Non-Abelian Minimal Closed Ideals of Transitive Lie Algebras. (MN-25)

by Jack Frederick Conn

Overview

The purpose of this book is to provide a self-contained account, accessible to the non-specialist, of algebra necessary for the solution of the integrability problem for transitive pseudogroup structures.

Originally published in 1981.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Product Details

ISBN-13:	9781400853656
Publisher:	Princeton University Press
Publication date:	07/14/2014
Series:	Mathematical Notes , #25
Sold by:	Barnes & Noble
Format:	eBook
Pages:	228
File size:	5 MB

Read an Excerpt

Non-Abelian Minimal Closed Ideals of Transitive Lie Algebras

---

By Jack F. Conn

PRINCETON UNIVERSITY PRESS

Copyright © 1981 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08251-6

---

CHAPTER 1

§1. Preliminaries

Throughout this section, we denote by K an arbitrary field of characteristic zero which is endowed with the discrete topology.

Let V be a Hausdorff topological vector space over K. Then V is said to be linearly compact if:

(i) V is complete; and

(ii) There exists a fundamental system {Vα} of neighborhoods of Q in V such that each Vα is a vector subspace of finite codimension in V.

An example of such a space is provided by the local algebra

F = K[[x1, ..., xn]]

of formal power series over K in n indeterminates x1, ..., xn. Denote by F0 the maximal ideal of F, which consists of those formal power series whose constant term vanishes. If Fl denotes the (l + 1) - rst power of F0, then the ideals {Fl}l ≥ 0 form a fundamental system of neighborhoods of 0 for the Krull topology on F; endowed with this topology, F becomes a linearly compact topological vector space over K. Furthermore, in this topology, multiplication in the algebra F is a continuous mapping F X F [right arrow] F; thus we see that F is a linearly compact topological algebra over K. If V is a finite-dimensional vector space over K, then V is linearly compact in the discrete topology. Moreover, since a finite-dimensional Hausdorff vector space satisfying (ii) above is necessarily discrete, the discrete topology provides the only linearly compact structure on a finite-dimensional space. In the sequel, a finite-dimensional vector space will often be implicitly endowed with the discrete topology.

The properties of linearly compact spaces which we shall require are subsumed in the following proposition.

Proposition 1.1. Let V and W be linearly compact topological vector spaces over K.

(i) If V1 is a closed subspace of V, then both V' and V/V' are linearly compact.

(ii) A subspace V' of V is open if and only if V1 is closed and of finite codimension in V.

(iii) If V' and V" are closed subspaces of V, then the sum V' + V" is closed in V.

(iv) The topological direct sum V [direct sum] W is linearly compact.

(v) If φ V [right arrow] W is a continuous linear mapping, then the image φ(V) is a closed subspace of W, and φ is an open mapping of V onto φ(V). In particular, any continuous linear bijection φ: V [right arrow] W is a topological isomorphism.

(vi) (Closed graph theorem). A linear mapping φ: V [right arrow] W is continuous if and only if the graph of f is closed in V X W.

(vii) If {Vα} is a family of linearly compact spaces, then the product [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is linearly compact. If {(Vα, fαβ an inverse system of linearly compact vector spaces and continuous linear mappings, then the projective limit [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is linearly compact.

(viii) (Chevalley's theorem). Let U be an open subspace of V, and let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

be a descending chain of closed subspaces of V such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, there exists an integer l0 ≥ 1 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In particular, if the spaces Vl are open, for l ≥ 1, then {Vl}l ≥ 1 forms a fundamental system of neighborhoods of 0 in V.

The proofs of these statements may be found in Section one of the paper ([11]) of Guillemin, and in Chapter two of the book ([19]) of Kothe.

