Lie algebras by nathan jacobson

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LIEALCEBRAS
Nathan
Jacobson
Lie group theory, developed by M. Sophus Lie in the lgth centur.|, ranks
among the more important
developments in modern mathematics. Lie
algebras comprise a significant part of Lie group theory and are being
actively studied today. This book, by Professor Nathan Jacobson of Yale,
is the definitive treatment of the subject and can be used as a textbook
for graduate courses.
Chapter I introduces basic concepts that are necessary for an understanding
of structure theory, while the following three chapters present the theory
itself: solvable and nilpotent Lie algebras, Cartan's criterion and its consequences, and split semi-simple Lie algebras. Chapter 5, on universal
enveloping algebras, provides the abstract concepts underlying represerrtation theory. Then the basic results on representation theory are given in
three succeeding chapters: the theorem of Ado-Iwasalva, classification of
irreducible modules, and characters of the irreducible modules. In Chapter
9 the automorphisms of semi-simple Lie algebras over an algebraically closed
field of characteristic zero are determined. These results are applied in
Chapter l0 to the problems of sorting out the simple Lie algebras over an
arbitrary field. The reader, to fully benefit from this tenth chapter, should

have some knowledge about the notions of Galois theory and some of the
results of the Wedderburn structure theory of associative algebras.
Nathan Jacobson, presently Henry Ford II Professor of Mathematics
at Yale University, is a well-known authority in the field of abstract
algebra. His book, Lie Algebras, is a classic handbook both for researchers
and students. Though it presupposes a knowledge of linear algebra, it is
not overly theoretical and can be readily used for self-study.

Unabridged, corrected (1979) republication of the original (1962)
Bibliography. Index. ix a 331pp. \s/B x 8/4. Paperbound.
A DOVER EDITION DESIGNED FOR YEARS OF USE!
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ISBN 0-486-63832-4

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LIEALGEERAS
by

I.TATHAN
JACOBSON
Henry Ford II Professor of Mathematics
Yale University, New Haven, Connecticut

DoverPublications,
Inc.
New'\brk

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Copyright O 1962 by NathanJacobson.
All rights reserved under Pan American
national Copyright Conventions.

and Inter-

Published in Canada by General Publishing Companf,
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Published in the United Kingdom by Constable aild
Company, Ltd., l0 Orange Street, London WCZH 7EG.
This Dover edition, first published in l9?9, is en
unabridged and corrected republication of the work
originally published in 1962 by Interscience Publishersl a
division ofJohn Wiley & Sons, Inc.
Int enut ional Standard B ooh Num b er : 0 - 486 -6i $ 2 -i
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PREFACE
The present book is based on lectures which the author has given
at Yale during the past ten years, especially those given during
the academic year 1959-1960. It is primarily a textbook to be
studied by students on their own or to be used for a course on
Lie algebras. Besides the usual general knowledge of algebraic

concepts, a good acquaintance with linear algebra (linear transformations, bilinear forms, tensor products) is presupposed. Moreover, this is about all the equipment needed for an understanding
of the first nine chapters. For the tenth chapter, we require also
a knowledge of the notions of Galois theory and some of the
results of the Wedderburn structure theory of associative algebras.
The subject of Lie algebras has much to recommend it as a
.subject for study immediately following courseson general abstract
algebra and linear algebra, both because of the beauty of its
results and its structure, and because of its many contacts with
other branches of mathematics (group theory, differential geometry,
differential equations, topology). In this exposition we'have tried
to avoid rnaking the treatment too abstract and have consistently
followed the point of view of treating the theory as a branch of
linear algebra. The general abstract notions occur in two groups:
the first, adequate for the structure theory, in Chapter I; and the
second, adequatefor representation theory, in Chapter V. Chapters
I through IV give the structure theory, which culminates in the
classification of the so-called "split simple Lie algebras." The
basic results on representation theory are given in Chapters VI
through VIII. In Chapter IX the automorphisms of semi-simple
Lie algebras over an algebraically closed field of characteristic
zero are determined. These results are applied in Chapter X to
the problem of sorting out the simple Lie algebras over an arbitrary
field.
No attempt has been made to indicate the historical development of the subject or to give credit for individual contributions
to it. In this respect we have confined ourselves to brief indications here and there of the names of those responsible for the
main ideas. It is well to record here the author's own indbbtedness to one of the great creators of the theory, Professor Hermann
Weyl, whose lectures at the Institute for Advanced Study in 1933lvl

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PREFACE

vl

1934 were truly inspiring and led to the author's research in this
field. It should be noted also that in these lectureB Professor
Weyl, although primarily concerned with the Lie theory of continuous groups, set the subject of Lie algebras on its own independent course by introducing for the first time the term "Lie algebra"
as a substitute for "infinitesimal group," which had been used exclusively until then.
A fairly extensive bibliography is included; howevef, this is by
no means complete. The primary aim in compiling the bibliography has been to indicate the avenues for further jstudy of the
topics of the book and those which are immediately rrelated to it.
I am very much indebted to my colleague George $eligman for
carefully reading the various versions of the maguscript and
offering many suggestions for improving the expo$ition. Drs.
Paul Cohn and Ancel Mewborn have also made valuable comments,
and all three have assisted with the proofreading. I take this
opportunity to offer all three my sincere thanks.
May 28, 1961
New Hauen, Connecticut

