Theorem 3.1. For a parabolic subalgebra q = gZ qS of a reductive Lie algebra g = gZ gS
over the complexlike field K, the derivation algebra Derq decomposes as the direct sum of ideals
Derq = L adq,
where L consists of all Klinear transformations on q mapping into qZ and mapping [q,q] to 0.
Explicitly, for any root system F with respect to which q is a standard parabolic subalgebra,
q decomposes as q = gZ ?+c ?+[q,q] and the ideal L consists of all Klinear transformations on q
that map gZ +c into gZ and map [q,q] to 0, whereby
Derq = Hom
K
(gZ ?+c,gZ) qS.
40
We must explain what we mean by HomK(gZ ?+c,gZ) as a Lie algebra, since it is
merely a space of linear maps and does not come equipped with a Lie bracket by de
fault. For vector spaces V1, V2, we consider the space HomK(V2, V1) an abelien Lie alge
bra. Then, HomK(V1 ?+ V2, V1) may be realized as the Lie algebra semidirect sum
Hom
K
(V1 ?+ V2, V1) = gl(V1)nHom
K
(V2, V1)
with the action of gl(V1) on HomK(V2, V1) defined by
f g = f g 8f 2gl(V1), g2Hom
K
(V2, V1).
This definition is canonical in the sense that if we fix bases for V1 and V2, then HomK(V1 ?+ V2, V1)
is identified with the subalgebra of gl(V1 ?+ V2) consisting of block matrices of the form
illustrated in figure 3.1 (compare to figure 1.2), and the Lie bracket defined by the action
above coincides with the standard Lie bracket of matrices, ie, [M, N] = MN NM.
V1 V2V
1
V2 0 0
Figure 3.1: Embedding of HomK(V1 ?+ V2, V1) in gl(V1 ?+ V2)
Proof of theorem 3.1. For clarity, the proof of the theorem is organized into a progression
of claims. The first three claims establish that an arbitrary derivation may be written as a
sum of an inner derivation and a derivation mapping gZ ?+c to gZ and [q,q] to 0. To this
end, let D be an arbitrary derivation of q.
Claim 1. There is an x 2 q such that D ad x maps c to gZ, annihilates t, and stabilizes
each root space Kxa.
Let h, k 2h be arbitrary and write D(h) = z + h0+ g ag(h)xg and D(k) = c + k0+
g ag(k)xg with z, c 2 gZ and h0, k0 2 h and ag(h), ag(k) 2 K. Recall [h, k] = 0 since
41
h, k2h and consider D([h, k]).
0 = D([h, k])
= [h, D(k)] [k, D(h)]
=
"
h, c + k0+
g
ag(k)xg
#
"
k, z + h0+
g
ag(h)xg
#
=
"
h,
g
ag(k)xg
#
"
k,
g
ag(h)xg
#
=
g
ag(k)[h, xg]
g
ag(h)[k, xg]
=
g
(ag(k)g(h) ag(h)g(k)) xg.
So
ag(k)g(h) ag(h)g(k) = 0 for all g2F0, h, k2h. (3.1)
Furthermore, for any pair h, k for which g(h)6= 0 and g(k)6= 0, we have that
ag(h)
g(h) =
ag(k)
g(k) .
This observation, along with the fact that g(h) 6= 0 for at least one h 2 h, allows us to
associate with each g2F0 the numerical invariant
dg = ag(h)g(h)
independently of our choice of h. Notice that ag(h) dgg(h) = 0 by definition when
g(h)6= 0. If g(h) = 0, the same equality still holds, as equation 3.1 becomes ag(h)g(k) =
0 for all k 2 h. Since at least one k 2 h satisfies g(k) 6= 0 we have ag(h) = 0 in case
g(h) = 0, giving
ag(h) dgg(h) = 0 for all h2h. (3.2)
42
Now, set x = g dgxg. Write D0 = D ad x. We will show that D0 maps c to gZ,
annihilates t, and stabilizes each root space Kxa.
We first show that D0 maps h to gZ ?+ h. Let h 2 h be arbitrary and again write
D(h) = z + h0+ g ag(h)xg. We have that
D0(h) = D(h) ad x(h)
= z + h0+
g
ag(h)xg
g
dg ad xg(h)
= z + h0+
g
ag(h)xg
g
dg ad h(xg)
= z + h0+
g
ag(h)xg
g
dgg(h)xg
= z + h0+
g
(ag(h) dgg(h)) {z }
=0 by 3.2
xg
= z + h0,
affirming the assertion.
Having established that D0 maps h into gZ ?+h, we have left to show that D0 annihi
lates t and stabilizes eachKxa. Let h2h and a2F0 be arbitrary, and write D0(h) = z + h0
and D0(xa) = c + k + g bgxg with z, c 2 gZ and h0, k 2 h and bg 2 K. Consider
D0([h, xa]). On one hand,
D0([h, xa]) = D0(a(h)xa)
= a(h)D0(xa)
= a(h)c +a(h)k +
g
a(h)bgxg. (])
43
On the other hand,
D0([h, xa]) = [D0(h), xa]+[h, D0(xa)]
= [z + h0, xa]+
"
h, c + k +
g
bgxg
#
= [h0, xa]+
g
bg[h, xg]
= a(h0)xa +
g
g(h)bgxg
= a(h0)+a(h)ba xa +
g6=a
g(h)bgxg. ([)
By equating ] and [ and by direct sum decomposition of q we obtain
a(h)c = 0, (3.3)
a(h)k = 0, (3.4)
a(h)bg = g(h)bg for g6= a, and (3.5)
a(h)ba = a(h0)+a(h)ba. (3.6)
Since h is arbitrary, equations 3.3 and 3.4 give c = 0 and k = 0 respectively. Second,
equations 3.5 give us bg(g a)(h) = 0 for all g 6= a. If any one bg 6= 0, then we would
have g = a, a contradiction, so each bg = 0, whence D0 stabilizes each root space.
Next, equation 3.6 gives us 0 = a(h0). Since a is arbitrary in F0 and F0 contains a
basis of h , h0 = 0, so D0(h) gZ. Since derivations in general stabilize [q,q], D0(t)
gZ\[q,q] = 0, so D0 annihilates t. The claim is verified.
Claim 2. There is an h2h whereby D ad x ad h annihilates [q,q].
We have the D0 = D ad x maps c to gZ, annihilate t, and stabilize each root space
Kxa. For each g2F0 write
D0(xg) = cgxg
44
with cg 2K. Taking each a2D, the scalars ca define a linear functional
?c : h !C.
We first verify that for each g2F0, cg = ?c(g).