Let V and W be linearly compact vector spaces over K, and suppose that {Vα} (resp. {Wβ}) is a set of open subspaces which forms a fundamental system of neighborhoods of 0 in V (resp. W). We define a structure of topological vector space on the tensor product V [cross product]K W by taking as a fundamental system of neighborhoods of 0 the subspaces

Vα [cross product]K W + V [cross product]K Wβ

of V [cross product]K W, where Vα (resp. Wβ) ranges over all elements of {Vα} (resp. {Wβ}); the topology thus defined depends only on the topologies of V and W. The space V [cross product]K W is not, in general, complete. The Hausdorff completion of V [cross product]K W is a linearly compact topological vector space which we denote by V [??]K W. and call the completed tensor product of V and W. If at least one of the spaces V and W is finite-dimensional, then V [??]K W coincides with V [cross product]K W. The completed tensor product V [??]K W may be canonically identified with the projective limit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of finite-dimensional discrete spaces, and with either of the projective limits

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of linearly compact spaces. Let K' be a field which is a finite extension of K; endow K' with the discrete topology. Then, from what we have said above, we see that the tensor product K' [cross product]K V has a natural structure of linearly compact vector space over K'. A different construction of the completed tensor product, based on Grothendieck's construction of the Hausdorff completion, is given in ([11]); the exercises to Chapter III of the book ([1]) of Bourbaki also treat this material.

Let V be a finite-dimensional vector space over K; for l ≥ 0, we denote by Sl (V) the 1 - th symmetric tensor power of V. The symmetric algebra of V*,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a graded unitary algebra, whose completion

(1.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a linearly compact local algebra over K. We shall call F{V*} the algebra J1 formal power series on the vector space V. We write F0{V*} for the maximal ideal of F{V*}; then, for m ≥ 0, the powers

Fm{V*} = (F0{V*})m+1

of F0{V*} form a fundamental system of neighborhoods of 0 in F{V*}. In terms of the isomorphism (1. 2), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all m ≥ 0; for convenience, we set

Fm{V*} = F{V*}, for m < 0.

Suppose that dim(V) = n, and let f1, ..., fn be elements of F0{V*} whose images form a basis for the vector space

F0{V*}/F1{V*} [equivalent] V*/

Then, there exists a unique isomorphism of linearly compact algebras

φ: K[[x1, ..., xn]] [right arrow] F{V*}

such that

φ(xl) = fl, for ≤ l ≤ n.

The elements f1, ..., fn will be called a formal coordinate system for V.

This paper is primarily concerned with the category of linearly compact Lie algebras and continuous homomorphisms, that is, topological Lie algebras whose underlying topological vector space is linearly compact. An example of such a Lie algebra is provided by any finite-dimensional Lie algebra, endowed with the discrete topology. We shall present examples in which the topology plays a more essential role below.

If A is an algebra over K, we denote by Der(A) the Lie algebra of derivations of A, that is, linear mappings D : A [right arrow] A such that, for all a, b [member of] A,

D(a x b) = D(a) x b + a x D(b).

The Lie bracket in Der(A) is defined, for all D1, D2 [member of] Der(A), by

[D1, D2] = D1 ο D2 - D2 ο D1.

As before, we denote by F the linearly compact local algebra K[[x1, ..., xn]], endowed with the Krull topology. The Lie algebra Der (F), with the topology defined below, furnishes another example of a linearly compact Lie algebra; this example will play a fundamental role in the sequel. Let p be an integer ≥ 0; then, from the definition of the ideals of {Fl}l ≥ 0 of F, we see that

F2p+1 = Fp x Fp.