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N.lrs.$.r Jlconsou


CONTENTS
Cnlrrpn

I

Basic Concepts
1. Definition and construction of Lie and associative
algebras
2. Algebras of linear transformations. Derivations
3. Inner derivations of associative and Lie algebras .
4. Determination of the Lie algebras of low dimensionalities
5. Representations and modules .
6. Some basic module oPerations
7. Ideals, solvability, nilpotency .
8. Extension of the base field .

2
5
9
11
14
t9
23
26

CulprnR II
Solvable and Nilpotent Lie Algebras
Weakly closed subsets of an associative algebra
Nil weakly closed sets
Engel's theorem
Primary components. Weight spaces
Lie algebras with serni-simple enveloping associative
algebras
6. Lie's theorems

7. Applications to abstract Lie algebras. Some counter
examples .

1.
2.
3.
4.
5.

31
33
36
37
43
48
51

Cn.l.rtsn III
Cartan's Criterion and Its Consequences
1.
2.
3.
4.
5.
6.
7.

Cartan subalgebras .
Products of weight spaces
An example

Cartan's criteria
Structure of semi-simple algebras
Derivations.
Complete reducibility of the representations of
semi-sinrple algebra
lvii l

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57
61
64
66
70
73
75


Viii

CONTENTS

8.

Representations of the split three-dimensional simple
Lie algebra.
9. The theorems of Levi and Malcev-Harish-Chandra .
10. Cohomology groups of a Lie algebra .
11. More on complete reducibility
. . .

83
86
93
96

Cn^l,rtpn IV
Split Semi-simple Lie Algebras
1. Properties of roots and root spaces.
2. A basic theorem on representations and its
consequencesfor the structure theory
3. Simple systems of roots
4. The isomorphism theorem. Simplicity .
5. The determination of the Cartan matrices
6. Construction of the algebras
7. Compact forms .

.i .

. 10g

. .; .
. .
. .

. Ilz
. 119
.IZT
. LZg
. 13S

. 146

Definition and basic properties
. .
The Poincar6-Birkhoff-Witt theorem
Filtration and graded algebra
Free Lie algebras
The Campbell-Hausdorff formula .
Cohomology of Lie algebras. The standard complex
Restricted Lie algebras of characteristic p
Abelian restricted Lie algebras .

151
156
163
t67
170
174
185
r92

Cn.c.prpn V
Universal Envetoping Algebras

CHlrrnn

VI

The Theorem of Ado-Iwasawa
1. Preliminary results .

2. The characteristic zero case
3. Thecharacteristicpcase.

..

. Z}L
.Z}J

,f

.207

CHAPTER.
VII
Classification of Irreducible Modules
1. Definition of certain Lie algebras .

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ix

CONTENTS

.212
.zLs

2. On certain cyctic modulesfor E
3. Finite-dimensionalirreducible modules
4. Existence theorem and isomorphism theorem

for semi-simpleLie algebras .
5. Existenceof E, and Ea .
6. Basic irreducible modules

.220
.223
.225

Cn.lprpn VIII
Characterg of the lrreducible

Modules

1. Some propertiesof the Weyl group
2.
3.
4.
5.

240
243
249
257
259

Freudenthal's formula
Weyl's character formula
Some examples .
Applications and further results
CHlrrnn

IX

Automorphisms
Lemmas from algebraic geometry
Conjugacy of Cartan subalgebras .
Non-isomorphism of the split simple Lie algebras
Automorphisms of semi-simple Lie algebras over
an algebraically closed field
5. Explicit determirtation of the automorphisms
for the simple Lie algebras

1.
2.
3.
4.

.zffi
.271
.274
.275
.281

Cru,prpn X
Simple Lie Algebras oyer an Arbitrary
1. Multiplication algebra and centroid of
a non-associative algebra .
2. Isomorphism of extension algebras .
3. Simple Lie algebras of types A-D
4. Conditions for isomorphism

5. Completeness theorems .
6. A closer look at the isomorphism conditions
7. Central simple real Lie algebras
Bibliography.