We begin with g2F0\F+. Let g = a1 + ... + ak with each ai 2D and where each
sequential partial sum a1 + ... +ai 2F0. Then
axg = [ [[xa1, xa2], xa3], , xak]
for some 06= a2K. Apply D0 to both sides. Since D0 is a derivations, we have
cgaxg = D0[ [[xa1, xa2], xa3], , xak]
= D0(xa1), xa2 , xa3 , , xak
+ xa1, D0(xa2) , xa3 , , xak
+ [xa1, xa2], D0(xa3) , , xak
+ + [[xa1, xa2], xa3], , D0(xak)
= ca1 [ [[xa1, xa2], xa3], , xak]
+ ca2 [ [[xa1, xa2], xa3], , xak]
+ ca3 [ [[xa1, xa2], xa3], , xak]
+ + cak [ [[xa1, xa2], xa3], , xak]
= ca1 axg + ... + cak axg
= ?c(a1 + ... +ak)axg
= ?c(g)axg
whereby cg = ?c(g) for all g2F0\F+.
45
Next let g2F0\F . Consider [xg, x g]2t, and apply D0.
0 = D0([xa, x a])
= [D0(xg), x g]+[xg, D0(x g)]
= cg[xg, x g]+ c g[xg, x g]
= (cg + c g)[xg, x g].
Since [xg, x g]6= 0, we have cg + c g = 0 so
cg = c g
= ?c( g) (since g2F0\F+)
= ?c(g)
as desired.
Next, we use the canonical isomorphism Y : h ! h [22, Ch. VII, ?4] to produce
Y(?c) = h2h. Notice that for each g2F0 we have the identity
?c(g) g(h) = 0 (3.7)
by the definition of the canonical isomorphism.
The claim is that D0 ad h annihilates [q,q]. Since [h,t] = 0, we need only check that
D0 ad h maps each xg to 0.
(D0 ad h)(xg) = ?c(g)xg g(h)xg
= (?c(g) g(h)) {z }
=0 by 3.7
xg
= 0
46
verifying the claim.
Claim 3. D = L + ad p for some p 2 q and some derivation L which maps gZ ?+c to gZ
and maps [q,q] to 0.
Set p = x + h as above and set L = D ad p = D0 ad h. Then D = L + ad p as
desired. We note that since L is the difference of two derivations, L is itself a derivation.
We know from claim 2 that L annihilates [q,q]. We must check that L maps gZ ?+c to gZ.
We have already seen that gZ is the center of q, and more, that a derivation of q must
stabilize the center of q. What is left to verify claim 3 is to check that L maps c into gZ. Let
c2c be arbitrary. We have
L(c) = D0 ad h (c)
= D0(c) [h, c]
= D0(c){z}
2gZ by claim 1
verifying claim 3.
Since D was arbitrary, we now have that Derq is spanned by adq and the subset of
Derq consisting of derivations that map gZ ?+c to gZ and [q,q] to 0. The next three claims
establish facts about the relationship between these two sets.
Claim 4. L Derq.
L is defined as the set of Klinear endomorphisms of q mapping into the center of q
and mapping [q,q] to 0. We will show that any such linear map is indeed a derivation of
q. Suppose L : q !q is any Klinear map satisfying L(q) gZ and L([q,q]) = 0. Then,
for any x, y2q we have
[
2gZz}{
L(x), y] {z }
=0
+[x,
2gZz}{
L(y)] {z }
=0
= 0 = L(
2[q,q]z}{
[x, y]) {z }
=0
47
so L is a derivation.
Claim 5. L is an ideal of Derq.
First we must show that L is closed under taking linear combinations of members.
Let L1, L2 2L. We must show that L1 + kL2 2L (where k2K). Let x2q. We have
(L1 + kL2)(x) = L1(x){z}
2gZ
+k L2(x){z}
2gZ
2gZ
so L1 + kL2 maps q into gZ. Next, let y2[q,q]. We have
(L1 + kL2)(y) = L1(y){z}
=0
+k L2(y){z}
=0
= 0
so L1 + kL2 sends [q,q] to 0.
Second, let L 2 L and D 2 Derq. We must show that [D, L] = D L L D 2 L,
ie, that [D, L] maps q into gZ and maps [q,q] to 0. Recall that D(gZ) gZ and D([q,q])
[q,q]. Let x2q. Consider [D, L](x).
[D, L](x) = D(
2gZz}{
L(x)) {z }
2gZ
L(
2qz}{
D(x)) {z }
2gZ
2gZ
so [D, L] maps q into gZ. Now, let y2[q,q] and consider [D, L](y).
[D, L](y) = D(
=0z}{
L(y)) {z }
=0
L(
2[q,q]z}{
D(y)) {z }
=0
= 0
so [D, L] maps [q,q] to 0?ie, [D, L]2L?verifying the claim.
Claim 6. L and adq intersect trivially.
48
Suppose D 2 L\adq. Since D 2 L, D maps q into gZ. Since D 2 adq, D maps q
into [q,q]. So, D maps q into gZ\[q,q] = 0, whereby D = 0, completing the proof of the
theorem.
As a simple application, we will use theorem 3.1 to derive a formula for the dimen
sion of Derq in terms of g and q.
Corollary 3.2. For q a parabolic subalgebra of the reductive Lie algebra g = Cn gS over K, and
with notation as above, the dimension of Derq is given by
dim Derq = n +jDj jD0j n + dimqS.
Proof. The corollary follows from the isomorphism Derq = HomK(gZ ?+c,gZ) adq, ie,
the facts that for any vector spaces V1, V2 the dimension of V1 ?+ V2 is the sum dim V1 +
dim V2 and the dimension of HomK(V1, V2) is the product (dim V1)(dim V2). Now, dimc =
jDj jD0j, and dim adq = dimqS because adq = q/qZ = qS. Applying the mentioned
general principles completes the proof.
We will note now that a similar dimensioncounting result will not be possible ?
in general ? in the real case, though formulas may be possible for classes of certain
parabolic subalgebras of specific real Lie algebras. Notice that the statement and proof
of corollary 3.2 rely heavily on the explicit description of the ideal L and knowledge of
the dimension of the subspace c q, which in turn rely on properties of the root space
decomposition for Lie algebras over algebraicallyclose, characteristiczero fields. Anal
ogous properties fail to hold in general for the restricted root space decomposition of a
real Lie algebra; however, the dimension of Derq in the real case may be calculated on a
examplebyexample basis.
49
3.2 The real case
In this section, g = gZ gS denotes a reductive real Lie algebra with center gZ and
maximal semisimple ideal gS. q = gZ qS is a parabolic subalgebra of g, where qS = q\gS
is a parabolic subalgebra of gS.
We begin by proving the limited sense of the central theorem in the context of real
Lie algebras. The proof will rely heavily on the complexification ?g of g, to which we will
apply theorem 3.1. Afterwards, we consider the restricted root space decomposition of g
and expand upon the central theorem.
Theorem 3.3. For a parabolic subalgebra q = gZ qS of a reductive Lie algebra g = gZ gS
over R, the derivation algebra Derq decomposes as the sum of ideals
Derq = L adq,
where L consists of all Rlinear transformations on q mapping into qZ and mapping [q,q] to 0.
Proof. We may assume without loss of generality that g is realized as a set of real matrices
by Ado?s Theorem (proposition 2.5). We fix the following notation:
i denotes the imaginary unit;
?g = g ?+ ig = (gZ ?+ igZ) (gS ?+ igS) denotes the complexification of g;
q = gZ qS denotes a parabolic subalgebra of g;
?q = q ?+ iq = (gZ ?+ igZ) (qS ?+ iqS) is a parabolic subalgebra of ?g; and
cgZ = gZ ?+ igZ = ?gZ denotes the center of ?g.
