Non-Abelian Minimal Closed Ideals of Transitive Lie Algebras. (MN-25) [Course Book ed.] 9781400853656

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NON-ABELIAN MINIMAL CLOSED IDEJUjS OF TRANSITIVE LIE ALGEBRAS

by

Jack F. Conn

Princeton University Press and University of Tokyo Press

Princeton, New Jersey 1981

Copyright(c) 1981 by Princeton University Press All Rights Reserved

published in Japan Exclusively by University of Tokyo Press in other parts of the world by Princeton University Press

Printed in the United States of America by Princeton University Press, Princeton, New Jersey

Library of Congress Cataloging in Publication Data will be found on the last printed page of this book

Table of Contents

Introduction

1

1.

Preliminaries

15

2.

Derivations of Transitive and Simple Lie Algebras

65

3.

Simple Algebras with Parameters

95

4.

Closed Ideals of Transitive Lie Algebras

110

5.

Minimal Closed Ideals of Complex Type

152

References

217

Preface

Apart from their inherent interest as algebraic structures, transitive Lie algebras play an essential role in any study of the integrability problem for transitive pseudogroup structures on manifolds. This monograph presents, in an essentially self-contained way, work on the structure of transitive Lie algebras and their non-abelian minimal closed ideals.

Many of the results contained here have simple

differential-geometric interpretations, and bear directly upon the integrability problem.

It is a pleasure to acknowledge the constant encouragement and advice given me by Hubert Goldschmidt and Donald C. Spencer; work owes much to their suggestions and trenchant criticism.

this Victor

Guillemin generously provided an unpublished manuscript which afforded some essential new results.

I

would also like to thank Vivian Davies

whose typing is responsible for the attractive appearance of the finished manuscript.

The Institute for Advanced Study provided

gracious hospitality to the author during a portion of this research, which was partially supported by Grant MCS77-18723 from the National Science Foundation. Jack F. Conn Pasadena, California

Introduction Transitive pseudogroups of local diffeomorphisms preserving geometric structures on manifolds have been studied by many authors; the origins of this subject are classical, and may be said to lie in the works of Sophus Lie and Elie Cartan.

The structure of such a pseudo-

group Γ acting on a manifold X is reflected in the structure of the Lie algebra of formal infinitesimal transformations of Γ, that is to say, those formal vector fields on X which are formal solutions to the linear partial differential equation which defines the infinitesimal trans­ formations of Γ.

The Lie algebras of formal vector fields obtained in

this way provide examples of what are now known as transitive Lie algebras: such Lie algebras are, in general, infinite-dimensional. The study of transitive Lie algebras was first placed on a strictly algebraic basis by the paper ([16]) of V. W. Guillemin and S. Sternberg. Subsequent work of Guillemin ([11]) characterized transitive Lie algebras as linearly compact topological Lie algebras which satisfy the descending chain condition on closed ideals, and established the existence of a Jordan-HSlder decomposition in such Lie algebras.

This latter result

is a weak analogue of the Levi decomposition for finite-dimensional Lie algebras.

Several authors have since adopted this abstract algebraic

viewpoint for the study of transitive algebras; one result of their work has been the rigorous and progressively simplified proofs ([12], [14], [15], [21], [23], [29]), in the category of transitive Lie algebras, of the classification of the infinite-dimensional primitive Lie pseudogroups given by E. Cartan ([3]). the present work.

We shall make use of this classification in

Transitive Lie algebras have been studied also to provide insight into the behavior of the integrability problem for transitive pseudogroup structures.

A precise formulation of this problem may be found in

([20]); for surveys of the principal results concerning this problem, we refer the reader to ([8]) and the introduction of ([9]).

The role played

by real transitive Lie algebras and their non-abelian minimal closed ideals in the integrability problem was elucidated by H. Goldschmidt and D. C. Spencer ([9])·

In our present work, we give a complete

algebraic description of the structure of these non-abelian minimal closed ideals.

Our study was undertaken a s a tool for the investigation

of the integrability problem, and i s an essential element in the proof of Conjecture III of ([9]) a s outlined in ([8]) and in greater detail in the introduction to ([31]). particular,

The proof of this conjecture implies, in

that the integrability problem is solved for all transitive

Lie pseudogroups acting on R n which contain the translations, a fortiori for all flat pseudogroups.

In an attempt to prove Conjecture I of ([9])

following the outline suggested there, we found that the geometry of pseudo-complex structures (induced structures on r e a l submanifolds of complex n-space (E n ) was expressed in the structure of non-abelian minimal closed ideals of complex type in real transitive Lie algebras. From this observation, we were able to construct simple counterexamples to Conjectures I and II of ([9]) involving such closed ideals; these counterexamples have appeared in our note ([4]).

Our presentation

follows through Section four the outline given in §13 of ([9]); this part of the present work contains our results on the structure of non-abelian minimal closed ideals of real type which a r e used by Goldschmidt in ([31]) to prove Conjecture I of ([9]) for these closed ideals.

In a

sequel to ([31]), Goldschmidt will present a proof of Theorem 9 of

([8]) which relies on our description of non-abelian minimal closed ideals of complex type in terms of pseudo-complex structures given in Section five; in this way, the proof of Conjecture III of ([9]) will be completed. In this work, we shall view transitive Lie algebras from an abstract viewpoint as topological Lie algebras, following the work of Guillemin and Sternberg ([11] , [16]) mentioned above.

Let K be a

field of characteristic zero, endowed with the discrete topology (even when K is equal to IR or ¢).

A transitive Lie algebra is a linearly

compact topological Lie algebra over K which possesses a fundamental subalgebra, that is, an open subalgebra L^ containing no ideals of L except {θ}; this is equivalent ([11]) to requiring that L satisfy the descending chain condition on closed ideals.

Any finite-dimensional

Lie algebra L over K becomes a transitive Lie algebra when endowed with the discrete topology, of L.

since {θ} is then a fundamental subalgebra

However, in the infinite-dimensional examples the topology plays

a more essential role.

If Γ is a transitive pseudogroup acting on a

manifold X, and L is the Lie algebra of formal infinitesimal trans­ formations of Γ at a point χ eX, then the isotropy subalgebra of L, that is, the subalgebra of formal vector fields in L which vanish at x, is a fundamental subalgebra I? of L.

Conversely, it is a theorem of

H. Goldschmidt ([6]) that any transitive Lie algebra L over JR and fundamental subalgebra L^ C L can be realized in this way.

The

abstract viewpoint of Guillemin and Sternberg which we adopt is thus seen to be completely consistent with the differential-geometric view­ point.

As we mentioned above, Guillemin proved ([11]) that a JordanHolder decomposition can be introduced in any transitive Lie algebra L. Such a decomposition consists of a finite descending chain L = I 0 D I i 3 · · · ^ f n = {0} of closed ideals of L, such that, for each integer ρ with 0< p< n-1, either (i) The quotient IpAp + j is abelian; or (ii) The quotient Ip/^p + i

i- s non-abelian, and is a minimal closed

ideal of L/l ,. ' P +1 Guillemin also showed that the number and type of quotients of type (ii), both as topological Lie algebras and as topological L-modules, is independent of the choice of Jordan-Holder sequence for L.

The

existence of such a decomposition had been conjectured (in the category of transitive pseudogroups) by E. Cartan.

The quotient of a transitive

Lie algebra by a closed ideal is again a transitive Lie algebra, since it also satisfies the descending chain condition on closed ideals; there­ fore, each of the quotients Ip/l

i- n

a

Jordan-Holder sequence for a

transitive Lie algebra L is a closed ideal in a transitive Lie algebra L/l

j.

Quotients of type (i); that is, closed abelian ideals of transi­

tive Lie algebras, have been extensively studied as part of the work of Goldschmidt and Spencer ([9] » [10]).