Hence, if [xi] is a derivation of F, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since p ≥ 0 was arbitrarily chosen, it follows that [xi] is continuous. Thus, we see that each element of Der(F) is a continuous linear mapping of F. From this, and the fact that any derivation of F must annihilate the scalars K [subset] F, it follows easily that Der(F) may be identified with the Lie algebra of formal vector fields, that is, expressions [xi] of the form

(1.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [xi]l [member of] F, for 1 ≤ l ≤ n. The action of such a formal vector field [xi] on F is given, for all f [member of] F, by

(1.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

to simplify expressions, we shall sometimes write as [xi](f) as [xi] x f. The Lie bracket in Der(F), under this identification, is expressed as

(1.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all [xi], η [member of] Der(F). From (1.3) we see that we may endow Der(F) with a structure of linearly compact vector space by identifying Der(F) with the Cartesian product F x ... x F of n copies of F. We define a descending sequence {Derp(F)}p ≥ 0 of subalgebras of Der(F) by setting

Derp(F) = {[xi] [member of] Der(F) | [xi](F) [subset] Fp},

for all p ≥ 0. From (1.4), we see that a formal vector field [xi] lies in Derp(F) if and only if each of the coefficients [xi]l lies in Fp, for 1 ≤ l ≤ n. It follows at once that the subalgebras Derp(F) are open in Der(F), and that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

thus, by Chevalley's theorem, the sequence {Derp(F)}p ≤ 0 forms a fundamental system of neighborhoods of 0 in Der(F). For convenience, we set Derp(F) equal to Der(F), for p < 0. Then it is easily seen from (1.5) that

[Derp(F). Derq(F)] [subset] Derp+q(F),

for all p, q [member of] Z . As an immediate consequence, we see that the Lie bracket is a continuous mapping

Der(F) x Der(F) [right arrow] Der(F);

thus, Der(L) is a linearly compact Lie algebra. We also note that, by (1.4), the action of Der(F) on F gives rise to a structure of linearly compact topological Der(F)-module on F. In fact, if we set Fp = F, for p < 0, then

Derp(F) x Fq [subset] Fp+q,

for all p, q [member of] Z.

Let L be a Lie algebra over K, and let S be a subspace of L. The derived subspace DL(S) is defined to be

DL(S) = {[xi] [member of] S | [xi], L] [subset] s}.

Using this, we define inductively a descending sequence subspaces of S, by

(1.6) D0(S) = S, Dp+1L(S) = DL(DpL(S));

for convenience, we set

DpL(S) = L, for p < 0.

We record the following properties of the derived subspace:

Proposition 1.2. Let L be a Lie algebra, and let S be a subspace of L.

(i) The derived subspace DL(S) is a subalgebra of L contained in S. Moreover, if S is a subalgebra of L, then DL (S) is an ideal of S.

(ii) If I is an ideal of L contained in S, then I [subset] DL (S).

Assume, in addition, that the Lie algebra L is linearly compact; then,

(iii) If S is a closed (resp. open) subspace of L, then DL(S) is a closed (resp. open) subalgebra of L contained in S.

Proof: We leave to the reader the straightforward verification of (i) and (ii). To prove (iii), we note that, because the Lie bracket in L is continuous, it is immediate that if S is closed, then DL(S) is closed in L. Assume that S is open. From what we have just shown, we see that to prove that is open, we need only demonstrate that it is of finite codimension in L. Because S is open, it follows from the continuity of the Lie bracket that there exists an open subspace S' of L such that

S' [subset] S, [S', S'] [subset] S.

From the definition of DL(S), one checks easily that DL(S) contains the kernel of the natural mapping

ψ: S' [right arrow] HomK (L/S', L/S)

induced by the Lie bracket. Because S' and S are open in L, the spaces L/S' and L/S are finite-dimensional. Thus, the kernel ker(ψ) is of finite codimension in S'. Since S' is open in L, we see that ker(ψ) is of finite codimension in L; because DL(S) contains ker(dj), it follows that the co-dimension of DL(S) in L is also finite.

As an immediate consequence of parts (ii) and (iii) of the above proposition and the inductive definition of the derived subspaces {DpL(S)}p ≥ 0, we have:

Corollary 1.1. Let L be a Lie algebra over K, and let S be a subspace of L. Then, the intersection

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is an ideal of L contained in S, and contains all ideals of L contained in S. If L is a linearly compact Lie algebra, and the subspace S is closed in L, then D∞L(S) is in addition, closed in L.