Given a derivation D of q, we have a corresponding derivation ?D of ?q given by ?D(x +
iy) = D(x)+ iD(y). As a derivation of ?q, ?D decomposes as ?D = L + ad(x + iy) with L
50
mapping ?q into ?gZ and mapping [?q, ?q] to 0 and with x, y2q. Note that L sends [q,q] to 0,
since [q,q] [?q, ?q].
Let u2q. ?D stabilizes q, so we have
D(u) = ?D(u) = L(u)+ ad(x + iy)(u) = L(u)+[x, u]+ i[y, u]2q.
Now, L(u)2 ?gZ = gZ + igZ, so we may write L(u) = v1 + iv2 with v1, v2 2gZ. Then
D(u) = v1 + iv2 +[x, u]+ i[y, u] = (v1 +[x, u])+ i(v2 +[y, u])2q
so v2 + [y, u] = 0, and by the directsum decomposition q = gZ + [q,q], v2 = 0 and
[y, u] = 0. In particular, we have L(u) = v1, so L maps q into gZ. Furthermore, since u
was arbitrary and [y, u] = 0, we have y 2 gZ. Since y 2 gZ, we have for any arbitrary
z = u + iv2g
ad(x + iy)(z) = ad x(z)+ i ad y(z)
= [x, u + iv]+ i[y, u + iv]
= [x, u] [y, v]{z}
=0
+i([x, v]+[y, u]{z}
=0
)
= [x, u]+ i[x, v]
= [x, u + iv]
= ad x(z)
thus we have
ad(x + iy) = ad x.
We now have D = Ljq +ad x with x2q and Ljq anRlinear transformation mapping
q to gZ and [q,q] to 0, as desired. We have left to check that arbitrary Rlinear maps
51
sending q to gZ and [q,q] to 0 are derivations, that L as described is an ideal of q, and
that L and adq intersect trivially, the proofs of which are identical to the proofs given of
claims 4, 5, and 6 of theorem 3.1, respectively.
We will now examine the relationship between the direct sum decompositon of Derq
and the restricted root space decomposition of g. Given a parabolic subalgebra q =
gZ qS of a reductive real Lie algebra g = gZ gS, we may choose a restricted root space
decomposition of g that is compatible with q in the sense that q is a standard parabolic
subalgebra of g. We may then decompose q into the sum of q = gZ ?+ c ?+ [q,q] where
c is an appropriatelychosen complimentary subspace, similar to the complexlike case.
To achieve this decomposition, we rely on Langland?s decomposition of qS, described in
chapter 2. We fix the following notation pertaining to the restricted root space decompo
sition of g:
g = gZ ?+a ?+m ?+ a2Fga;
D a base of F;
D0 D corresponding to qS;
F0 = F+[(F\SpanD0); and
q = gZ ?+a ?+m ?+
g2F0
gg
 {z }
qS
.
Write Langland?s decomposition of qS:
l = a ?+m ?+ g2F0\ F0gg and
n = g2F0n F0 gg
so that qS = l ?+n with l reductive and n nilpotent. Write
c for the center of l and
52
lS for the unique semisimple ideal of l
so that l = c ?+lS.
Claim. [q,q] = lS ?+n
Proof. Let x, y 2 q. We must show [x, y] 2 lS ?+ n. Without loss of generality, we may
assume x, y2qS, since their projections onto gZ are lost upon applying bracket.
Write x = xl + xn and y = yl + yn with xl, yl2l and xn, yn2n. Then
[x, y] = [xl + xn, yl + yn]
= [xl, yl]{z}
2lS
+[xl, yn]+[xn, yl]+[xn, yn] {z }
2n
2lS ?+n
since l is reductive and n is an ideal.
Thus we arrive at the desired decomposition,
q = gZ ?+l ?+n{z}
gS
= gZ ?+c ?+lS ?+n
= gZ ?+c ?+[q,q].
Theorem 3.4. For any root system F with respect to which q is a standard parabolic subalgebra, q
decomposes as q = gZ ?+c ?+[q,q] and the ideal L of Derq consists of all Rlinear transformation
on q that map gZ ?+c to gZ and map [q,q] to 0, whereby
Derq = Hom
R
(gZ ?+c,gZ) adq.
Proof. The proof is essentially done. The majority is merely the description of the decom
position of q, already done above. We have left to show only that L = HomR(gZ +c,gZ),
53
which is obvious in light of the decomposition q = gZ ?+c ?+[q,q]. The sceptical reader is
referred to figure 1.2, illustrating the block form of matrices in L.
Because of the coarseness of the restricted root space decomposition, the dimension
of c is not readily available in the real case, in contrast to the complexlike case. dimc
may be calculated if given a specific real Lie algebra g and a specific standard parabolic
subalgebra q.
3.3 Corollaries
The following three corollaries represent extremal cases of the central theorem. Corol
lary 3.5 applies to arbitrary parabolic subalgebras of a semisimple Lie algebra (ie, the case
gZ = 0). Corollaries 3.6 and 3.7 apply specifically to minimal parabolic subalgebras (ie,
Borel subalgebras) and maximal parabolic subalgebras (ie, the entire Lie algebra g), re
spectively.
Corollary 3.5. For a parabolic subalgebra q of a semisimple Lie algebra g over the field F (where
F is complexlike or R), the derivation algebra Derq satisfies
Derq = adq.
Proof. By the central theorem, Derq = L adq, and since gZ = 0, we have L = 0.
Corollary 3.5 was proven for Borel subalgebras of semisimle Lie algebras over an
arbitrary field by Leger and Luks in [19]. Tolpygo found the same result for parabolic
subalgebras of complex Lie algebras [29]. Our contribution is to include parabolic sub
algebras of real Lie algebras.
54
Corollary 3.6. For a Borel subalgebra b = gZ ?+ g0 ?+ a2F+ ga of the reductive Lie algebra
g = gZ gS over the field F (where F is complexlike or R), the derivation algebra Derb satisfies
Derb = Hom
F
(gZ ?+(g0)Z ,gZ) adb.
Proof. Write bS = g0 ?+ a2F+ ga. Since a2F+ ga is clearly the nilpotent radical of bS, the
Levi factor l = g0. Applying the central theorem gives the result.
Farnsteiner proved corollary 3.6 over a complexlike field in [10]. As with corollary
3.5, our contribution is to extend this result to Borel subalgebras of real Lie algebras.
Corollary 3.7. For a reductive Lie algebra g = gZ gS over the field F (where F is complexlike
or R), the derivation algebra Derg satisfies
Derg = gl(gZ) adg.
Proof. gS is its own Levi factor. Being semisimple, the center of gS is trivial, so L consists of
linear maps stabilizing gZ and sending gS = [g,g] to 0, which is isomorphic to gl(gZ).