We shall concentrate here upon

the structure of quotients of type (ii), that is, non-abelian minimal closed ideals of transitive Lie algebras.

The investigations of

Goldschmidt and Spencer cited above reduce the integrability problem for a transitive pseudogroup Γ to a series of questions concerning the

structure of the quotients I /

both as topological Lie algebras and

topological L-modules, appearing in a Jordan-HSlder sequence for the Lie algebra L of formal infinitesimal transformations of Γ.

As a

consequence, our results bear directly upon the integrability problem for transitive pseudogroup structures. We now describe the main results of this work; to simplify our outline, we assume, unless otherwise specified, that all Lie algebras considered below are defined over the field IR of real numbers.

Many

of our results are obtained for linearly compact topological Lie algebras without the assumption of transitivity.

For the sake of clarity, we

make several preliminary observations before beginning our outline itself. Let L be a linearly compact topological Lie algebra, and suppose that I is a non-abelian minimal closed ideal of L.

Then it is known

([11]) that I possesses a unique maximal closed ideal J; moreover, the quotient l/J is a non-abelian simple transitive Lie algebra R. commutator ring

The

of R, that is, the algebra of IR-Iinear mappings

c : R —*• R such that, for all ξ, η e R, c(U.l]) = [c(£), η] is, according to ([11]), actually a field which is a finite algebraic extension of ]R.

Thus, the field

is equal to IR or to (E; we shall,

then, say that the non-abelian minimal closed ideal I of L is of real or complex type, respectively.

The simple real transitive Lie algebra

R may be viewed naturally as a transitive Lie algebra over its commutator field K^, and every real-linear derivation of R is actually Kj ^-Iinear.

Unless R is finite-dimensional, it need not be true that

every derivation of R is inner.

However, the space Der(R) of deriva­

tions of R has a natural structure of transitive Lie algebra over K^, and the adjoint representation of R allows us to identify R with a closed For η an integer > 0, consider

ideal of finite codimension in Der(R). the local algebra

F

=

K

r[[xI·

"'· xJ]

of formal power series in η indeterminates over mean that F = Kp); endow F with the Krull topology.

(when η = 0, we The maximal

ideal F 0 of F consists of those formal series which vanish at the origin; the powers

comprise a fundamental system of neigh­

borhoods of zero in F, which is a linearly compact topological algebra. The space Der(F) of derivations of F has a natural structure of transi­ tive Lie algebra over K^, with the Lie bracket given by the usual commutator of derivations; the stabilizer Der 0 (F) = {ξ e Der (F) j £(F°) C F 0 }

of F° is a fundamental subalgebra of Der(F). structures of topological Lie algebra over

There are natural and topological Der(F)-

module on the tensor product

Der(R) (X)

F ; R

/\

the Hausdorff completion Der(R)

F of this space inherits linearly R compact structures of topological Lie algebra and topological Der(F)module.

Furthermore, the transitive Lie algebra Der(F) acts by Λ derivations on the Lie algebra Der(R) ( X) k - F. We can, then, form the R semi-direct product

(Der(R) ®

F) © Der(F) , R /S

which is a transitive Lie algebra over

and R

F is then a nonR

abelian minimal closed ideal in this Lie algebra.

We come now to the actual outline of our results on the structure of non-abelian minimal closed ideals.

Although our results

are of greater interest and novelty in the case of ideals of complex type, it will be convenient to treat the real case first.

We maintain

the notational conventions of the previous paragraph.

Assume that the non-abelian minimal closed ideal I of L is of real type.

Then the normalizer N = N l (J)

in L of the maximal closed ideal J of I is a subalgebra of finite codimension in L, as is proved in ([11]).

Set η = dim(L/N), and

F = JRttx 1 , . . . , x n ]] . In Theorem 4.2 we prove that there exists a morphism of real topolo­ gical Lie algebras Φ : L - (Der(R) ( S ) j r F ) © Der(F) , such that the restriction of Φ to I is an isomorphism Φ

I1

: X - R ®

e

F .

The kernel of Φ is equal to the commutator of I in L, and the projection ir(®(L)) of $(L) onto Der(F) is a transitive closed subalgebra of Der(F),

8 in the sense that = Der(F) Guillemin proved in ( [ 1 1 ] ) that I and

. are isomorphic as

abstract Lie algebras; our proof of Theorem 4 . 2 consists mainly of a close examination of Guillemin's work,

combined with the observation

(Lemma 2. 6) that the topology of Der(R) as a transitive Lie algebra coincides with the weak topology Der(R) inherits as a subspace of the continuous linear transformations of R. We now assume that the non-abelian minimal closed ideal 1 of L is of complex type.

A s above,

the normalizer

a subalgebra of finite codimension n in L .

of J in L is

In Section five,

we show

that I may be viewed naturally as a complex topological Lie algebra with J a maximal closed complex ideal, representation,

and that L acts,

via the adjoint

on I by continuous complex-linear mappings.

This

action of L on I may be complexified to a representation of the complex Lie algebra

on I; the normalizer N u

representation is a complex subalgebra of mension m

is a transitive Lie subalgebra of Der(IR[[x^,

...

,

xn

] ])

in the sense that Der(IRt[x1 , . . .

, x j ]) = A + Der°(3R[ [ x j

X Q ] ] ),

and A0

= A

n Der°(IR[[x1,

is a fundamental subalgebra of A .

...

Upon defining the s e m i - d i r e c t product

(Der (R) (gl^H) © as before,

, x j ])

A

we obtain a real transitive Lie algebra in which R (x) ^H

f o r m s a non-abelian minimal closed ideal.

In Theorem

5.2,

we prove

that there exists a morphism of real topological Lie algebras i|j : L — (Der(R) ®

a

H)

©

A

such that the restriction of i[i to I is an isomorphism

•Hj : I -

R ® ^

.

The kernel of I|J is equal to the commutator of I in L,

and the projection

•^(^(L)) of i|;(L) onto A is a transitive subalgebra of A ,

in the sense that

A = TT(i|J(L)) + A 0 We also associate to L and N " (N"/N_)

X ( N ' ' / )

-

.

a Hermitian mapping (L_/(N"

+ N"))

10 which we call the Levi f o r m of I.

The vanishing of

is shown in

Proposition 5 . 4 to be equivalent to the existence of an isomorphism

of real local algebras

such that the complexification

when restricted to H,

is an i s o m o r p h i s m of H onto the subring In Proposition 5. 6,

we prove

that if L is a real transitive Lie algebra with fundamental subalgebra

I?

and abelian subalgebra V such that

then the Levi f o r m of each non-abelian quotient of complex type occurring in a Jordan-Holder

sequence for L must vanish.

This

proposition is an essential result for the proof of Conjecture III of ( [ 9 ] ) . A key step in the proof of Proposition 5 . 6 is provided by an unpublished result of

Guillemin,

which states that if I is a non-abelian minimal

closed ideal of a transitive Lie algebra L (over any field of characteristic zero) and N-^(J) is the normalizer in L of the maximal closed ideal J of I,

then

contains every fundamental subalgebra L^ of L .

We

have reproduced Guillemin's result here as Proposition 4 . 5.

Returning to our discussion of ideals of complex type,

the

monomorphism

which we construct f r o m L and N " ,

may be viewed geometrically as the

pullback mapping associated to the formal expansion φ at 0 e K n of an embedding φ : U — (E m ,

of a neighborhood U of

p(0) = 0

0 in IR n as a generic real submanifold of (E m .