Field
. 290
.295
. 298
.303
. 308
. 311
. 313
..319

In d e x

.329

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LIEALGEBRAS

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CHAPTER I

Basic Concepts

The theory of Lie algebras is an outgrowth of the Lie theory of
continuous groups. The main result of the latter is the reduction of
"local" problems concerning Lie groups to corresponding problems
on Lie algebras, thus to problems in linear algebra. One associates
with every Lie group a Lie algebra over the reals or complexes
and one establishes a correspondence between the analytic subgroups
of the Lie group and the subalgebras of its Lie algebra, in which
invariant subgroups correspond to ideals, abelian subgroups to abelian
subalgebras, etc. Isomorphism of the Lie algebras is equivalent to
local isomorphism of the corresponding Lie groups. We shall not
discuss these matters in detail since excellent modern accounts of
the Lie theory are available. The reader may consult one of the
following books: Chevalley's Theorl of Li,e Groufs, Cohn's Lie
Groups, Pontrjagin's Topological Groups.
More recently, two other types crf group theory have been aided
by the introduction of appropriate Lie algebras in their study. The
first of these is the theory of free groups which can be studied by
means of free Lie algebras using a method which was originated
by Magnus. Although the connection here is not so close as in
the Lie theory, significant results on free groups and other types
of discrete groups have been obtained using Lie algebras. Particu'
larly noteworthy are the results on the so-called restricted Burnside
problem: Is there a bound for the orders of the finite groups with
a fixed number r of generators and satisfying the relation tr* : I,
m a fixed, positive integer? It is worth mentioning that Lie algebras
of prime characteristic play an important role in these applications
to discrete group theory. Again we shall not enter into the details
but refer the interested reader to two articles which give a good
account of this method in group theory. These are: Lazard tzl
and Higman [1].

The type of correspondence between subgroups of a Lie group
and subalgebras of its Lie algebra which obtains in the Lie theory
tll

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2

YIE ALGEBRAS

has a counterpart in chevalley's theory of linear algebraic groups.
Roughly speaking, a linear algebraic group is a subdroup of the
group of non-singular n x n matrices which is specified,by a set of
polynomial equations in the entries of the matrices. An example
is the orthogonal group which is defined by the set of equations
Xirl?i:l,\idiiditt:A,
i + h , j , k - 1 , . . . t l t t o n t h e e h t r i e se ; ; o f
the matrix (a;1). with each linear algebraic group chevalley has
defined a corresponding Lie algebra (see Chevalley t2l) hrhictr gives
useful information on the group and is decisive in the theory of
linear algebraic groups of characteristic zero.
I
In view of all this group theoretic background it is inot surprising that the basic concepts in the theory of Lie algdbras have a
group-theoretic flavor. This should be kept in mindr throughout
the. study of Lie algebras and particularly in this chdpter, which
gives the foundations that are adequate for the mafin structure
theory to be developed in Chapters II to IV. euestioris on foundations are taken up again in Chapter V. These concern some concepts that are necessary for the representation theoryl which will
be treated in Chapters VI and VII.
1.

Defrnition and construction of Lie
and aasociatiae algebras

We recall the definition of a non-associative algebra (:not
sarily associative algebra) ?I over a field, a. This is jirst a
space ll over o in which a bilinear composition is defined.
for every pair (r, !), r,y in ?I, we can associate a pfoduct
and this satisfies the bilinearity conditions

(1)
(2)

(xr+xil:h!*rz!
a(xy):

necesvector
Thus
rJ ell

x(y, + !r): rlt * *!z
(ax)y - x (a y ) ,
a€O .

A similar definition can be given for a non-associative algebra over
a commutative ring t0 having an identity element (unit) 1. This is
a left O-module with a product ry € ?I satisfying (1) iand (2). We
shall be interested mainty in the case of algebras over fields and,
in fact, in such algebras which are finite-dimensiorlal as vector
spaces. For such an algebra we have a basis (er,er,.)..,en)and we

can write €r€i:}I=ffure*
where the 7's are in A. The zt faiy arte,
called the constants of multiplication of the algebra (relative to the
chosen basis). They give the values of every prduet e;e1,

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I. BASIC CONCEPTS

1,2, . . , , n. Moreover, these productsdetermineevery productin ?I.
Thus let r and y be any two elementsof tr and write r: ZEi€;,
!:>Tiei, Errti€O. Then, bV (1) and (2),
xy - ( T,{'erXl,n iei) : l,(E ie)Qte i)
.
r,t
t

- \E;@{nP)):

\E;nr(e&t) ,

and this is determined by the e&t.
This reasoning indicates a universal construction for finite-dimensional non-associative algebras. We begin with any vector space
lI and a basis (er) in t. For every pair (l,l) we define in any way
we please *gt ds an element of lt. Then if x : LTEee; y - Zlv$t
we define

(3)

try: f

Erro@oun)

i, i:L

One checks immediately that this is bilinear in the sense that (1)
and (2) are valid. The choice of e1e1is equivalent to the choice
of the elements Tux in @ such that e&t:}Taixan.
The notion of a non-associative algebra is too general to lead to
interesting structurai results. In order to obtain such results one
The
must impose some further conditions on the multiplication.
most important ones-and the ones which, will concern us here, are
the associative laws and the Lie conditions.
Dprrurrrorv 1. A non-associative algebra lI is said to be associatiue
if its multiplication satisfies the associative law

x(vz).