The final corollary provides a highlevel abstract description of Derq useful for dim
ensioncounting arguments. It is also satisfying on a theoretical level, since it relies on
simple constructions that can be carried out on any Lie algebra, suggests that the result
here for reductive Lie algebras might be generalized to larger classes of Lie algebras.
Recall that q/[q,q] is the minimal abelian quotient of q. Since q = gZ ?+c ?+[q,q], we
have gZ ?+c = q/[q,q]. Also, gZ = qZ, and adq = q/qZ, thus:
Corollary 3.8. For a parabolic subalgebra q of a reductive Lie algebra g over a complexlike field
or over R, we have
Derq = Hom(q/[q,q],qZ) (q/qZ).
Proof. Above.
55
Chapter 4
Zero Product Determined Derivation Algebras
Let g be a Lie algebra over an arbitrary field F. g is called zero product determined if for
each Fbilinear map j : g g !V into an Fvector space V, the condition that
j(x, y) = 0 whenever [x, y] = 0 (4.1)
implies the existence of an Flinear map f : g !V satisfying
j(x, y) = f [x, y] for all x, y2g. (4.2)
We will drop the reference to the base field F when the context is clear. However, the
reader should remember that the condition that a Lie algebra is zero product determined
is tied to the understood base field. Explicitly, a given Lie algebra g may consist of com
plex matrices and may be considered as either a real Lie algebra or a complex Lie algebra.
It is conceivable that g may be zero product determined as a real Lie algebra, but not zero
product determined as a complex Lie algebra, or vice versa.
A few remarks. First, some terminology. A bilinear map satisfying property 4.1 is
said to preserve zero products. Second, property 4.2 can be thought of as a map factoring
property. The bracket [ , ] can be thought of as a bilinear map m defined by
m :
8
>><
>>:
g g !g
(x, y)7![x, y]
56
Then the definition can be understood in terms of function composition as saying that
the bilinear map j factors as the composition of a linear map f and the Lie bracket m; in
symbols, g is zero product determined if an only if
j(x, y) = 0 whenever m(x, y) = 0 implies9f , j = f m.
In words, g is zero product determined if and only if every bilinear map j that preserves
zero products factors through the bracket map m.
Third remark: the setting as described above is not entirely desirable, since it com
bines notions of bilinearity and linearity, making the study problematic in certain settings.
An equivalent definition can be phrased entirely in terms of linear maps by considering
tensor products of vector spaces [5]. One may replace the Cartesian product g g with
the tensor product g g without ambiguity, since linear maps on the tensor product g g
are in onetoone correspondence with bilinear maps on the Cartesian product g g [22,
Ch. IX, ?8]. See figure 4.1 for a diagrammatic expression of the factorization j = f m in
the tensorproduct setting.
Finally, the above definition (along with the reformulated definition in terms of ten
sor products) works equally well for associative algebras?such as the algebra Rn n of n
by n real matrices with the usual matrix multiplication?and nonassociative, nonLie al
gebras if the bracket is replaced by an appropriate multiplication. In fact, the initial work
on zero product determined algebras was done in the context of matrix algebras, consid
ering the standard associative matrix product, the nonassociative Lie bracket, and the
nonassociative Jordan product [4].
For the remainder of this chapter, let K denote a complexlike field.
57
4.1 Zero product determined algebras
Definition 4.1. A Kalgebra is a pair (A, m) whereAis a Kvector space and
m :A KA !A
is an Klinear map. The image Im m is denotedA2.
Definition 4.1 encompasses Lie algebras when m is defined by m(x y) = [x, y]. (We
note in the Lie algebra case that A2 = [A,A].) The definition also includes associative
algebras (eg, matrix algebras under the usual matrix product, Banach algebras) and other
nonassociative algebras, such as Jordan algebras. In the sequel, we will suppress the
mention of the scalar field K when there is no danger of ambiguity.
The definition of zero product determined as applied to algebras is originally due to
Bre?ar, Gra?i?c, and S?nchez Ortega; it was motivated by applications to analysis on Ba
nach algebras [1, 4]. Definition 4.2, given below, is equivalent to that found in [4] but
rephrased in terms of linear maps on tensor products rather than in terms of bilinear
maps on Cartesian products [5]. A purely linear approach offers the advantage of con
sidering kernels and images of linear maps, alleviating certain difficulties found in the
bilinear approach. Consider that for a bilinear map j, the image of j is not necessarily a
subspace of the target vector space, and the notion of a kernel of j is nonexistent.
Definition 4.2. An algebra (A, m) is called zero product determined if for each vector space
V and for each linear map j :A A !V, if
j(a1 a2) = 0 whenever m(a1 a2) = 0,
then there is a linear map f :A2 !B whereby j factors through m as
j = f m.
58
A A j //
m
V
A2
f
77oo
ooo
ooo
Figure 4.1: Tensor definition of zero product determined
The figure diagrammatically expresses the factorization j = f m. All maps depicted are linear.
Existence of j and m is assumed a priori. f exists if j preserves zero products.
We reference several results on zero product determined algebras that will be used
in the sequel.
Proposition 4.1 (Theorem 2.3 in [5]). An algebra (A, m) is zero product determined if and only
if Ker m is generated by elementary tensors.
Some terminology will be helpful for understand the statement of the theorem. Ele
ments ofA Aare called tensors. An elementary tensor is a member ofA Aof the form
a1 a2 for a1, a2 2A. In general, an arbitrary tensor t2A Ais a linear combination of
elementary tensors, ie, t = ni=1 a(i)1 a(i)2 for some positive integer n.
WhileA Ais generated by elementary tensors through taking linear combinations,
an arbitrary subspace of A A may fail to be generated by the elementary tensors it
contains. In fact, there are nontrivial subspace of A A that contain no elementary
tensors other than the zero tensor 0 = 0 0.
Proposition 4.2 (Theorem 3.1 in [5]). Let I be an arbitrary set and for each i2 I let (Ai, mi) be
an algebra. Consider the algebra direct sum (A, m) = (Li2IAi,Li2I mi). (A, m) is zero product
determined if and only if each summand (Ai, mi) is zero product determined.
4.2 Parabolic subalgebras of reductive Lie algebras
For this section we adopt several of the notational conventions of [33] for clarity. Let
g = gZ gS be a finitedimensional reductive Lie algebra over K with gZ abelian and gS
semisimple. Let h be a Cartan subalgebra of gS and let F be the root system of gS relative
59
to h. Let D be a base of F, and denote the positive roots relative to D by F+. Then gS
decomposes as gS = h ?+ b2Fgb.
From each b 2 F+ we select a nonzero vector eb 2 gb. Then for each b there is a
unique e b 2g b with the property that eb, e b, and hb = [eb, e b] span a subalgebra of
gS isomorphic to sl2(K). Each gb for b 2F is spanned by the vector eb, and the vectors
ha for a 2D form a basis of h. For each b 2F and each h 2h we have [h, eb] = b(h)eb,
and for each pair b, g 2 F we have [eb, eg] 2 gb+g, from which we define Nb,g 2K by
[eb, eg] = Nb,geb+g.
