ANSELM EGGERT

EXTENDING

THE

CAMPBELL-HAUSDORFF

MULTIPLICATION Dedicated to Karl Heinrich Hofmann on the occasion of his 60th birthday

ABSTRACT. Let i be an ideal of the finite-dimensional Lie algebra g such that 2hi is not an eigenvalue of ad x : 9 ~ g for every x E J- It is shown that there exists an open ball V around 0 and an analytic extension of the Campbell-Hausdorff multiplication o :(i + V) x (i + V) ~ g. This generalizes an old result by Dixmier significantly, and gives a new and easier proof.

1.

INTRODUCTION

Locally, the exponential function of a Lie group G is a diffeomorphism. Thus the g r o u p multiplication • : G × G ~ G gives rise to a local group multiplication •: U x U ~ g where U denotes a sufficiently small n e i g h b o r h o o d of zero

in the Lie algebra 9 of G. It is possible to describe this local multiplication * in terms of the Lie algebra structure, i.e. the Lie bracket: There is an universal power series in two n o n c o m m u t i n g variables X and Y, called the B a k e r - D y n k i n - C a m p b e l l H a u s d o r f f series, or CH-series for short, which is built from m o n o m i a l s in Lie brackets of X and Y: X * Y = X + Y + ½[X, Y] + ~-½(~[[X, Y], Y33 + E[ r, X], X]) + . . . N o w let g denote the Lie algebra of some finite dimensional Lie group G and assume g to be provided with a n o r m II-Jl satisfying [][x, Y] II 4 Ilxll ' IIyll for all x, y ~ g. Let U = {x ~ g : I[x If ~ ½ log 2}. Then for x, y s U, the power series x * y converges and we have e x p ( x , y) = (exp x)(exp y). An open ball B a r o u n d 0 such that the CH-series converges for x, y s B will be called a CH-neighborhood. The (partial) binary operation , : B x B--* g obtained by evaluation is called CH-multiplication. The following t h e o r e m summarizes some standard stuff. (Proofs m a y be found in most books on Lie theory, e.g. B o u r b a k i [23 or Hochschild [5].) 1.1. T H E O R E M . Let B be a CH-neighborhood in g. Then we have (i) (ii) (iii) (iv)

The CH-multiplication is analytic. With this multiplication, B becomes a local group. There is a CH-neighborhood C of g such that C * C ~_ B. I f 9 is nilpotent, then g itself is a CH-neighborhood.

Geometriae Dedieata 46: 35-45, 1993. © 1993 Kluwer Academic Publishers.

Printed in the Netherlands.

36

ANSELM EGGERT

In spite of case (iv), usually the CH-series x * y converges only for x and y sufficiently small. However, often one wants to multiply larger elements of g (cf. e.g. [4, Th. 4.28]). And indeed, in suitable circumstances, it is possible to find an analytic extension of the CH-multiplication onto much larger domains. The aim of this paper is to prove the existence of such an extension (Theorem 4.3).

2.

A SHORT COLLECTION

OF S T A N D A R D

MATERIAL

The basic idea of what follows involves the description of the CH-multiplication in terms of a differential equation whose solutions are used to define the required analytic extension. Before we can do this, we need some more material (cf. [2, Chs II and III] for example). 2.1. D E F I N I T I O N . Let f and g denote the power series

(-1)" X. f ( X ) = ,,=o (n + 1)~

and g(x) = 1

1

b2n

+

x

2n

with the Bernoulli numbers b2,. The power series f converges everywhere, whereas g has 2re as radius of convergence. If A is a Banach algebra, then f and g define analytic functions on their respective domains of convergence. These will be denoted again by f and g. 2.2. REMARK. In the case that both sides are defined, for an element T of a Banach algebra, the following equations hold: 1 -e -r f(T)=(1-e

T)T ~--

T

and T g(T) = f ( T ) -1 - 1 - e -T"

2.3. P R O P O S I T I O N . Let B be a CH-neighborhood in g and x, y ~ B. Let ~ > 0 be small enough to ensure that (1 + e)y ~ B. Then the curve 7 : ] - e , 1 +8[ ~ g, t ~ x * ty is a solution o f the initial value problem

(*)