Using results of Goldschmidt ([6]) on the analytic realization of transi­ tive Lie algebras,

one can show that φ can be chosen to be convergent,

in which case y(U) becomes a real-analytic generic real submanifold of 0,

(M, M®) is a transitive Lie algebra. Let V be a finite-dimensional vector space over K.

linearly compact Lie algebra over K,

If L is a

then a transitive representation of

37 L on F { v ' }

is a morphism

of topological Lie algebras such that the image X(L) is a transitive algebra of D e r ( F { V } ) . (1.21)

sub-

We call the open subalgebra

of L the isotropy subalgebra of X,

and,

to express

( 1 . 2 1 ) we

sometimes

say that \ is a transitive representation of (L, M) on D e r ( F { V } ) ; the vector space V is then isomorphic to L / M . X. i s such a transitive representation,

One verifies easily that if

then

(1.22)

for all p > 0; in particular, L.

Thus,

the kernel of X is the closed ideal

if we endow L with the filtration

and filter D e r ( F { V } ) by the subalgebras a homomorphism of filtered Lie algebras. morphism of graded Lie algebras

which is

of

seen f r o m ( 1 . 2 2 ) to be infective.

, Hence,

then X is

we obtain an associated

38 If

0.

Assume,

in addition that the vector

space W is linearly compact;

42 then,

since V is finite-dimensional,

are linearly compact.

the spaces

Let ^Tbe a topological space,

is a continuous mapping,

with (p, q) + ( 0 , 0 ) .

and suppose that

If

then there exists a continuous mapping

such that

Proof: Choose a basis v . , I

the dual basis for V " .

for all p,

q > 0,

...

, v

n

for V,

and let v . ,

Define a linear mapping

by setting

for all w eW and a

and

where we have used the

symbol • to denote multiplication in the algebra S ( V r ) . putation using ( 1 . 2 3 )

for all

. . . , v be i n

shows

An easy c o m -

that

The f i r s t part of our lemma follows

43 immediately. If U is a topological vector

space over K,

we denote by

the vector space of continuous mappings that,

for all p e Z

P

and q > 0,

One verifies

easily

there is a natural isomorphism

:

such that

The second part of the lemma is then obtained

f r o m the first part, Lemma spaces over K.

upon replacing W by

1.5 ([25]).

Let V and W be finite-dimensional

vector

Suppose that

is a graded S(V)-submodule of integer p^ > 0 such that,

Then,

there exists an

for all p > p^,

Remark: There exists an integer and the dimensions of V and of W,

such that,

depending only upon PQ for all

and

(1.25)

A proof

of this statement may be found in the paper ( [ 2 7 ] )

of Sweeney.

The weaker statement that there exists an integer p^ such that ( 1 . 2 5 ) holds for all p > p. and q > 0 can be proved by dualization, f r o m facts

44 concerning the Kozul complex (cf.

[24],

[25]).

Proof: Because W is finite-dimensional,

is finitely generated. polynomial ring K [xi'

If n '" dim(V),

the graded S(V)-module

the ring S(V) is isomorphic to the

, x ], and is thus Noetherian; since WQ9K S(V) n

is finitely generated, it is a Noetherian module over S(V). The spaces

for

PE:~,

sum to form a graded S(V)-submodule

G

1

:::

®

G

1

p

PE:~

of

W'~

Q9 S(V). K

Denote by IT the natural mapping p

for

given by multiplication in the S(V)-module G

1

PE:~

,

Since G

1

is a submodule

of the Noetherian module W* Q9 S(V), it is finitely generated; thus, there K exists an integer PO;::: 0 such that, for all P;::: PO' the mapping ITp is surjective.

One verifies easily that IT

P

is surjective if and only if

HP,i(G) ::: {O}, which completes the proof. If G is a subspace of W Q9K SP(V*),

the subspace

for p;::: 0, we denote by g(i)

45

of W ®K Sp+1(V*); using this, we define subspaces

for j

~

1, inductively by setting

(1. 26)

for

j

~

The space gU) is called the j-th prolongation of g.

G

G

1 .

If

P

is a graded S(V)-submodule of W ®K SlY':'), we see from Lemma 1. 4 that if p

~ 0, then HP,1(G) = {O} if and only if G

(G )(1). p

p+1

We now discuss the notions of simplicity and primitivity for transitive Lie algebras.

A Lie algebra L is said to be simple if L is

non-abelian and the only ideals of L are linearly compact Lie algebra, simplicity: we

say that

L

and L itself.

If L is a

there is another possible notion of is topOlogically simple if

abelian and the only closed ideals of Proposition 4.3),

{O}

L

are

{O}

L

and

is nonL.

In ([ 11 ])

Guillemin proved that these two notions coincide:

.i!:. linearly compact Lie algebra L topOlOgically simple.

~

K is siITlple

if

and only

if i1

is

There continues to be confusion concerning this

point in the literature.

We note that a simple linearly compact Lie

algebra L is necessarily transitive, since any proper open subalgebra of L is fundamental; such subalgebras exist by (iii) of Proposition 1.2. To emphasize the point made above, we shall sometimes call a maximal closed ideal I of a linearly compact Lie algebra L strictly maximal, to indicate that such an ideal is maximal among all ideals of L. Let L be a transitive Lie algebra; then L is said to be primitive if there exists a proper open subalgebra L^ of L such that

(i) (ii)

L^ is maximal among the subalgebras of L, and, the subalgebra L^ is fundamental.

Such an open subalgebra L® will be called a primitive subalgebra of L. Our definition does not exclude the one-dimensional abelian Lie algebra, which forms the only example of an abelian primitive Lie algebra over K. We note that a simple transitive Lie algebra is necessarily primitive. Indeed, if L is a linearly compact Lie algebra other than {θ}, there exists a proper open subalgebra M of L, by Proposition 1. 2, (iii); since M is of finite codimension in L, we can find a maximal subalgebra L^ of L containing M.

This maximal subalgebra L^ is open, since L^ D M;

moreover, if L is simple, then L^ is fundamental, and is thus a primi­ tive subalgebra of L.

We shall use most of this argument below, in the

proof of Lemma 1.6, (ii). Lemma 1.6 ([11]).

(i)

Let L be a primitive Lie algebra over K, and let L^ be a

primitive subalgebra of L.

If I is a non-zero closed ideal of L, then

47 the codimension of I in L is finite. (ii)

If L is any n o n - z e r o linearly compact Lie algebra over K,

then there exists a proper closed ideal J of I which is strictly m a x i m a l . Proof: To prove (i),

f i r s t note that,

subalgebra L^ of L does not contain I.

since I =£-[()},

Thus,

algebra of L which properly contains L^;

the fundamental

the sum I + L^ is a sub-

since L is maximal,

we

conclude that

F r o m this,

we see that there is a natural

of topological Lie algebras.

epimorphism

By Proposition 1 . 3 ,

the spaces

f o r m a descending chain of closed ideals of L^; moreover, fundamental subalgebra of L, neighborhoods of 0 in L^.

since l "

is a

this sequence is a fundamental system of

It follows that the images

f o r m a descending chain of closed ideals of L / I such that

(1.27)

F r o m Theorem 1 . 1 ,

Corollary 1 . 2 ,

exists an integer m > 0 for which

and ( 1 . 2 7 ) ,

we conclude that there

since D^(L^) is open in L, it follows that the codimension of I in L is finite. To prove (ii), we first observe that, according to the argument given before the statement of the lemma, if L is any non-zero linearly compact Lie algebra over K, then there exists a maximal subalgebra M of L which is open.

The quotient

L' = L/D"(M)

is then seen to be primitive, using Corollary 1.1.

If L 1 is either simple

00 or one-dimensional abelian, we may take D^(M) as J.

Otherwise, it

follows easily from part (i) that there exists a non-zero maximal closed ideal J 1 of L 1 ; the preimage J of J' in L is then a strictly maximal closed ideal of L. Let L be a primitive Lie algebra over K.