Of any arbitrary parabolic subalgebra q of g, we assume without loss of generality
that it is a standard parabolic subalgebra of g corresponding to some subset D0 D.
More explicitly, the assumption is that that q = gZ
h ?+ b2F0 gb
, where F0 = F+[
(F\SpanD0).
In [33], the authors prove ? with minor error ? that the parabolic subalgebras of the
finitedimensional simple Lie algebras over K are zero product determined. Specifically,
lemma 2.2 of [33] contains the following unproven claim: For any a, g2F0 where a+g is
a root, at least one of a+2g or 2a+g is not a root. The root system of the simple Lie algebra
G2 provides a counterexample, as illustrated in figure 4.2.
6
?
3
+
QQ
QQ
Qs
QQ
QQ
Qk

A
A
AAU
A
A
AAK
a
g 2a+g
a+ 2g
Figure 4.2: G2 root system
Both a+ 2b and 2a+ b are roots.
60
Lemma 2.2 of [33] does however provide a suitable base case for an induction argu
ment, which we give below. We are given a parabolic subalgebra q and a bilinear map
j : q q ! V (V is some arbitrary vector space) of which we assume j(x, y) = 0
whenever [x, y] = 0. For each g2F0 we chose a dg 2h so that g(dg) = 1.
Following Wang et al, we define a linear map f : [q,q] !V by
f(ha) = j(ea, e a) for each a2D,
f(eg) = j(dg, eg) for each g2G,
and extending linearly. The basic theorem of [33] is to show that f([x, y]) = j(x, y) for all
x, y 2q. Lemma 2.2 of [33] gives a special case that Wang et al. use to facilitate the proof
of the basic theorem.
Lemma 4.3 (Lemma 2.2 in [33]). For a, g2F0 , if a+g6= 0, then
f([ea, eg]) = j(ea, eg).
We will need Wang et al.?s Lemma 2.1 for the proof of lemma 4.3. We state it now for
use later: for the proof we refer the reader to [33].
Proposition 4.4 (Lemma 2.1 in [33]). For all h2h and all b2F0, we have
f([h, eb]) = f(h, eb).
Proof of lemma 4.3. Let k be the largest integer such that ka + g is a root. The proof of
Lemma 2.2 in [33] shows that the proposition holds in case k = 0, 1. We proceed by
assuming for induction that the proposition hold for all pairs of roots b, d such that b +
d6= 0 and b+d is not a root.
Pick h2h so that g(h) = 0 and a(h) = 1. Let a0 = 1 and for each i from 1 to k let
ai = 1i ai 1Na,(i 1)a+g.
61
Notice that
iai ai 1Na,(i 1)a+b = 0 (4.3)
for each i from 1 to k. Then we have
"
h ea,
k
i=0
aieia+g
#
=
k
i=0
ai h, eia+g
k
i=0
ai ea, eia+g
=
k
i=0
ai(ia+g)(h)eia+g
k
i=0
aiNa,ia+ge(i+1)a+g
= g(h){z}
=0
eg +
k
i=1
iai ai 1Na,(i 1)a+g
 {z }
=0 by 4.3
eia+g ak [ea, eka+g] {z }
=0
= 0
and since j preserves zero products by assumption we arrive at
j
h ea,
k
i=0
aieia+g
!
= 0. (4.4)
By bilinearity of j we have
0 = j
h ea,
k
i=0
aieia+g
!
=
k
i=0
aij(h, eia+g)
k
i=0
aij(ea, eia+g),
and by proposition 4.4 and the definition of ai and Na,(i 1)a+g
aij(h, eia+g) = ai f([h, eia+g])
= ai f((ia+g)(h)eia+g)
= ai f(ieia+g)
= ai iN
a,(i 1)a+g
f([ea, e(i 1)a+g])
= ai 1 f([ea, e(i 1)a+g]).
62
for each i from 1 to k. Then equation 4.4 becomes
0 = a0 j(h, eg) {z }
=0
+
k
i=1
ai 1 f([ea, e(i 1)a+g])
k
i=0
aij(ea, eia+g)
=
k 1
i=0
ai f([ea, eia+g])
k 1
i=0
aij(ea, eia+g) ak j(ea, eka+g) {z }
=0
=
k 1
i=0
ai f([ea, eia+g]) j(ea, eia+g) .
For i 1, applying the inductive hypothesis to the pair a, ia + g gives f([ea, eia+g])
j(ea, eia+g) = 0, so the sum reduces to the i = 0 term:
0 = f([ea, eg]) j(ea, eg),
which is what we set out to show.
We now state the Basic Theorem of Wang et al. for use later. The remainder of the
proof is of course found in [33].
Proposition 4.5 (Basic Theorem in [33]). A parabolic subalgebra q of a simple Lie algebra g over
K is zero product determined.
The results of [5] allow us to generalize proposition 4.5 to reductive Lie algebras and
their parabolic subalgebras.
Lemma 4.6. If g is an abelian Lie algebra, then g is zero product determined.
Proof. To say g is abelian is to say Ker m = g g, which is generated by elementary tensors
x y 8x, y2g. By proposition 4.1, g is zero product determined.
Theorem 4.7. Let q = gZ qS be a parabolic subalgebra of a reductive Lie algebra g = gZ gS
over the field K. Then q is zero product determined.
63
Proof. gZ is zero product determined by lemma 4.6, qS is zero product determined by
proposition 4.5. The direct sum q = gZ qS is zero product determined by proposition
4.2.
In particular, the Borel subalgebra b = gZ ?+ b2F+ gb and the reductive Lie algebra
g are zero product determined.
4.3 Derivations of parabolic subalgebras
We now return to the original motivation of this dissertation. We apply the study of
zero product determined algebras to the derivation algebra Derq of a parabolic subalge
bra q = gZ qS of the reductive Lie algebra g = gZ gS over the field K.
Definition 4.3. Let n, k2Z 0. Denote by L(n, k) the subalgebra of gl(Cn+k) consisting of
matrices whose (n + i)th rows are zero for 1 i k.
L(n, k) consists of complex matrices with block form
0
B@
n k
n
k 0 0
1
CA.
As a Lie algebra, L(n, k) = mnn where: m = gl(Cn); n is abelian, consisting of n k ma
trices with trivial bracket; and the action of m on n is given by usual matrix multiplication
? la [x, y] = xy for all x2m, y2n.
Notice that L(n, 0) = gl(Cn), so L(n, k) is zero product determined by 4.7 when k = 0.
Lemma 4.8. L(n, k) is zero product determined.
Proof. Without loss of generality, we assume k 1. Write L = L(n, k), and define m :
L L // // [L, L] by m(x y) = [x, y]. By the ranknullity theorem, we have
dim Ker m = n4 + 2n3k + n2k2 n2 nk + 1.
64
We will exhibit a basis for Ker m consisting of elementary tensors.
Notation as above, L = mnn. m is zero product determined by theorem 4.7. By
proposition 4.1, Ker mjm m admits a basis consisting of n4 n2 + 1 elementary tensors of
the form x y with x, y2m and [x, y] = 0.