(4)

A non-associative algebra lI is said to be a Li,e algebra if its
mu'ltiplication satisfies the Lie conditions

(5)

x,:0,

( x y ) z + ( y z ) x +( z r ) y- 0 .

The second of these is called the Jacobi identity.
Since these types of non-associative'algebras are defined by
identities, it is clear that subalgebras and homomorphic images
are of the same type, i.e., associative or Lie. If lI is a Lie algebra
andx,yell, then 0 -.(r* y)': x'+ xy + ltc*!':
xl +yr so that

(6)

-yx

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4

LIE ALGEBRAS

holds in any Lie algebra. Conversely, if this condition holds then
2tr' :0, so that, if the characteristic is not two, then rz't: 0. Hence
for algebras of characteristic * 2 the condition (6) canj be used for
the first of (5) in the definition of a Lie algebra.
Pnoroslrlox 1. A non-associatiue
algebra W,with basis(er,er,. ..,eo)
ouer O is associ,atiaeif and only if (ep)er-er(e$)
for i,j,k:
1,2, . -.,n. If eaei:\,Tu,e, theseconditionsare equiaalentto
(7 )

l T u rT 1 ,, ,

\ T ur T , * r:

i , j , k , s : 1, 2, " ., ? .

The algebra 2l is Lie if and only if e',t:0,

e&i:

- ei01,,

(ete)er * (eie)er * @p)e1 : g

for i,j,k:1,2,
(8)

. . ., n.

These conditions

Tur:0
Z(TotTrr,

,

Tux:

are equiaalentt to
-

T i t : r,
* Ti*,Tru * T*rrTr.rr): 0 .

Proof: If ?I is associative, then (ep)er,: e;(epx). Conversely,
assume these conditions hold for the et. lf x: ),E;ev, ! :Zqpt,
z : 2($t,
then (xy)z : 2E iv{n(e&)er and x(yz) : 2iniCpr(ep*).
Hence (ry)z - x(yz) and lI is associative. If €;€1: ET;t&,, then
(ep)e*: Xr, eT;irT*cqcand et(ep*) : X", tTirtTirr€t. Hence the linear
independence of the ei implies that the conditions (e;b)er,: e;(eq*)
are equivalent to (7). The proof in the Lie case is slmilar to the
foregoing and will be omitted.
In actual practice the general procedure we have indicated is not
often used in constructing examples of associative and of Lie algebras
except for algebras of low dimensionalities. We shall employ this in
determining the Lie algebras of one, two, and three dimpnsions in g4.
There are a couple of simplifying remarks that can be made in the
Lie case. First, we note that if e?:0 and e;ei : -' ,ru, in an algebra, then the validity of (e;e)en* (ep*)er + @p)et: 0 for a particular triple i, j, k implies (eie;)erc
* (e;e)et * (eret)e;: 0, Since cyclic
permutations of i, j, k are clearly allowed it follows that the Jacobi
identity for (e;,e;')4, is valid for it, j',k', a permutation of. i, j,h.
Next let i : j . Then e?e*+ (e;e*)e;* (ene;)e;: 0 * (e;et)er- (e;er)e;- O.
Hence e?: 0, ei01: - ri€i or, what is the same thing, x' : 0 in ?I
implies that the Jacobi identities are satisfied for e;, e;,,€i. In particular, the Jacobi identities are consequencesof x' :0 if dim A < 2

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5

I. BASIC CONCEPTS

and if dim t[ :3, then the only identity we have to check is
= 0.
(ep)es* (eze)er* (eser)ez
2.

Algebrat

of linear tranaformations.

Deriaations

Actually, it is unnecessary to sit down and construct examples
of associative and Lie algebras by the method of bases and multiplication tables since these algebras occur "in nature." The prime
examples of associative algebras are obtained as follows. Let lll
be a vector space over a field O and let @denote the set of linear trans'
formations of Dt into itself. We recall that if. A, BeE and ae0,
then A + B, aA and AB are defined by x(A + B) : xA * xB,
x(aA): s.(rA), x(AB): (rA)B for r in lll. Then it is well known
that € is a vector space relative to * and the scalar multiplication
and that multiptication is associative and satisfies (1) and (2). Hence
6 is an associative algebra. It is well known also that if tJt is
rz-dimensional, n, 1 @, then O is m'-dimensional over O. lf. (er,
€2,...,2^) is a basis for fi over o, then the linear transformations
if. r+i, i,i -1, "',/rt, form a
.E;l such that etEti:ei,0rE;i:0
basis for O over O. If AeE, then we can write 0;A - }ia;Pt
i -- l,- . . t fri, and (a) : (ar) is the matrix of. A relative to the ba'