A primitive realization

of L is a faithful, transitive representation λ of L on F-[V }, for some finite-dimensional vector space V, such that the isotropy subalgebra of λ is a primitive subalgebra of L.

In ([12]), Guillemin proved that an

infinite-dimensional primitive Lie algebra L oyer K has a uniq ue primitive subalgebra L^; thus we see from Theorem 1.2 that a primitive realization of such a p r i m i t i v e L i e a l g e b r a i s uniquely determined, up t o the action of an isomorphism of F{(L/L^)'}.

A primitive subalgebra of

a finite-dimensional primitive Lie algebra is not, in general, unique. If L is a Lie algebra over K, then the commutator ring

of L

is the subring of End^(L) consisting of all K-Iinear mappings c : L —• L such that, for all ξ, η eL,

49

c( [ i , ^ ) )

= [e,c(n)] .

The scalars K are naturally embedded in the ring K ^ . elements

of K . ,

then,

for all

r| e L,

( < V c 2 )([S.ri]) = [ C j d ) ,

therefore,

if

If c ^ and c^ are

c2fa)] = (c2- c^US.T!])

[L, L ] = L then K ^ is a commutative ring.

;

The following

extension of the c l a s s i c a l Schur's lemma is proved in ( [ 1 1 ] ,

Proposition

4.4). Proposition

1.4.

Let L be a simple transitive Lie algebra over K

Then the commutator ring K ^ is a field which is a finite algebraic extension of K. phism of L .

If c is an element of K ^ ,

Thus,

algebra over K ^ ,

then c is a linear

homeomor-

L has a natural structure of simple transitive Lie

which extends the transitive Lie algebra structure of L

over K.

Let L be a simple transitive Lie algebra over K. space over K, K. .

then K ^ (X)^ Y is naturally a vector

We define a natural mapping

(1.28)

by setting,

for all

and

and

If V is a vector

space over the field

50 then v is an isomorphism proposition,

of vector spaces over K.

The following

which also extends a c l a s s i c a l result for finite-dimensional

simple Lie algebras, Proposition

is proved in ( [ 1 1 ] ,

1.5.

Proposition

5.1).

Let L be a simple transitive Lie algebra over K.

Suppose that V is a finite-dimensional vector

space over K,

and denote by

\ the representation of L on L (X)^, V given by

for all

>7 f L and v e V .

Then,

a subspace W of L (X)^ V is invariant

under the action of y if and only if there exists a K ^ - s u b s p a c e k

L ®K

v

such

U of

that

where v is the mapping

(1.28).

We now describe the c l a s s i c a l examples of simple infinite-dimensional Lie algebras.

Let n be an integer >

transitive 1;

as before,

we set

Recall that we have identified Der(F) with the Lie algebra of formal vector fields ( 1 . 3 ) . algebra over d x , , i

We denote by . . . , dx , n

the formal

with coefficients in F; then

exterior

51 An element u of

may be uniquely

expressed

in the f o r m

(1.29)

where the coefficients identified with F .

lie in F; the space

Thus,

to { o } ; for convenience,

for p > n,

the spaces

we set

element of

is are

equal

We call an

a f o r m a l differential p - f o r m .

We

have then

the operations of f o r m a l exterior differentiation

and interior

for all p > 0,

multiplication

differential f o r m s on a differentiable manifold (the symbol common use for TV).

is then defined,

( 1 . 3 0all ) for

operations on

defined in the s a m e way as the corresponding

is also in

The operation of formal Lie differentiation

for all p > 0,

Der(F) and

J

by

F r o m ( i . 29) we see that each

52 of the spaces

. . . , x ) may be given a linearly compact

as the Cartesian product of ( j j ) each of the operations d and 7;, to the topologies of A * 3 ^ , Suppose that n > 2,

...

structure

copies of F; it is easily verified that and hence

H,

is continuous with respect

, x ) and D e r ( F ) .

and let fi be the formal differential n - f o r m

We denote by sjf^fn, K) the transitive subalgebra of Der(F) consisting of those f o r m a l vector fields £ which satisfy

(1.31)

The algebra sj?00(n, K) is sometimes called the algebra of f o r m a l divergence-free vector fields,

since,

in terms of ( 1 . 3 ) ,

the condition

( 1 . 3 1 ) is expressed as

A s s u m e that n = 2m is even,

with m > 2; denote by Q the f o r m a l

differential 2 - f o r m

The algebra sp 0 0 (2m, K) of f o r m a l symplectic vector fields is the transitive subalgebra of Der(F) consisting of the f o r m a l vector fields £ such that

53 Now suppose that n = 2m + 1 is odd,

with m > 1,

and let to be the

formal 1 - f o r m

(1.32)

The contact algebra cfc(2m + 1, K) is the transitive subalgebra of Der(F) consisting of those f o r m a l vector fields | which satisfy the condition

this is equivalent to saying that

(1.33)

for some f e F . F o r consistency of notation,

if n i s any integer >

1,

we

sometimes

denote by K) ofthethe Lietransitive algebra Der(F) of formal vector fields. If Lg-f^fn, is one Lie algebras

defined above,

then L is infinite-dimensional and simple,

and

54

(1.35)

is the unique primitive subalgebra of L . is irredundant,

i.e.,

the papers ( [ 1 2 ] ,

Moreover,

the list given above

no two algebras on this list are isomorphic.

[14],

[15],

[21],

[23],

[29]),

From

there results the

following classification: Theorem 1 . 3 . teristic z e r o . algebra over K,

defined above.

Let K be an algebraically closed field of charac-

If L is a simple transitive infinite-dimensional

Lie

then L is isomorphic to one of the Lie algebras

Over the field IR of real numbers,

infinite-dimensional Lie algebras

the simple

transitive

are

considered as transitive Lie algebras over IR. We now proceed to list those properties of the classical algebras defined above which we shall require in the sequel.

Lie

If L is

55 one of the algebras

(1. 36)

spoo(Zm, K) ,

and LO is the unique primitive subalgebra (1. 35) of L,

then (L, LO) is

flat, as one sees easily from the defining expres sions for these algebras. Set V = L/LO, and let $

be the filtration {Dt(LO)}PEZ of L.

Since

(L, L 0) is flat, we have a natural isomorphism

L '"

IT

grP(L, /y)

PEZ in conjunction with the natural mapping (1. 16), this yields an identification of L with a flat,

transitive subalgebra of D(V).

We shall call this

the standard realization of L; note that the standard realization of gl oo (n, K) is just the identity mapping

°

The subalgebra gr (L, $) of L is identified with a Lie subalgebra of the space

of linear transformations of V.

If L is equal to sl oo (n, K),

then grO(L, .Y)

is the special linear algebra sl(n, K), in its standard representation as the subalgebra of linear transformations of trace zero in HomrztV, V). L is equal to spoo(Zm, K), sp(Zm, K),

then grO(L, £7) is the symplectic algebra

in its standard representation as the subalgebra of linear

If

56 transformations which preserve infinitesimally an antisymmetric

bilinear

f o r m of maximal rank on the 2m-dimensional vector space V .

If L is

equal to

then

is the general linear algebra

identified with H o m ^ ( V , V ) . which we are discussing,

Moreover,

in each of the cases

the graded t e r m s

gi(n,K),

(1.36)

for p ^

1,

are

given by the prolongations

of

as defined in ( 1 . 2 6 ) ;

again,

this is a direct consequence of

the f o r m of the defining equations for these algebras.

If L is one of the

algebras

(1.37)

then it is a well-known result of classical invariant theory ([ 30 ]) that the action of irreducible,

on each of its prolongations

is

for p > 1; we shall also use the easily verified result that

(1.38)

for p > 1,

in each of these cases

(1.37).

Denote by I the identity

transformation

of V .