Since n is abelian, n is zero product determined by lemma 4.6, and proposition 4.1
provides n2k2 more elementary tensors of the form x y with x, y2n and [x, y] = 0. We
require 2n3k nk more linearly independent elementary tensors in Ker m.
Consider the 2n3k 2n2k tensors
Ti,j,l,q = ei,j el,n+q 2m n
and
Ti,j,l,q = el,n+q ei,j 2n m
for i, j, l n and q k with j6= l. Additionally, we have 2n2k 2nk tensors
Si,j,q = ei,j ei,j+1 ej,n+q + ej+1,n+q 2m n
and
Si,j,q = ej,n+q + ej+1,n+q ei,j ei,j+1 2n m
with i n, j n 1, and q k. Finally, we have nk tensors of the form
R(i, q) = ei,i + ei,n+q ei,i + ei,n+q 2(m ?+n) (m ?+n)
for i n and q k, giving the desired 2n3k nk elementary tensors in Ker m. We have
left to show that these tensors are linearly independent.
65
Expanding Si,j,q we see that
Si,j,q = ei,j ej,n+q ei,j+1 ej+1,n+q {z }
/2SpanfTi,j,l,qg
+ ei,j ej+1,n+q ei,j+1 ej,n+q {z }
2SpanfTi,j,l,qg
is not in the span of the Ti,j,l,q. A similar observation shows that Si,j,q is not in the span of
the Ti,j,l,q tensors.
Expanding R(i, q) we have
R(i, q) = ei,i ei,i {z }
2m m
+ ei,n+q ei,n+q {z }
2n n
+ ei,i ei,n+q + ei,n+q ei,i {z }
2m n ?+n m
.
Since ei,i ei,i and ei,n+q ei,n+q are in m m and n n, respectively, we may subtract
them, and we have left to consider R0(i, q) = ei,i ei,n+q + ei,n+q ei,i. R0(i, q) is not in
the span of Ti,j,l,q, Ti,j,l,q since individually ei,i ei,n+q and ei,n+q ei,i are not among the
T
i,j,l,q, Ti,j,l,q
. Now, consider S
i,i,q + Si,i,q if i < n (in case i = n we are done, since we
require j n 1 in Si,j,q). We have
Si,i,q + Si,i,q = ei,i ei,n+q + ei,n+q ei,i {z }
=R0(i,q)
+T ei,i+1 ei+1,n+q + ei+1,n+q ei,i+1
with T 2Span Ti,j,l,q, Ti,j,l,q , so we have
R0(i, q) = Si,i,q + Si,i,q T + ei,i+1 ei+1,n+q + ei+1,n+q ei,i+1.
Write R00(i, q) = ei,i+1 ei+1,n+q + ei+1,n+q ei,i+1. If i = n 1 we are done. If
i < n 1 we may reduce R00(i, q) using the same method as above, and so by induction
we are done.
Thus we have a basis for Ker m consisting of elementary tensors, and L(n, k) is zero
product determined by proposition 4.1.
66
Recall from theorem 3.1 that q decomposes as
q = gZ ?+c ?+[q,q]
and Derq decomposes as
Derq = L adq = Hom
K
(gZ ?+c,gZ) qS.
Since qS is known to be zero product determined by proposition 4.5, we direct our at
tention to L = HomK(gZ ?+c,gZ), which is zero product determined in light of lemma
4.8.
Recall that our study in chapter 3 on the direct sum decomposition of Derq was
originally motivated by the question of whether the derivation algebras of certain Lie
algebras were zero product determined. We conclude this chapter with the answer to this
original question in the affirmative.
Theorem 4.9. Let q be a parabolic subalgebra of a reductive Lie algebra g over a complexlike field
K. The derivation algebra Derq is zero product determined.
Proof. We begin with the decomposition
Derq = L adq
established by theorem 3.1. We have adq = qS, and qS is zero product determined by
proposition 4.5. Furthermore, L is zero product determined. To verify this, write n =
dimgZ and k = dimc, and observe that L = L(n, k). By lemma 4.8, L is zero product
determined. By proposition 4.2, Derq, as the direct sum of zero product determined Lie
algebras, is zero product determined.
67
Chapter 5
Examples and Future Research
We close this dissertation by taking note of directions that future research could take
and how such research would fit into the existing body of results, and by providing
worked examples and tabular data for reductive Lie algebras of types A5, G2, and F4.
5.1 Examples
We provide an algorithmic method for computing the center lZ of the Levi factor
in the Langland?s decomposition of a parabolic subalgebra corresponding to any given
subset of the base D of the root system. We then enumerate all standard parabolic subalg
ebras and give the dimensions of L, qS (which, recall, is isomorphic to adq), and Derq in
tabular form.
5.1.1 Type A5
Let g = Cn gS where gS = sl(C6). gS has the root space decomposition
gS = h ?+
i6=j
Cei,j
where h consists of traceless diagonal 6 6 complex matrices. Chose D =fa1, ..., a5gas a
base where gai = Cei,i+1. Then F+ =
n
5i=1 aiai
ai 2f0, 1g
o
and F = F+[ F. Write
xi = ei,i+1, yi = ei+1,i, and hi = [xi, yi] = ei,i ei+1,i+1. For each i, let ti be the coroot dual
to ai, so ai(tj) = di,j. T = ft1, ..., t5gis a basis for h. Partial multiplication table for g in
terms ofH=fh1, ..., h5gandT are provided in tables 5.1 and 5.2 respectively.
68
x1 x2 x3 x4 x5
h1 2x1 x2 0 0 0
h2 x1 2x2 x3 0 0
h3 0 1x2 2x3 x4 0
h4 0 0 x3 2x4 x4
h5 0 0 0 x4 2x5
Table 5.1: Partial multiplication table for sl(C6) in terms ofH
x1 x2 x3 x4 x5
t1 x1 0 0 0 0
t2 0 x2 0 0 0
t3 0 0 x3 0 0
t4 0 0 0 x4 0
t5 0 0 0 0 x5
Table 5.2: Partial multiplication table for sl(C6) in terms ofT
For any D0 D with corresponding parabolic subalgebra q = gZ ?+h ?+ b2F0gb, we
make three observations. First, the derived algebra [q,q] is determined by D0 as
[q,q] = Span hi ai 2D0 ?+
b2F0
gb.
Second, the center lZ of the Levi factor l is given by
lZ = Span ti ai 2DnD0 .
Third, the matrix whose columns are the members of T written as vectors in terms of
the basisHis the inverse of the transpose of the Cartan matrix of g. Figure 5.1 gives the
Cartan matrix and the inverse transpose of g, and table 5.3 gives members ofT in terms
ofHand as matrices.
Utilizing the three above observations allows one to explicitly compute a basis for
the ideal L of Derg. Table 5.4 contains data on for all standard parabolic subalgebras of g
with respect to h.