sis (ec). The correspondence A-+ (a) is an isomorphism of 0 onto
the algebt? O^ of. m x m matrices with entries ait in O.
The atgebra @is called the (associatiae)algebra of linear transform'
ations in llt over O. Any subalgebra l[ of 0, that is, a subspace
of O which is closed under multiplication, is called an algebra of
linear tr ansformations.
If lt is an arbitrary non-associative algebra and a e lI, then the
mapping ar which sends any x into ra is a linear transformation.
It is well known and easy to check tl:pt (a * D)" : aB * bn, (aa)p:
aan
if U is associative, (ab)": anba. Hence if U is an as'
"nd' algebra, the mapping a -> an is a homomorphism of ?I into
sociative
the algebra @ of linear transformations in the vector space ?I. If
lI has an identity (or unit) 1, then a -+ an is an isomorphism of lI
into @. Hence 2t is isomorphic to an algebra of linear transform'
ations. If lI does not have an identity, w€ can adjoin one in a
simple way to get an algebra, ?I* with an identity such that dim
lI* : dim ?I + 1 (cf. Jacobson l2l, vol. I, p. 84). Since lI* is
isomorphic to an algebra of linear transformations, the same is
true for lt. If U is finite-dimensional, the argument shows that

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LIE ALGEBRAS

l[ is isomorphic to an algebra of linear transformation$ in a finitedimensional vector space.
Lie algebras arise from associative algebras in a ltrery simple
way. Let lI be an associative algebra. lf x,yell, thqn we define

the Li,e product or (additive) commutator of. x and y AS
( 9)

txyl: xy - lx .

One checks immediately that
ln * rcz,
lf : [x'yj * Ixryl,
lr, !, * yrl : Ixyrl * [x!r] ,
alxyl: lax, yf : lr, dyl .
Moreover,
lxrl:tct-x':0,

llxylzl+ [yzlxl + llzxlyl
: (xy- yr)z- z(xy- yr) + (yz - zy)r
- r(yz - zt) * kx - rz)y tkx

-',t"z) - 0 .

fhus the product Ixyl satisfies all the conditions on tire product in
a Lie algebra. The Lie algebra obtained in this way Iis called, the
Lie algebra of the associative algebra lI. We shall defrote this Lie
algebra as 2Is. In particular, we have the Lie algebrd @z obtained
from G. Any subalgebra I of CIz is called a Li,e algebra of linear
transformations. we shall see later that every Lie algebra is
isomorphic to a subalgebra of a Lie algebra ?Ir, g[ associative. In
view of the result just proved on associative algebras this is equivalent to showing that every Lie algebra is isomordric to a Lie
algebra of linear transformations.
we shall consider now some important instances od subalgebras
of Lie algebras @r, @ the associative algebra of linehr transformations in a vector space llt over a field O.

orthogonal Lie algebra. Let !)t be equipped with a nbn-degenerate
symmetric bilinear form (x, y) and assume !)t finite-dimensional.
Then any linear transformation A in !)l has an adjoint ..4* relative
to (x,y); that is, Ax is linear and satisfies: (rd, y) : (r, tA*). The
mapping A -. A* is an anti-automorphism in the algebra G:
(A+ B ) *: A * + B* , (a A 1 * : a A * , (AB )* : B* A * . l l -et € denote
the set of Ae @which are skew in the sense that A* * - A. Then
is a subspace of @ and if. A* - - A, B* : - B, fhen lABl* :

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7

I. BASIC CONCEPTS

-LABI. Hence
(AB - BA)* -- B*A* - A*B* : BA - AB: lBAl:
tABle 6 and 6 is a subalgebra of 0r.
If @ is the field of real numbers, then the Lie algebta 6 is the
Lie algebra of the orthogonal group of llt relative to (r' y). This
is the group of linear transformations O in 9]t which are orthogonal
r,! in !It. For this reason we
in the sense that (xO,yO):(x,!),
the orthogonal Li.e algebra relative to (r, y).
shall call
Syruplectic Lie algebra. Here we suppose (r, y) is a non-degenerate
and again dim l}t < oo. We recall that
alternate form: (r,x):0
these conditions imply that dim tn - 2l is even. Again let ,4* be

the adjoint of ,4( e O) relative to (r, y). Then the set 6 of skew
(A* : - A) linear transformations is a subalgebra of @2. This is
related to the symplectic group and so we shall call it the symPlectic
Lie algebra 6 of the alternate form (x, y).
c!]l-*Dt
Triangular linear transfarmations. Let 0c1]tr cl}tzc'''
let ! be
I
and
IJtt:
dim
9Jt
such
that
of
of
subspaces
be a chain
the set of linear transformations T such that lltrT G TJtt. It is clear
that E is a subalgebra of the associative algebra @: hence E; is a
'We
can choose a basis (xr, rr, "', x*) for lll so
subalgebra of @2.
Then if TeT, Tft;T S !]h
that (trr,trr, "', x;) is a basis for fii.
implies that the matrix of. T relative to (xr, Nr, "', x*) is of the
form

(10)

,:[::
:'

:]

Such a matrix is called triangular and correspondingly we shall
call any TeT a triangular linear transformation.
Deriaation algebras. Let ?I be an arbitrary non-associative algebra.
A deriaation D in ?I is a linear mapping of lI into ?I satisfying

(11)

(xy)D:

(xD)y * x(yD) .