Then,

considering I as an element of D(V),

acts as the degree derivation of D(V); that i s ,

the mapping ad(I)

if 4 is an element of the

57 p - t h graded term

of D(V),

then

(1.39)

as is seen f r o m ( 1 . 1 7 ) .

If L is one of the algebras

have identified L with a flat subalgebra of D(V); thus, normalizer of L in D(V), is equal to

(1.36),

then we

I lies in the If L

and ( 1 . 3 9 ) holds for all

then I lies in

If L is the contact algebra ct(2m + 1 , primitive subalgebra ( 1 . 3 5 ) of L,

K) and L^ is the unique

then (L, L^1) is not flat.

Nevertheless,

there exists a graded Lie algebra

where each of the summands g*5 is finite-dimensional, p < -2,

such that L is isomorphic,

and

as a topological Lie algebra,

for to the

c ompletion

(1.40)

Under this isomorphism,

L^ corresponds

to the subalgebra

and there exists an element I e g ^ which acts as the degree derivation on L,

that i s ,

( 1 . 3 9 ) holds for all

We shall indicate briefly how

the isomorphism ( 1 . 4 0 ) is constructed after proving the following l e m m a :

58 Lemma

1.7.

Let F be the ring

and let L

be the contact algebra

Denote by to the formal differential 1 - f o r m ( 1 . 3 2 ) ,

be the subspace of Der(F) annihilated by

and let

Then Der(F) is the direct

sum

(1.41)

of subspaces invariant under the action of L . Proof:

We f i r s t show that

(1.42)

If i lies in

3 ,

then,

by ( 1 . 3 0 ) ,

(1.43)

if,

in addition,

lies

in L,

then we see f r o m ( 1 . 3 3 ) that

59 Comparing coefficients of dx , we conclude that if i

for some fEF.

SE

I]))

n L,

then

S 7\

0,

dw

°

since w is a i-form of maximal rank, (1. 42).

Let

g

S=0,

which proves

be the submodule

.r;:

of

we infer that

= {f w I

1 A (dx , .,. , dx + ) over F. 1 2m 1

f EF }

To prove that

Der(F) ,

we must show that, if

according to (1. 33).

given, for all

SE

I])) ,

S E Der(F), then there exists S E (J) such that

It obviously suffices to show that the mapping

by

p(s)

is surjective.

m.od

It is clear from the definition of

g,

I]))

that

9J

is an F-

subm.odule of Der(F), and from. (1. 43) we see that p is a hom.orphism.

60 of

F-modules.

of D e r ( F ) l i e in

for

The

elements

, and w e

S i n c e the f o r m a l

span

we conclude

the d i r e c t s u m d e c o m p o s i t i o n

to s h o w that

by (1.33);

1-forms

over F,

This proves

exists

have

i s i n v a r i a n t under

s u c h that

then,

for all

that p is

(1.41).

the a c t i o n of L .

surjective.

A l l that r e m a i n s If

is

then t h e r e

61 which proves

that

is

L-invariant.

It i s an i m m e d i a t e c o n s e q u e n c e

of L e m m a

1 . 7 that the m a p p i n g

defined by

i s an i s o m o r p h i s m of t o p o l o g i c a l v e c t o r the t o p o l o g i c a l L i e a l g e b r a

f o r a l l f, g e F ,

From

it f o l l o w s

in F

this,

the

degree

transfer

s t r u c t u r e of c t ( 2 m + 1, K) to F via

straightforward computation shows expressed,

s p a c e s ; we m a y thus

that the L i e b r a c k e t in F i s

by

that if w e a s s i g n to e a c h

monomial

A then

6Z

+

Zp 1

Pz

+

P3

+ ... + PZm+1

- Z ,

then the resulting graduation on the polynomial ring

gives rise to a structure of graded Lie algebra

K[X1' ... , x Zm +1 ]

=0

gP

PEZ

on K[x1' ... , x

+ ], with gP Zm 1

= {O}

for P < -Z.

The topological Lie

algebra F is isomorphic to the completion of this graded Lie algebra. It is in this way that the isomorphism (1. 40) is constructed.

degree derivation I

E

gO corresponds,

as one sees immediately from (1.44).

under cp,

The

to the element

-Z and g-1

The dimensi ons of g

over K are given by

-Z )

dim(g

the Lie algebra g

o

- 1 dim(g )

1,

Zm

-1 is faithfully and irreducibly represented on g as

the symplectic algebra sp(Zm, K), a one-dimensional center {cI

in its standard representation, plus

ICE K}

We have shown that if L is one of the classical simple transitive Lie algebras lis ted in (1. 34),

then L is isomorphic to the completion

63

(1.45)

of a g r a d e d L i e a l g e b r a

with f i n i t e - d i m e n s i o n a l

s u c h that algebra

of L .

of L ,

summands,

L e t L0 b e the unique p r i m i t i v e and d e n o t e b y

F r o m (1.45),

9

the a s s o c i a t e d

w e obtain a s e c o n d

sub-

filtration

filtration

(1.46)

of L .

If (L, L^) i s f l a t ,

i s e q u a l to g r ( L ,

).

then t h e s e f i l t r a t i o n s c o i n c i d e ,

In a l l of the c a s e s

(1.34),

we

and

have

(1.47)

f o r all

and

(1.48)

f o r a l l p,

F o r c o m p l e t e p r o o f s of a l l of the a s s e r t i o n s w e

m a d e c o n c e r n i n g the c l a s s i c a l the r e a d e r to the p a p e r s Finally,

([12],

simple transitive Lie algebras, [21])

of G u i l l e m i n and

w e state a w e a k v e r s i o n of L e m m a 4 . 2 of

we

have refer

Morimoto-Tanaka. ([11]):

Lemma 1. 8.

Let L be a simple transitive Lie algebra over K.

There exists a finite-dimensional subspace W of L such that

[W, L] = L

65 Derivations

of T r a n s i t i v e and S i m p l e L i e

T h r o u g h o u t this s e c t i o n ,

Algebras

we d e n o t e by K a f i e l d of

characteristic

zero.

L e t L b e a L i e a l g e b r a o v e r K, B y a d e r i v a t i o n of A into L ,

s u c h that,

and l e t A be a s u b a l g e b r a

we shall m e a n a K - l i n e a r

of

mapping

for all

f r o m the J a c o b i identity in L , d e r i v a t i o n of A into L . s i m p l y a d e r i v a t i o n of L .

w e s e e that if ^ e L ,

When A = L ,

then ad(^) i s a

such a linear mapping

R e c a l l that,

in the l a s t s e c t i o n ,

is we

e s t a b l i s h e d the n o t a t i o n D e r ( R ) f o r the L i e a l g e b r a of d e r i v a t i o n s algebra

on the c o m m u t a t o r

ring

for all D e D e r ( L )

and

Let

b e the l i n e a r m a p p i n g d e f i n e d b y setting,

(2.1)

of an

R.

W e n o w d e f i n e an a c t i o n of D e r ( L ) L.

L.

of

66 w h e r e the b r a c k e t i s taken with r e s p e c t to the L i e a l g e b r a

Lemma Lie algebra

Assume

2.1.

T h e m a p p i n g |jl takes v a l u e s in D e r

of K,

that the r i n g

in a d d i t i o n ,

i s an i n t e g r a l d o m a i n . over

the L i e a l g e b r a

If c i s an e l e m e n t of

that K 1 i s a f i e l d w h i c h i s an a l g e b r a i c

and that

T h e n e v e r y d e r i v a t i o n of L i s

structure

o v e r K,

structure over

Proof: Let

which

and in w h i c h the i n n e r

f o r m an i d e a l .

have

and i s a

then f o r a l l

and D e r ( L ) has a n a t u r a l L i e a l g e b r a

we

of

homomorphism

which is algebraic

Assume,

structure

b e an e l e m e n t of D e r ( L ) .