69
ti ti in terms ofH ti as a diagonal matrix
t1 (5/6, 2/3, 1/2, 1/3, 1/6) diag(5/6, 1/6, 1/6, 1/6, 1/6, 1/6)
t2 (2/3, 4/3, 1, 2/3, 1/3) diag(2/3, 2/3, 1/3, 1/3, 1/3, 1/3)
t3 (1/2, 1, 3/2, 1, 1/2) diag(1/2, 1/2, 1/2, 1/2, 1/2, 1/2)
t4 (1/3, 2/3, 1, 4/3, 2/3) diag(1/3, 1/3, 1/3, 1/3, 2/3, 2/3)
t5 (1/6, 1/3, 1/2, 2/3, 5/6) diag(1/6, 1/6, 1/6, 1/6, 1/6, 5/6)
Table 5.3:T in terms ofHand as matrices
A =
2
66
66
4
2 1 0 0 0
1 2 1 0 0
0 1 2 1 0
0 0 1 2 1
0 0 0 1 2
3
77
77
5
(AT) 1 =
2
66
66
4
5/6 2/3 1/2 1/3 1/6
2/3 4/3 1 2/3 1/3
1/2 1 3/2 1 1/2
1/3 2/3 1 4/3 2/3
1/6 1/3 1/2 2/3 5/6
3
77
77
5
Figure 5.1: Cartan matrix and transpose inverse for Type A5
5.1.2 Type G2
Let g = Cn gS where gS is simple of type G2. The same observations in the previous
example apply to any parabolic subalgebra corresponding to a D0 D. In particular,
lZ = Span tiai 2DnD0. Figure 5.2 gives the Cartan matrix for Type G2 and gives the
inverse transpose, whose columns are ti in terms of the hi. Table 5.5 gives data for all the
standard parabolic subalgebras of g.
5.1.3 Type F4
Let g = Cn gS where gS is simple of type F4. Again, to any parabolic subalgebra
corresponding to a D0 D lZ = Span tiai 2DnD0. Figure 5.3 gives the Cartan matrix for
Type F4 and gives the inverse transpose, whose columns are ti in terms of the hi. Table 5.6
gives data for all the standard parabolic subalgebras of g.
A =
2 3
1 2
(AT) 1 =
2 1
3 2
Figure 5.2: Cartan matrix and transpose inverse for Type G2
70
D0 dimlZ dimL dimqS dim Derq
? 5 n2 + 5n 20 n2 + 5n + 20
10000 4 n2 + 4n 21 n2 + 4n + 21
01000 4 n2 + 4n 21 n2 + 4n + 21
00100 4 n2 + 4n 21 n2 + 4n + 21
00010 4 n2 + 4n 21 n2 + 4n + 21
00001 4 n2 + 4n 21 n2 + 4n + 21
11000 3 n2 + 3n 23 n2 + 3n + 23
10100 3 n2 + 3n 22 n2 + 3n + 22
10010 3 n2 + 3n 22 n2 + 3n + 22
10001 3 n2 + 3n 22 n2 + 3n + 22
01100 3 n2 + 3n 23 n2 + 3n + 23
01010 3 n2 + 3n 22 n2 + 3n + 22
01001 3 n2 + 3n 22 n2 + 3n + 22
00110 3 n2 + 3n 23 n2 + 3n + 23
00101 3 n2 + 3n 22 n2 + 3n + 22
00011 3 n2 + 3n 23 n2 + 3n + 23
11100 2 n2 + 2n 26 n2 + 2n + 26
11010 2 n2 + 2n 24 n2 + 2n + 24
11001 2 n2 + 2n 24 n2 + 2n + 24
10110 2 n2 + 2n 24 n2 + 2n + 24
10101 2 n2 + 2n 23 n2 + 2n + 23
10011 2 n2 + 2n 24 n2 + 2n + 24
01110 2 n2 + 2n 26 n2 + 2n + 26
01101 2 n2 + 2n 24 n2 + 2n + 24
01011 2 n2 + 2n 24 n2 + 2n + 24
00111 2 n2 + 2n 26 n2 + 2n + 26
11110 1 n2 + n 30 n2 + n + 30
11101 1 n2 + n 27 n2 + n + 27
11011 1 n2 + n 26 n2 + n + 26
10111 1 n2 + n 27 n2 + n + 27
01111 1 n2 + n 30 n2 + n + 30
D 0 n2 35 n2 + 35
Table 5.4: Parabolic subalgebras of type A5
D0 lZ dimL dimqS dim Derq
? h n2 + 2n 8 n2 + 2n + 8
fa1g Spanfh1 + 2h2g n2 + n 9 n2 + n + 9
fa2g Spanf2h1 + 3h2g n2 + n 9 n2 + n + 9
D 0 n2 14 n2 + 14
Table 5.5: Parabolic subalgebras of type G2
71
A =
2
66
4
2 1 0 0
1 2 2 0
0 1 2 1
0 0 1 2
3
77
5 (AT) 1 =
2
66
4
2 3 2 1
3 6 4 2
4 8 6 3
2 4 3 2
3
77
5
Figure 5.3: Cartan matrix and transpose inverse for Type F4
D0 dimlZ dimL dimqS dim Derq
? 4 n2 + 4n 28 n2 + 4n + 28
1000 3 n2 + 3n 29 n2 + 3n + 29
0100 3 n2 + 3n 29 n2 + 3n + 29
0010 3 n2 + 3n 29 n2 + 3n + 29
0001 3 n2 + 3n 29 n2 + 3n + 29
1100 2 n2 + 2n 31 n2 + 2n + 31
1010 2 n2 + 2n 30 n2 + 2n + 30
1001 2 n2 + 2n 30 n2 + 2n + 30
0110 2 n2 + 2n 32 n2 + 2n + 32
0101 2 n2 + 2n 30 n2 + 2n + 30
0011 2 n2 + 2n 31 n2 + 2n + 31
1110 1 n2 + n 37 n2 + n + 37
1101 1 n2 + n 32 n2 + n + 32
1011 1 n2 + n 32 n2 + n + 32
0111 1 n2 + n 37 n2 + n + 37
D 0 n2 54 n2 + 54
Table 5.6: Parabolic subalgebras of type F4
72
5.2 Directions for future research
The theorems of chapter 4 apply primarily to reductive Lie algebras over complex
like fields. An immediate extension would be to prove these results for reductive Lie
algebras over R. Our results in chapter 4 extend work carried out by Wang et al. in [33],
where the authors prove that the parabolic subalgebras of a simple Lie algebra over a
complexlike field are zero product determined. The arguments employed by Wang et
al. do not appear to be easily extended to the real case, as the computational method
employed relies on the fact that root spaces are onedimensional, where restricted root
spaces can be arbitrarily large in dimension. An abstract argument, however, similar to
our proof of theorem 3.3 from theorem 3.1 might produce the desired result.
Along the same lines, we may extend the class of Lie algebras under consideration
by including Lie algebras over primecharacteristic fields [19, 26] or over general commu
tative rings [23, 30, 31]. Alternatively, we may consider infinitedimensional Lie algebras.