?'(t) = g(ad ?(t))(y),

7(0) = x,

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MULTIPLICATION

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and the power series g(ad y(t)) converges for all t ~ ] - e , 1 + el. (Defining y on an interval larger than [0, 1] is just a technical refinement, for we want the domain of y to be open.) N o w assume we already knew an extension o of the CH-multiplication. If for x, y • g the element x o y is defined, then we m a y describe the curve 7 : ] - e, 1 + e[--* g, t ~ x o ty by the initial value p r o b l e m (*). In the following we will take the inverse way: Given x, y • g we solve the initial value p r o b l e m and we define x o y = 7(1). There is one delicate point, namely, it is a priori not at all clear whether such a solution exists all the way to t = 1. Most of this article is devoted to overcoming this obstacle. 2.4. D E F I N I T I O N . A Lie algebra g is called exponential, if there is an analytic extension of its CH-multiplication to all of g × g. 2.5. D E F I N I T I O N . We say that an e n d o m o r p h i s m T e gl(g) is exp-regutar, if no eigenvalue of T is contained in 2TciZ\{0}. An element x • g is said to be exp-regular, if ad x is exp-regular. (Of course, the term 'exp-regular' results from the fact that an element x of a Lie algebra g is exp-regular if and only if the exponential function exp : g ~ G is regular at x, but we shall not m a k e use of this fact. F o r detailed information on regular and exp-regular points, the reader is referred to [6].) 2.6. P R O P O S I T I O N . Let g be a finite-dimensional vector space. The set M of all exp-regular elements T e gl(g) is open and dense in gl(g) and the function g of Definition 2.1 may be analytically extended to M. Furthermore, g(T) is invertible for every T e M. Proof The eigenvalues of T depend continuously on T, hence M is open. If S e gl(g) is chosen arbitrarily, there is a sequence ( r , ) , ~ in ~ converging to 1, such that r,S is exp-regular. Hence M is dense, too. We claim that for T • M the o p e r a t o r f ( T ) is invertible: Just write down the complex J o r d a n n o r m a l form of T and use that for z ¢ 2 ~ i Z \ { O } we have f (z) ~ O. N o w define g( T) = f ( T ) - a. [] (Of course, we will denote this extension again by g.) 2.7. C O R O L L A R Y . Let g be a Lie algebra. Then the function x ~-~ g(ad x) is defined on the open and dense set {x ~ g : x is exp-regular} and it is analytic.

3.

ABELIAN

EXTENSIONS

N o w we turn to the p r o b l e m of solving our initial value p r o b l e m (.) up to t = 1. We start with a short excursion.

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ANSELM EGGERT

3.1. D E F I N I T I O N . A short exact sequence of Lie algebras 0-oi~gA81

~0

is called an abelian extension, if i is abelian. 3.2. R E M A R K AND D E F I N I T I O N . In the following we may and shall identify i with its image i(i) ~_ g and 8a with 8/L Then i is an abelian ideal in 8 and the ideal i becomes a 81-module via

,p(x + i)(a) := Ix, aS. 3.3. LEMMA. Let i be a solvable ideal in g. Then g may be obtained from g/i by a finite sequence of abelian extensions. I f it") denotes the nth element of the commutator series of i, then these extensions are given by

with the obvious canonical homomorphisms. (By hypothesis, we have j~N)= {0} for N sufficiently large, hence g/j tin _ g.) Together with the preceding lemma, the following proposition will yield the induction argument for the proof of the main Theorem 4.3. 3.4. P R O P O S I T I O N . Let

0~iAsAg

~~0

be an abelian extension of 81 by i. Moreover, let C x ~_ gl be a neighborhood of 0 such that both the elements of ad C1 - g l ( g 0 and those of ¢p(C1)~ gl(i) (recall the definition of ~o from Definition 3.2) are exp-regular. Let Ba be a neighborhood of 0 in gl and let * : B1 × Ba ~ 81 be an analytic extension of the CH-multiplication of gl. Finally, assume B1 * Ba ~- C1. Then there is an analytic extension o of the CH-multiplication of g onto B := p-I(B1). Furthermore, we have B o B ~_ p-1(C1). For the proof of this proposition, we will need a variety of preliminaries and we will decompose it into a sequence of lemmas. The notation from the proposition will be kept on. First, we choose a vector space complement ~ for i _~ 8. Now we want to decompose a linear mapping T : 8 - ~ 8 with respect to the direct decomposition 8 = i O ~. After having chosen bases in i and ~, respectively, one may also think in terms of block matrices. 3.5. D E F I N I T I O N . Let P~ : 8 -o ~ and Pi: 8 ~ i denote the projections of 8