F o r all

extension K'-linear, extends

derivations

67 therefore,

and s o asserted.

l i e s in

i s a d e r i v a t i o n of

Further,

if

; thus |J. takes v a l u e s in

as

It n o w f o l l o w s i m m e d i a t e l y f r o m the J a c o b i i d e n t i t y in that |J. i s a L i e a l g e b r a

Assume over

that

homomorphism.

i s an i n t e g r a l d o m a i n .

If

is

algebraic

let

b e the m i n i m a l p o l y n o m i a l of c in K [ t ] , Upon a p p l y i n g

Since

w e have

Let

to the e q u a t i o n p ( c ) = 0,

has n o z e r o d i v i s o r s and K i s

we

b e an e l e m e n t of D e r ( L ) . obtain

of c h a r a c t e r i s t i c

zero,

we

c o n c l u d e f r o m the m i n i m a l i t y of p(t) that

L e t K 1 b e a f i e l d w h i c h i s an a l g e b r a i c

e x t e n s i o n of K s u c h that

then w e h a v e j u s t s e e n that

annihilates K ' .

From

the definition of μ, we see that this is equivalent to saying that D is a K 1 -Iinear derivation of L.

Finally, we define a vector space structure

over K 1 on Der(L) by setting

C -D = C o D

D eDer ( L ) , CEK 1 ;

that c · D so defined actually lies in Der(L) follows from the fact that D is K 1 -Iinear.

The reader will easily verify that this endows

Der(L)

with a Lie algebra structure over K 1 which satisfies the remaining assertions of the lemma.

Lemma 2. 2. subalgebra L^.

Let L be a transitive Lie algebra with fundamental

If I is a closed ideal of L such that

I + L0 = L ,

(2.2)

then the codimension of I in L is finite.

Proof: From (2.2), it follows that the quotient mapping

π : L0

> L/I

is an epimorphism of topological Lie algebras.

Since the subalgebra L?

is fundamental, it follows, from Corollary 1.1 and Proposition 1.3, i

0

that the derived subspaces D (L ), for I > 0, form a descending chain of closed ideals of L^ such that

Π ί >0

D i (L°) = {0} ,

69 and that the s p a c e s neighborhoods

a l s o a r e a fundamental s y s t e m

of 0 in L .

Thus,

of

the i m a g e s

f o r m a d e s c e n d i n g c h a i n of c l o s e d i d e a l s

of L / l

satisfying

(2.3)

Because

L i s a t r a n s i t i v e L i e a l g e b r a and I i s a c l o s e d i d e a l of L ,

quotient L / l is a l s o transitive,

by Corollary

1.2.

It f o l l o w s

that

s a t i s f i e s the d e s c e n d i n g c h a i n c o n d i t i o n on c l o s e d i d e a l s ; using w e c o n c l u d e that t h e r e e x i s t s an i n t e g e r

the

(2. 3),

s u c h that

that i s ,

Since finite,

is

open in L ,

c o n c l u d i n g the Lemma

2. 3.

subalgebra

we see

that the c o d i m e n s i o n of I in L i s

proof.

L e t L b e a t r a n s i t i v e L i e a l g e b r a with f u n d a m e n t a l

and l e t M b e a t r a n s i t i v e

the n o r m a l i z e r

is also a transitive

subalgebra

of

subalgebra

of

,

Then and M

f o r m s a c l o s e d i d e a l of f i n i t e c o d i m e n s i o n in N T ( M ) . P r o o f : S i n c e the L i e b r a c k e t on L i s c o n t i n u o u s and M f o r m s a closed subspace

of L ,

we may write

a s the

intersection

70 of c l o s e d

s u b s p a c e s o f L ; thus the s u b a l g e b r a

Obviously,

we

i s c l o s e d in

L.

have

and h e n c e ,

therefore,

is a transitive

f o r m s a c l o s e d i d e a l in

of

s u b a l g e b r a of The

is f u n d a m e n t a l ,

"We c o n c l u d e f r o m L e m m a

Clearly

M

subalgebra

and the c l o s e d i d e a l M s a t i s f i e s the

relation

2 . 2 that the c o d i m e n s i o n o f M in N ^ ( M )

is

finite.

Proposition 2 . 1 . K,

L e t V be a f i n i t e - d i m e n s i o n a l v e c t o r

and let L be a t r a n s i t i v e

fundamental

of L .

Lie

subalgebra of D(V).

subalgebra

S u p p o s e that

i s a d e r i v a t i o n o f L into D ( V ) s u c h that

space

Denote by

over the

71

(2.4)

is continuous,

Then

and there

exists

a unique element

of

such that

(2.5)

P r o o f : Since the L i e b r a c k e t in D ( V ) is c o n t i n u o u s ,

to p r o v e that

D is continuous we n e e d only show the e x i s t e n c e of an e l e m e n t satisfying ( 2 . 5 ) . of

We

shall c o n s t r u c t a s e q u e n c e o f

elements

w h i c h s a t i s f i e s the c o n d i t i o n s

(2.6)

(2.7)

f o r all

and

Taken t o g e t h e r ,

( 2 . 6 ) and ( 2 . 7 ) w i l l

imply

that the l i m i t

e x i s t s and is an e l e m e n t of (2.7) holds trivially for integer

since

and s u p p o s e that e l e m e n t s

have b e e n c o n s t r u c t e d f o r mapping

satisfying ( 2 . 5 ) .

Set

then

Now let m be an s a t i s f y i n g ( 2 . 6 ) and C h o o s e an i n j e c t i v e

(2.7)

linear

72 s u c h that,

f o r all v e V ,

then the i m a g e o f i f o r m s a c o m p l e m e n t to satisfies

(2.7),

by setting,

and,

for

for

we may define linear

in L .

Because

mappings

all

all

A s a consequence of ( 2 . 4 ) , In the u s u a l w a y ,

the m a p p i n g b v a n i s h e s i d e n t i c a l l y if m = - 1 .

w e r e g a r d c as an e l e m e n t

W e n e x t s h o w that c i s a c o c y c l e in the S p e n c e r c o m p l e x ( 1 . 2 4 ) , W = V.

U s i n g ( 1 . 2 3 ) and the d e f i n i t i o n of c ,

we compute,

for

all

with

73

the f i n a l s t e p f o l l o w s f r o m the a s s u m p t i o n that L is a s u b a l g e b r a

of

D ( V ) and ( 2 . 7 ) , w h i l e in the p r e c e d i n g s t e p w e u s e d the a s s u m p t i o n that i s a d e r i v a t i o n and the J a c o b i i d e n t i t y . have,

Reasoning

similarly,

we

f o r all

in the f i n a l s t e p we h a v e u s e d ( 2 . 7 ) and the f a c t that u n d e r the a c t i o n o f

If

is

we c o n c l u d e f r o m L e m m a

stable 1.4

that the m a p p i n g b v a n i s h e s i d e n t i c a l l y ; w e have p r e v i o u s l y noted that this a l s o h o l d s in the r e m a i n i n g

case m = - 1 .

Since

we

also

74 see f r o m L e m m a

1 . 4 that t h e r e e x i s t s an

element

s u c h that

Set

this c h o i c e c l e a r l y s a t i s f i e s elements

and

hence,

and

(2.6). s u c h that

If £ l i e s in L ,

t h e r e e x i s t unique

75 Thus,

our c h o i c e of

is s e e n to s a t i s f y ( 2 . 7 ) .

c o m p l e t e s the c o n s t r u c t i o n of the s e q u e n c e previously observed,

this,

in turn,

By induction,

this

; as we have

s u f f i c e s to complete

the

construction

of T o p r o v e that

is u n i q u e ,

s a t i s f y i n g ( 2 . 5 ) ; then

Suppose that

l i e s in

the e l e m e n t

l i e s in have

By induction, l i e s in

Since

Thus

it f o l l o w s that, is equal to

proof.