Let D(?I) denote the set of derivations in !I.
then

lf Dr,DzeS([),

(xy)(D, * D,) - (ry)D1 + @y)Dz- (rD,)t
* r(!D) * (xD)y * x(yD,)
: (r(Dr * Dr))y* x(y(h + Dr))

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LIE ALGEBRAS
Hence Dt * Dz e D(U;.

a,e0. We have

Similarly,

one checks that

a,DrQS(U) if

(xy)D,Dz: (@Db * x(yD,))Dz
: (xD,Dz)y * (xD,1lyD') + (xDr)(tD,) + x(vD,Dr) '
Interchange of l, 2 and subtraction gives
(ry)lD,Dzl:

i

@lDD'l)Y + r(YlD'D,l) .

Hence lDrDzle E(lI) and so S(U) is a subalgebra of Gz, where € is
the algebra of linear transformations in the vector space lI. We
shalt call this the Lie algebra of deriaations or deriadtion algebra

of a.
The Lie algebra lD(U) is the Lie algebra of the group of automorphisms of lI if lt is a finite-dimensional algebra over the field of
real numbers. We shall not prove any of our asseftions on the
relation between Lie groups and Lie algebras but refef the reader
to the literature on Lie groups for this. However, in rthe present
instance we shall indicate the link between the group;of automor'
phisms and the Lie algebra of derivations.
Let D be a derivation. Then induction on n gives the Leibniz
rule:

(12)

(ry)D" :

D i)(vD'- t)
V^("r)rx

If the characteristic of O is 0 we can divide bY n! and obtain
(12')

,"O#:E(+.D)(hyD".).

If ll is finite-dimensionalover the field of reals, then it is easy to
prove (cf. Jacobson[2], vol. II, p. 197)that the seriesl
(13)

r+D*+++ +"'

converges for every linear mapping D in l[, and the linear mapping
(12',),
exp D defined by (13) is 1: 1. Also it is easy to see, using
:
(rGXyG).
(ry)G
:
exP D satisfies
that if D is a derivation, then G
Hence G is an automorphism of ?I.
A connection between automorphisms and derivations can be

established in a purely algebraic setting which has lmportant ap'
plications. Here we suppose the base field of U iS arbitrary of

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I. BASICCONCEPTS
characteristic 0. Let D be a nilpotent derivation, sY,
Consider the mapping

DN :0.

G:expD:L+D++*...+d+

(14)

+(Dn-'l(N-1)!)
Z:D+(D'zpD+...
We write this as G:L*2,
t h e i n v e r s eL - Z +
HenceG:1*Zhas
and notethat ZN:0.
Z' + ... -f Z"-' anld so G is 1: 1 of lI onto lI. We have

: (E+X,!,#)
(rG)(yG)
zry_z/

r_/
yo"-' 11

:,FoEr
"D'11
it 11n-tyt))
zJy_2

: E @ y,' , D"
_
nl
n=o
.lv-l

D"

n=o

n!

(by r?t)

-- (xv)GHence G is an automorphism of lI.
3.

fnner deriiations

of aesoeiatiue and Lie algebras

If a is any element of a non-associative algebra lI, then a deter'
mines two mappings az: x -+ ax and ani x -, xQ of lI into itself.
These are called the left multiplication and right multiplication deter'
mined by a. The defining conditions (1) and (2) for an algebra

show that at and aa are linear mappings and the mappings a + (zL,
an are linear of !t into the space 0 of linear transformations
a --->
in !t. Now let ?t be associative and set D" : (zn - av Hence Do
is the linear mapping x--+tut - &r. We have
(15)
rja - axy -- (ra - ax)t * x(ya - ay) ;
hence D" is a derivation in the associative algebra ?I. We shall
call this the i,nner deriaation determined by a.
Next let I be a Lie algebra. Because of the way Lie algebras
arise from associative ones it is customary to denote the product
in 8 by [ry] and we shall do this from now on. Also, it is usual
to denote the right multiptication an (: - az since lxal - - lax)) by
ad a and to call this the adioint mapping determind by a. We
have

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IO

LIE ALGEBRAS

tlxyial + llyalx) + llaxJyl : 0,
llxy)a) - - llyalrl - lLar) yl - lrlyall + llxa)y1 ;
hence ad 6 v -,fxaf is a derivation. We call this also the i'nner
deriaation determined by ae8,.
A subset E of a non-associative algebra ?I is called i an ideal if
(1) 8 is a subspace of the vector space ?I, (2) ab, ba€ E for any a
in ll, 6 in E. Consider the set of elements of the forfn l,a;bt ai,