Let L be a linearly compact Lie algebra over K. An open subalgebra L0 of L is said to be fundamental if the only ideal of L contained in L0 is {0}; from Corollary 1. 1, we see that a proper open subalgebra L0 of L is fundamental if and only if

D∞L(L0) = {0}.

A linearly compact Lie algebra L is called transitive if there exists a fundamental subalgebra L0 of L; when we wish to indicate a choice of fundamental subalgebra, we shall sometimes say that (L, L0) is a transitive Lie algebra. From (iii) of Proposition 1.2, we see that a linearly compact Lie algebra L is transitive if and only if there exists a neighborhood O of 0 which contains no ideals of L except {0}, for, if S is an open subspace of L contained in O, then DL(S) is a fundamental sub-algebra of L; the converse is obvious.

Lemma 1. 1. If (L, L0) is a transitive Lie algebra, then the sequence {DpL(L0)}p ≥ 1 forms a fundamental system of neighborhoods of 0 in L.

Proof: According to Proposition 1. 2, (iii), the spaces DpL(L0), for p ≥ 1, form a descending chain of open subalgebras of L, and, by Corollary 1.1 we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, we may apply Chevalley's theorem ((viii) of Proposition 1.1) to conclude that {DpL(L0}[greater than or equal to] 1 is a fundamental system of neighborhoods of 0.

We note that any finite-dimensional, discrete Lie algebra L over K is transitive, with L0 = {0} a fundamental subalgebra of L. If we again denote by F the algebra K[[x1, ..., xn]] of formal power series, then the Lie algebra L = Der(F) which we studied earlier provides another example of a transitive Lie algebra over K. If L0 is the proper open subalgebra Der0(F) of L, then one verifies easily, from the formula (1.4) for the Lie bracket, that

(1.7) DpL(L0) = Derp(F),

for all p ≥ 1; thus, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which shows that L0 is fundamental.

The following alternative characterization of transitive Lie algebras is proved, as Theorem 3.1, in the paper ([11]) of Guillemin (see also Theorem 14.1 of ([10])).

Theorem 1.1. Let L be a linearly compact Lie algebra over K. Then, the following are equivalent:

(i) L is transitive;

(ii) L satisfies the descending chain condition on closed ideals.

Corollary 1.2. Let L be a transitive Lie algebra, and let I be a closed ideal of L. Then, the topological Lie algebra L/I is transitive.

Proposition 1.3. Let L be a Lie algebra over K, and let M be a subalgebra of L. Define a filtration {Lp}p [member of] Z of L by subalgebras, with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, the filtration {Lp}p [member of] Z {endows L with a structure of filtered Lie algebra, that is, if p ≤ q, then

(1.8) LP [contains] Lq,

and, for all p, q [member of] Z,

(1.9) [Lp, Lq] [subset] Lp+q.

Furthermore, if L is a linearly compact Lie algebra, and M is an open (resp. closed) subalgebra of L, then each of the subalgebras Lp is open (resp. closed) in L, for all p [member of] Z.

Proof: The first inclusion (1.8) is obvious. We shall prove (1.9) by induction on p + q; the case p + q < 0 is trivial, since then Lp+q = L. From the definition of the filtration {Lr}r [member of] Z we see that, for all

(1. 10) [L, Lr] [subset] Lr-1.

(Continues...)

---

Excerpted from Non-Abelian Minimal Closed Ideals of Transitive Lie Algebras by Jack F. Conn. Copyright © 1981 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

---

Table of Contents

• FrontMatter, pg. i
• Table of Contents, pg. iii
• Preface, pg. v
• Introduction, pg. 1
• §1. Preliminaries, pg. 15
• §2. Derivations of Transitive and Simple Lie Algebras, pg. 65
• §3. Simple Algebras with Parameters, pg. 95
• §4. Closed Ideals of Transitive Lie Algebras, pg. 110
• § 5. Minimal Closed Ideals of Complex Type, pg. 152
• References, pg. 217