Corollary 2.1. mental subalgebra

T h e n the i d e a l

(2.8)

and

and that

s u b a l g e b r a of D ( V ) , w e

w h i c h shows that

ending the

be a s e c o n d e l e m e n t of

l i e s in

is an i n t e g e r

L is a t r a n s i t i v e

f o r all

let

L e t L be a t r a n s i t i v e L i e a l g e b r a with f u n d a ,

Denote by

the L i e

algebra

76 i s of finite c o d i m e n s i o n in D e r Q ( L ) . s u c h that ( c f .

Lemma

L e t p b e the

smallest integer

> 0

1.5)

(2.9)

for

all

is a derivation of L

s u c h that

(2.10)

then

is

inner.

Proof; According

Let V be the f i n i t e - d i m e n s i o n a l v e c t o r

to

Theorem

a l g e b r a of D(V)

As

1.2,

i d e n t i f i e d w i t h the

of

Proposition

2.1,

the L i e

o f L in D ( V ) ;

under this identification.

it f o l l o w s f r o m

Lemma

sub-

may

that

the i d e a l

is of finite

(2.8)

be

To

p r o v e the

is a derivation of

L

Thus,

in

the i d e a l

second part of

satisfying

corre-

codimension

2 . 3 that the c o d i m e n s i o n o f L ^ is finite.

i s o f f i n i t e c o d i m e n s i o n in D e r assume

algebra

clearly,

Since

a f o r t i o r i of

corollary,

L as a transitive

subalgebra

o f the n o r m a l i z e r

in L ,

realize

s u c h that

a consequence

s p o n d s to

we may

space

(2.10).

(2.8) the

Under

77

D

the identifications established above, the derivation element

T)

of

o ND(V)(L).

corresponds to an

Using (2.10) and the fact that L is a transitive

subalgebra of D(V), it is easily seen that D is not inner, i. e., that

T)

lies in DP(V).

does not lie in L.

T)

Then,

Assume that

since L is closed

J. and the subspaces D (V) form a system of neighborhoods of 0 in D(V) , there exists an integer J. such that

r/J.

(2.11)

Let J. be the smallest such integer; clearly J.:::. P + 1, since Then there exists an element

®

Let a be the element of V

a

S of J.

L such that 17 -

S

T) E

DP(V).

lie s in DJ. -1 (V).

,>,

S (V'") defined by

(YJ -

S)

J. mod D (V).

We denote the filtration induced on L from ::.1.8 embedding in D(V) by

for j 2 -1.

From the relation

it follows easily that

JL Φ Using (2.9) and the fact that δ is injective on V 0 S (V ), we see that a lies in

i-1

.

Thus, there exists an element ζ of L

η - (ξ + ζ) = 0

i-1

such that

mod D i (V).

Since ξ + ζ lies in L, we have reached a contradiction to our hypothesis (2.11); hence D is inner. Lemma 2.4.

Let L be a non-abelian primitive Lie algebra, and

let L 0 be a primitive subalgebra of L.

Suppose that V is the finite-

dimensional vector space L/L°, and that (L,L°) is realized primitively as a transitive subalgebra of D(V).

Then the commutator of L in D(V)

is equal to {θ}. Proof: Assume that ζ is a non-zero element of the commutator of L in D(V).

Applying the uniqueness assertion of Proposition 2.1, with

D=O, we see that Ϊ, does not lie in D 0 (V).

Since L is a transitive

subalgebra of D(V), there exists an element ξ of L such that

ξ = ζ

mod D 0 (V) ;

the element ζ does not lie in D 0 (V), thus ξ is not an element of L 0 . Because ζ commutes with L, we see that

(2.12)

[ I, L 0 ] C L 0 ;

in particular, the direct sum

M = L 0 @ { € ξ I ceK}

79

is a subalgebra of L which p r o p e r l y contains primitive

subalgebra of L ,

f o l l o w s f r o m ( 2 . 1 2 ) that is fundamental,

Since

is a

w e i n f e r that M i s e q u a l to L . i s an i d e a l o f L ; t h u s ,

it is e q u a l to

algebra L is o n e - d i m e n s i o n a l ,

It now

s i n c e the

We c o n c l u d e that the

subalgebra Lie

h e n c e a b e l i a n ; this c o n t r a d i c t i o n ends the

proof.

Theorem 2.1.

L e t L be a s i m p l e t r a n s i t i v e

L i e a l g e b r a of o n e o f

the t y p e s

and d e n o t e b y

the unique p r i m i t i v e

finite-dimensional vector

space

unique e l e m e n t

L e t V be the

and s u p p o s e that

r e a l i z e d p r i m i t i v e l y as a transitive

i s a d e r i v a t i o n o f L into D ( V ) ,

s u b a l g e b r a of L .

then

subalgebra of D(V).

is c o n t i n u o u s ,

is If

and t h e r e e x i s t s a

o f D ( V ) s u c h that

(2.13)

In p a r t i c u l a r ,

the f i r s t H o c h s c h i l d - S e r r e

cohomology group

vanishes. Proof: by

We w r i t e

f o r the f i l t r a t i o n ( 1 . 4 6 ) o f L ,

and denote

80

for p

€

II': ,

the fintration induced by the fundamental subal$ebra of L.

Recall that,

by (1.47) and (1.48), we have

for all p

€

11':, and

[L ,L ] P q for all p,q,:: 1.

L

p+q

Let D be a derivation of L into D(V}; then, for all p::t 1,

D(L

Zp

)

=D([L ,L]} p P C

[D(L ),L ] P

P

Thus. fOJ:" all p .:: 1 ,

D(L 4p+6) c

[D(V), L

+ ] zp 3

c [D(V),LP+1]

since each of the filtrations {L} 'Wand {LP} "H is a fundamental p p€ "'" p€ "'" system of neighborhoods of 0 in L, we conclude that D is continuous. We

81 s h a l l p r o v e that t h e r e e x i s t s an e l e m e n t

of D ( V ) s u c h that

(2.14)

It w i l l then f o l l o w f r o m P r o p o s i t i o n 2 . 1 that t h e r e e x i s t s element

of

a unique

s u c h that

w h i c h w i l l e s t a b l i s h the e x i s t e n c e of an e l e m e n t £, s a t i s f y i n g R e c a l l f r o m ( 1 . 4 5 ) that L m a y be w r i t t e n as the p r o d u c t of dimensional discrete

(2.13). finite-

spaces

(2.15)

s u c h that,

f o r all integers

the f u n d a m e n t a l s u b a l g e b r a

i s the

space

(2.16)

The graded components

vanish f o r

the c o n t a c t a l g e b r a c t ( 2 n + l , K ) ,

the s p a c e

A s s u m e that L is e q u a l to Section one,

i.e.,

s u c h that

L is e q u a l to

is a l s o e q u a l to

o r c t ( 2 n + l , K ) ; then,

t h e r e e x i s t s an e l e m e n t I o f

degree derivation,

and u n l e s s

as w e n o t e d in

w h i c h a c t s o n L as the

82

f o r all

and

i s g i v e n by ( 2 . 1 6 ) , Thus,

therefore,

Since L is a transitive the s p a c e

subalgebra of D(V)

forms

a complement

there exist elements

setting

S i n c e I and a b o t h l i e in

s u c h that

we have

S u p p o s e that p i s an i n t e g e r there exist elements

to

and that a l i e s in and

a f o r t i o r i in

As

s u c h that

, we

have

above,

and

83

S i n c e K is o f c h a r a c t e r i s t i c that b o t h

z e r o and

vanish.

we

It f o l l o w s that the

conclude

equation

(2.17)

h o l d s f o r a l l r\ l y i n g o n the d e n s e

subspace

and a d ( £ ) a r e c o n t i n u o u s m a p p i n g s , This establishes

s i n c e both

( 2 . 1 7 ) m u s t t h e r e f o r e hold f o r

( 2 . 1 4 ) in the c a s e s

N o w a s s u m e that L is e q u a l to graded components

considered. or

.