Dt in ?I. We denote this set as lI2 and we can check ,that this is
an ideal in lI. If ?I : 8 is a Lie algebra, then it is customary to
write 8' for 8' and to call this the deriued algebra (or:ideal) of 8.
If 8 is a Lie algebra, then the skew symmetry of the multiplication implies that a subspace E of I is an ideal if and only if labl
(or [Da]) is in E for every aeSJ,DeE. It follows that the subset
G of elements c such that [ac] - 0 for all a e I is an lideal. This
is called the center of 8. 8 is called abelian if I ==6, which is
equivalent to 8' : 0.
Pnorosrrrox 2. If lI is associatiue or Lie, then the innQr deriuations
l
forrn an ideal 3(ll) iz the deriaation algebra D(?l).
Proof: In any non'associative algebra we have (a + b)r,: ar, * bt,
(aa)": dazt @ * b)n : aR * bn, (aa)*-- dan. Hence if ' D"-- an - at;
then D,+ a: D, * Dt, Dro - aDo and the inner deriv]ations of an
associative or of a Lie algebra form a subspace of S(U). Let D be
a derivation in lI. Then (ax)D - (aD)x * a(xD), ot (ax)D - a(xD):
(aD)x. In operator form this reads (xar)D - (xD)az',: x(aD)r' or
atD - Da, - (aD)r Similarly, [a*Dl-@D)n and tconsequently
lapl:
also [D"D] : D,n. These formulas show that if ll is dssociative or
Lie and .I is an inner derivation and D any derivatiori, then UDI is
an inner derivation. Hence S0I) is an ideal in O(U).
Erample. Let I be the algebra with basis (e,f) such that [efl - ffel and all other products of base elements are 0. Then
e
laa):O in Il and since dim I :2, 8 is a Lie algebra. The derived
algebra g' : Ae. If D is a derivation in any algebra ?I, then ?IzDg ?Iz.
Hence if D is a derivation in I then eD: 6e. Also a[(D/) has the
property e(adD/) : le,6fl -- 6e. Hence if E: D - adp/, then E is
a derivation and eE: A. Then e - lefl gives g - [e,fpl. It follows
that fE -- Te. Now ad(- re) satisfies e ad(- re) :0' /ad(- Te) :

re. Hence E-ad(- re) is inner and D: E*adDf
lf, - yel-ylefl:
is inner. Thus every derivation of [l : Ae + Af is infier.

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I. BASICCONCEPTS

11

In group theory one defines a group to be complete if all of
its automorphisms are inner and its center is the identity. If H
is complete and invariant in G then ^Ff is a direct factor of G.
By analogy we shall call a Lie algebta complete if its derivations
are all inner and its center is 0.
Pnorostuon 3. If R is complete and an ideal in t., then 8': S @ E
where E is an ideal.
Proof: We note first that if .f is an ideal in IJ, then the centralizer
E of s, that is, the set of elements D such that [&D]: 0 for all
&eS is an ideal. E is evidently a subspace and if DeE and
a € g, then lhlbal) - - [alkb]l - lbtakll : 0 - [b, k'7, ft' : fak) c Ri
Hence E is an ideal.
hence tklball:0 for all k e 0 and [ba]e$.
Now let ,R be complete. If c e S n E, then c is in the center of fr
and so c : 0. Hence S n E - 0. Next let a € 8. Since S is an
ideal in 8, ad a maps F into itself and hence it induces a deriva'
tion D in S. This is inner and so we have a & e R such that
E and a:b*k,
r D - - l r a l - - l x k l f . o re v e r y r € S . T h e n b : a - k e

r
e
q
u
i
r
e
d.
a
s
n
@
E
S *E
beE, ft e n. Thus I
is complete.
example
Af
last
Ae
the
of
The algebra
+
Eram\le.
4,

Determination of the Lie algebras
of low dimcnsionalities

We shall now determine all the Lie algebras IJ such that dim
8 S 3 . l f ( e r , € 2 , . . . , e , )i s a b a s i s f o r a L i e a l g e b r a S , t h e n l e ; e r ] : 0
and le;ei\:. - leie;\. Hence in giving the multiplication table for
the basis, it suffices to give the products fe;ei f-or i < i. We shall
use these abbreviated multiplication tables in our discussion.
I. dim I : 1. Then 8 :Oe, leel - g.
II. dimS-2.
(a) 8' : 0, I is abelian.
(b) 8' + 0. Since I -- Oe + Af,8' - a[ef] is one-dimensional. We
may choose e so that 8/ : Oe. Then lefl - ae * 0 and replacement
of.f by a-'f permits us to take lefl - e. Then I is the algebra of
the example of $ 3. This can now be characterized as the non'
abelian two-dimensional Lie algebra.
III. dim 13- 3.
(a) 8' :0, I abelian.
(b) dim 8/ : 1, 8' S S the center. If 8' -- oe we write 8,: oe *
Hence we may suppose lfgl - e. Thus
Af + Ag. Then 8,' :0[fg7.

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L2

LIE ALGEBRAS

8 has basis (e,f,g),

(16)

with

multiplication

lfsf- e,

table

lefl-0,

I e g -l 0 .