KacMoody algebras are infinitedimensional generalizations of the (finitedimensional)
semisimple Lie algebras, and they share many of the properties of semisimple Lie alge
bras especially as they relate to root space decomposition [16]. Farnsteiner in 1988 inves
tigated the derivations of Borel subalgebras of KacMoody algebras, and perhaps similar
techniques can be employed to extend these results to parabolic subalgebras [10].
The methods we employ in our investigation have several noteworthy precedents in
the literature. Recall, for instance, the discussion in chapter 1 of the work of Jacobson in
1955 in [14] and the related work by Dixmier and Lister in 1957 in [9]. Dixmier and Lister
show that a converse to a result proved by Jacobson is not possible by constructing an ex
ample of a nilpotent Lie algebra and explicitly describing its derivation algebra. Dixmier
and Lister employ methods in their concrete example that mirror the abstract methods
we use in this dissertation. The interesting point is this: parabolic subalgebras are never
nilpotent. This suggests that perhaps the methods empolyed here may be extended to a
much wider classes of Lie algebras.
73
In contrast, we may consider the methods of Leger and Luks in [19] and the meth
ods of Tolpygo in [29]. In these papers, the authors prove special cases of our results,
though they use completely different methods. Leger?s and Luks?s results imply that all
derivations of a Borel subalgebra of a simple Lie algebra are inner (over any field with
characteristic not 2) [19], and similarly Tolpygo?s results (applicable specifically over the
complex field) imply that all derivations of a parabolic subalgebra of a semisimple Lie
algebra are inner [29]. In fact, these results are more general and stated in the language of
cohomology: The authors prove that all cohomology group Hn(g,g) are trivial for their
respective classes of Lie algebras g under consideration [19, 29].
Very briefly, cohomology groups are computable invariants of a Lie algebra that pro
vide information about the Lie algebra under consideration. (For context, the reader is
reminded that the familiar Calculus derivative f0 of a realvalued function f is a com
putable invariant that provides information about f .) For instance, the first cohomology
group H1(g;g) of a Lie algebra g satisfies the isomorphism
H1(g;g) = Derg/ adg.
From this isomorphism, it follows that H1(g;g) = 0 implies that all derivations of g are
inner. A further application of cohomology is to extensions of a Lie algebra b by an ideal
a. We have the isomorphism
H2(b;a) = Ext(b;a)
meaning that the second cohomology group H2(b;a) parametrizes the set of all possible
extensions of b by a (cf. definition 2.11). If we happen to know that H2(b;a) = 0, then
we know that the only extension of b by a is the trivial extension defined by the action
8b 2b,8a 2a, b a = 0. Such an action results in a componentwise bracket rule, so the
extensions is in fact the Lie algebra direct sum b a.
74
In light of these two isomorphisms, the language and methods of cohomology pro
vide a strong framework for discovering structural properties of Derg as they relate to
properties of g. Our results on derivations apply to reductive Lie algebras, trivial ex
tensions of semisimple Lie algebras by an abelian Lie algebra. Cohomology might be
employed to study the derivations of general extensions of Lie algebras. Our results on
direct sums of zero product determined algebras might likewise be generalized to exten
sions of algebras. Our present results, in turn, can enrich the study of cohomology by
providing further examples of classes of Lie algebras that exhibit nontrivial first coho
mology groups.
Another direction for extending our results is to generalize the notion of derivation
itself. A straightforward way of doing this is to drop the requirement that a derivation be
linear, an approach studied in by Chen and Wang in [6] and [7]. The authors use the term
nonlinear map satisfying derivability for a notnecessarilylinear map D : g !g satisfying
8x, y2g D [x, y] = D(x), y + x, D(y) . (5.1)
Alternatively, one can generalize the notion of a derivation by relaxing the product rule
(equation 5.1), studying linear maps D : g !g satisfying the weaker condition
8x, y, z2g D
[x, y], z
=
h
D(x), y , z
i
+
h
x, D(y) , z
i
+
h
x, y , D(z)
i
. (5.2)
Such a map is called a Lie triple derivation, and these maps are studied in [20], [35], and [32]
among others. The two approaches can be combined, studying maps D that are not neces
sarily linear and satisfy condition 5.2 rather than condition 5.1. Chen and Wang take this
approach in [8], naming such a map a nonlinear Lie triple derivation. Figure 5.4 illustrates
the logical connection between the various types of derivationlike maps considered.
We offer a brief summary of the results in [6] and [8]. In [6], Chen and Wang study
nonlinear maps satisfying derivability on parabolic subalgebras of simple Lie algebras
75
Derivations Nonlinear triple derivations
Triple derivations
Nonlinear maps satisfying derivability
=)
=)
=)
=)
Figure 5.4: Logical relations among various types of derivationlike maps
over a complexlike field. The authors show that any such map is the sum of an inner
derivation and what the authors call an additive quasiderivation (a map that, notably,
fails to be homogeneous). As an aside, this gives an alternate proof that Der(q) = ad(q)
when q is a parabolic subalgebra of a simple Lie algebra. In [8], the same authors study
nonlinear Lie triple derivations in the same setting, parabolic subalgebras of a simple Lie
algebra g over a complexlike field. What is worth noting, though, is that their result is ex
actly the same: a nonlinear Lie triple derivation is the sum of an inner derivation and an
additive quasiderivation. In other words, the weaker assumption of requiring condition
5.2 rather than condition 5.1 resulted in no new maps ? the nonlinear Lie triple deriva
tions and the nonlinear maps satisfying derivability on a parabolic subalgebra exactly
coincide in case g is simple.
The results of Chen and Wang motivate the following questions: Are there nonlinear
triple derivations that are not nonlinear maps satisfying derivability? If so, what classes
of Lie algebras must we consider in order to differentiate between the two types of maps?
It would be interesting to extend these results to parabolic subalgebras of reductive Lie
algebras for a number of reasons. Considering derivations of reductive algebras has pro
vided examples of derivations that are non inner ? a sort of nontriviality result about
76
outer derivations. In parallel, considering nonlinear maps satisfying derivability and
nonlinear triple derivations of parabolic subalgebras of reductive algebras could perhaps
lead to examples of nonlinear triple derivations that are not nonlinear maps satisfying
derivability.
A final vehicle for future research that we will discuss deals with the abstract form
of the decomposition of Derq established by theorems 3.1 and 3.3. If we denote by g a
parabolic subalgebra, we have that the derivation algebra Derg decomposes as
Derg = Hom(g/[g,g],gZ) adg. (5.3)
by corollary 3.8. The constructions adg, gZ, g/[g,g], and Hom(g/[g,g],gZ) can be carried
out for any Lie algebra g, motivating the following question: for which Lie algebras g
does isomorphism 5.3 hold?
We remind the reader that a Lie algebra g is called complete if gZ = 0 and g has only
inner derivations. Analogously, we propose the following definition: a Lie algebra g is
almost complete if isomorphism 5.3 holds. We see that the almost complete Lie algebras are
an intermediate class between the complete Lie algebras and general Lie algebras, and as
an area for future investigation we may wish to characterize almost complete Lie algebras
in order to refine the classification of Lie algebras in general.
77
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