T h e n the

a p p e a r i n g in ( 2 . 1 5 ) m a y be i d e n t i f i e d with the

spaces

sub-

i n d u c e d by the f i l t r a t i o n

and w e h a v e an

isomorphism

o f g r a d e d L i e a l g e b r a s ; the s u b a l g e b r a algebras

or

sp(2n,K).

a c o m p l e m e n t to

by setting,

Since

all

We

Since

is o n e of the s i m p l e vanishes,

define a linear

Lie

the s p a c e

forms

mapping

f o r all

i s c o n t a i n e d in

. f o r t i o r i in

-we h a v e ,

for

all

84

p([a,b])

m.od DO(V)

=:

D([a,b])

=:

[D(a),b]

+

[a,n(b)]

[p(a),b]

+

[a,p(b)]

Since g-1 is a finite-dim.ensional

hence p is a derivation of gO into g-1

m.odule over the sim.ple Lie algebra gO, the clas sical first lem.m.a of

J. H. C. Whitehead ([ 18]) im.plies that there exists an elem.ent such that, for all a

€

S

of g-1

°

g ,

pta)

[t;,a] ,

that is,

(n - ad(t;»)(a)

Suppose that p is an integer'::: 1.

by setting, for all b

€

°

°

m.od D (V) •

As above, we define a linear m.apping

gP,

>-.(b)

(0 -

ad(s»(b)

°

m.od D (V) .

From. (1.38), we see that the strict inequ'l.lity

85 holds; therefore, and

thus,

the m a p p i n g

a r e c o n t a i n e d in

the k e r n e l N o f

immediately before

has a n o n - t r i v i a l k e r n e l N . I, w e h a v e ,

is a

(1.38),

continuity, and,

l y i n g in the d e n s e

f o r all

satisfying (2.13).

As we

is i r r e d u c i b l e ;

noted since

Thus,

It again f o l l o w s that the e q u a t i o n subspace

This establishes

a s w e have n o t e d ,

on

, it is t h e r e f o r e equal to

vanishes identically.

holds f o r all

and b e N,

s u b s p a c e of

the a c t i o n of

the s p a c e N i s not e q u a l to mapping

invariant

f o r all a

Since

the

(2.17)

and h e n c e , in the r e m a i n i n g

by cases,

thus p r o v e s the e x i s t e n c e o f an e l e m e n t

The u n i q u e n e s s o f t, f o l l o w s i m m e d i a t e l y f r o m

Lemma

2.4. T h e f o l l o w i n g r e s u l t w a s f i r s t o b t a i n e d by F r e i f e l d Corollary

2.2.

the f l a t t y p e s

o r the c o n t a c t

algebra

L e t L be a s i m p l e t r a n s i t i v e L i e a l g e b r a o f one

of

86 and denote by L

0

the unique primitive subalgebra of L.

finite-dimensional vector space L/L O•

Let V be the

Suppose that (L, L 0) is given a

primitive realization as a transitive subalgebra of D(V); if (L, L 0) is flat, assume that this realization is standard.

Then the following

relations hold between L and the normalizer ND(V)(L) of L in D(V):

ND(V)(gi oo (n, K» ND(V)(sioo(n,K»

= sioo(n,K) CD{cI!

= ct(2n+ 1,K),

N D (V)(ct(2n+ 1,K»

where I denotes the identity transformation

There exists an isomorphisITl of Lie algebras

cp : Der(L)

such that, for all

D

ND(V) (L) ,

€ Der(L),

(2.18)

and, for all s€ ND(V)(L) ,

(2.19)

----l>

ad(cp(I5»

IL

'

c€K},

87 in particular, the image under «J of the inner derivations is L.

The

isomorphism «J endows Der{L) with the structure of a transitive Lie algebra in which the inner derivations form a closed ideal of codimension at most one.

Upon identifying L with the closed ideal of inner

derivations in Der{L) , there exists a unique fundamental subalgebra DerO{L) of Der{L) such that

Der{L)

°

L + Der (L) ,

and there exists a closed abelian subalgebra of DerO{L) which forms a complement to L in Der(L).

Proof: We first verify the assertions relating ND(V){L) to L.

If

L is equal to g.lco(n,K), there is nothing to prove, since g.lco{n,K) is equal to D{V).

Suppose that L is equal to s£oo{n,K) or spco{2n,K); then

L is realized as a flat subalgebra of D(V).

Let t;, lie in ND(V)(L).

may write t;, uniquely in the form 00

6

s£

.l +1 ,~ withs.eEV@S (V).

P. =-1

Since V is contained in L, we have, for all v E V

and P. .::: -1,

(2.20)

The Spencer cohomology groups II,1(L,LO) vanish for cases; thus, we infer from (2.20) that

£.2: 1, in these

We

88

for all

The l i e s in L .

element

It f o l l o w s

a l s o l i e s in L, that

and s o w e

lies in

see

that

; in particular,

we

have

that i s ,

is a derivation

f i n i t e - d i m e n s i o n a l and s i m p l e , that t h e r e

The

exists

of

it follows f r o m a c l a s s i c a l

a unique e l e m e n t a of

element

k n o w n that,

S i n c e the L i e a l g e b r a

the r e p r e s e n t a t i o n s

exists a scalar

such

result

s u c h that

then c o m m u t e s for

is

involved,

with

it i s

this i m p l i e s

that

wellthere

that

W e c o n c l u d e that

the o p p o s i t e i n c l u s i o n i s o b v i o u s , spaces structure

s i n c e the a c t i o n of I on e a c h of

i s m u l t i p l i c a t i o n b y the s c a l a r of

is as a s s e r t e d .

to the c o n t a c t a l g e b r a exists a direct

sum

decomposition

Finally,

From

assume

Lemma

1.7,

Thus

the

that L i s we

the

equal

s e e that

there

89

of D ( V ) i n t o L - i n v a r i a n t s u b s p a c e s , the c o m m u t a t o r

of L in D ( V ) i s equal to

that the n o r m a l i z e r We d e f i n e the

a s f o l l o w s : if

and f r o m L e m m a

2 . 4 we s e e

It f o l l o w s

i s equal to L ,

as

that

immediately

specified.

isomorphism

i s an e l e m e n t of D e r ( L ) ,

t h e r e e x i s t s a unique e l e m e n t

then a c c o r d i n g

to T h e o r e m

2. 1

s u c h that

We set

i t i s i m m e d i a t e l y v e r i f i e d that this d e f i n e s an i n j e c t i v e h o m o m o r p h i s m Lie algebras, ,

and s a t i s f i e s

(2.18).

then the m a p p i n g

n e s s a s s e r t i o n of T h e o r e m 2 . 1 ,

thus,

( 2 . 1 9 ) is satisfied,

derivations

onto L .

and

If

i s an a r b i t r a r y e l e m e n t

l i e s in D e r ( L ) , we must

and,

b y the

of

of unique-

have

i s an i s o m o r p h i s m w h i c h m a p s the i n n e r

T h e f i n a l a s s e r t i o n s f o l l o w i m m e d i a t e l y f r o m the

r e l a t i o n s w h i c h w e have p r e v i o u s l y d e t e r m i n e d b e t w e e n

and

L.

90 Lemma

2.5.

e x t e n s i o n of K .

Let

be a field which is a finite

algebraic

S u p p o s e that L i s a s i m p l e t r a n s i t i v e L i e a l g e b r a

of

o n e of the types

c o n s i d e r e d a s