We have only one Lie algebra satisfying our conditions. (If we
have (16), then the Jacobi condition is satisfied.)
(c) dim 13': 1, !' E 0 the center. If 8' - O€,then,there is an
/ such that lef) + 0. Then [ef] - Fe + 0 and we may suppose
lefj - e. Hence Oet Af is the non-abeliantwo'dimensionalalgebra
n . S i n c e R = 8 ' , S i s a n i d e a ta n d s i n c e S i s c o m p l e t e8,- S O E ,
HenceIJ has basis (e,f,g) with multiplicatiorltable
E:Og.
(17)

lefl : e,

lesl- o,

[fsl -- o .

(d) dim 8' :2.
8' cannot be the non-abeliantwo-dimensional

L i e a l g e b r a S . F o r t h e n8 : S @ E a n d 8 ' : S ' - A . B u t S / c S .
Hencewe have 8' abelian. Let 8/ : Oe+ Af and I - Ae + af + Og.
Then 8t:alegl+o17g1 and so adg inducesa 1:1 linear mapping
in 8'. Hence we have basis (e,f, g) with
(18)

lefl - 0,

[esl: ae * Ff ,

lfsl - re * 8f

: (;
Convergely, in any
t. a non-singular matrix.
il
"
space I with basis (e,f, g) we can define a product [aD] so that
[aa] - 0 and (18) holds. Then tteflsl + [tfg]el + llselfl = 0 and hence
I is a Lie algebra. What changes can be made in the multiplica'
tion table (18)? Our choice of basis amounts to thiS: We have
chosen a basis (e,f) for It and supplemented this witt'. a 9 to get
a basis for 8. A change of basis in 8' will change A to a similar
matrix M-'AM.
The type of change allowable f.or g [s to replace
Thenle,pg*xl:plegl,'lf,pg *rl:
it by pg+x, p+OinO,rinS/.
plfgl so this changes A to pA. Hence the different matrices .4
which can be used in (18) are the non-zero multiples of ithe matrices
similar to A. This means that we have a 1 : 1 correspondence be'

tween the algebras I satisfying dim 8 : 3, dim 8' :2 and the con'
jugacy classes in the two dimensional collineation grotrp.
If the field is algebraically closed we can choose A in one of the
following forms:
where

(l )

a*0,

(; f) o+o

These give the multiplication tables

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:


T3

I. BASIC CONCEPTS

lefl:0,
lefl -0,

lfgl:af
Ieg):e,
lfgl- f '
legl: e + Bf ,

Difierent choices of a give different algebras unless ad' : \. Hence
we get an infinite number of non-isomorphic algebras.
Gj dim 8' : 3. Let (eb €2,€s) be a basis and set fere'l: f',
-4 - fr, ferezl: fr. Then (fr, fr,/t) is a basis' Write f i [i='eiiet'
Vrrrl
-- (a;i) non-singular. The only Jacobi condition which has to be
imposed is that lf 'erl * lfrerl * [/gea]: 0. This gives
* asrlererl* aazlezesl
* azrLe4zl* azslegezf
0 : arzlezer)* argleserf
- - anfg * arrf, * dnfs - azsft - asrfz * atrft '
so A is a symmetric matrix. Let (7yEz,ds)_be
Hence aii = a4;
"nd,
:
a second basis where 4 : Lpitrt, M : (piil non'singular. Set /1
per'
(i,
cyclic
any
i, k)
l,rdrl, fr: [d&rl, ir: [er,i- We have f'or
mu ta ti on of .( L, 2, 3)
- Epi,ttn,f0,0,l
7, - ldprl - [l,rr &,,I,pr,eJ
: (pizltrs ' tttsltrr)f ,' * (pisttt , :}v;' f '
'

piillnsJfz* (pnp"' - ttizpt)fg

The matrix N: (vi.i) - adj M' : (M')-' det Mt . The matrix relat'
ing the /'s to the a's is A and that relating the e's to the 7's is
-M-1. Hence if A is the matrix (au) such that f t, \d.;id1, then
(19)

{ - (det M'Y.M')-'AM-'

.

Two matrices A, Bare called multi.pticatiuelycogredient if' B:pN' AN
where N is non-singular and p + 0 in O. In this case we may
write B - po'(o-tN)t A@-tN), o : p det N and if the matrices are
- oN-' and fi of three rows and columns, then we take llrf
p(M-')'AM-t,
7c: poz: det M. _Thus we have the relation (19).
Thus the conditions on A and A are that these symmetric matrices
are multiplicatively cogredient. Hence with each I satisfying our
conditions we can associate a unique class of non'singular multi'
plicativety cogredient symmetric matrices. We have as many alge'
bras as there are classes of such matrices. For the remainder of
this section we assume the characteristic is not two. Then each cogredience class contains a diagonal matrix of the form diag {4, LL},
aF + 0. This implies that the basis can be chosen so that

(20)

fere2l: es ,

-legel) Pes.

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