CAMPBELL--

HAUSDORFF

MULTIPLICATION

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onto t~ and i, respectively, along the other summand. For a linear operator T: g ~ g, we define the following mappings: (i) (ii) (iii) (iv)

T~t=P~or]f:[~[ Ttl = Pt ° Tli: i --, t T~r = P~o Y[t:t --, i Tii = Pi o Tli: i ~ i

Furthermore, for an element x of g, we set xt = P~(x) and xi = Pi(x). In the following, we shall identify T and x with

T=(

Ttt \Tit

Tti) ~i

and

(xt~, X = \ xi/

respectively. N o w calculations are done exactly as in the case of block matrices. 3.6. L E M M A . Let T a gl(g) be an operator that leaves i invariant (e.g. ad x for some x e g). Then (i) T~i = O. (ii) T induces canonically an operator Ti: g/i ~ g/i satisfying Ti = (pID o Tt~o(p I D -1.

In other words: I f we identify t and gl = g/i via p] t, then Ttt corresponds to the induced mapping Ti. (Note that p i t is an isomorphism of vector spaces.) (iii) For x ~ g and a e i we have (ad(x + a))tt = (ad x)tt = (ad xr)tt

and (ad(x + a))ii = (ad x)il = (ad x[)ii ,

i.e. (ad x)~t and (ad x)ii depend only on x~. (iv) Each x e p - I ( C 0 is exp-regular. In particular, g(ad x) is defined. Proof. (i) and (ii) are immediate, and (iii) follows easily from the fact that i is an abelian ideal of g. (iv) F r o m x e p ~(C~) we conclude p(x)eC1. Thus by hypothesis, both ad p(x)egl(gl) as well as qo(p(x))~gl(i) are exp-regular. But these two linear mappings are equal to (ad x)t~ (if we identify t and gl according to (ii)), and to (ad x)ii, respectively. Moreover, the eigenvalues of ad x are just the eigenvalues of (ad x)t~ and of (ad x)ii, for we may consider ad x as a lower block triangular matrix. []

40

ANSELM EGGERT

Now we have established the language we need to formulate the following lemma. Although easy to prove, it will turn out to be the essential insight in the course of our argument. 3.7. LEMMA. Let A ~ gl([) and B ~ gl(i) be 9iven and assume that A and B are exp-regular. We consider the mapping

( o))

• : Lin(l, i) ~ Lin(~, i), X ~-~ g

B

ii

Then @ is well defined and linear. Proof. Since A and B are exp-regular, • is well defined. For A, B and X sufficiently small, the power series of 9 around 0 converges. We will denote its coefficients by a o, a l , . . . . A short calculation yields:

Hence the assertion is true at least for A, B and X sufficiently small. Finally we observe that the set of all T e gl(g), that leave i invariant and that are exp-regular is connected. (Just consider the real Jordan normal form.) Now the assertion follows from the uniqueness theorem for analytic functions. [] 3.8. LEMMA. For any k~[, the following mappings are affine: (i) a ~ (ad(k + a))i~: i --+ Lin([, i) (ii) a ~ 0(ad(k + a))i~: i --+ Lin(~, i) Proof. (i) This is clear, since ad is linear. (ii) From Lemma 3.6(iii) we conclude that (ad(k+a))rt =(adk)tr and (ad(k + a))a = (ad k)il do not depend on a. Now the assertion follows from (i) and Lemma 3.7. []

At this point we have collected all the material we need to prove Proposition 3.4. Let x, y e B be arbitrary but fixed. We formulate the initial value problem (*) in detail now. Let yt and Yl denote the projections of ~ under Pt and Pi, respectively. (See Figure 1.) Then (*) reads (y~(t)) ( ( a d ( , ~ ( t ) + 7i(t))~t 0 ))(y~), 71(t),/= g \ad(71(t) + yi(t))i[ ad(7~(t)+ ~]i(t))ii Yi

CAMPBELL-- HAUSDORFF

MULTIPLICATION

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P

__-.)

Fig. 1.

which is, in view of L e m m a 3.6(iii), equivalent to (**)

(y~(t)x~ = ( \y[(t)]

g(ad(7~(t))~0

\(g(ad(yr(t) + 7i(t))))ir

0 "](y~']. g(ad(~'r(t))ii)J\yi/

The initial condition reads

~i(o)/

x~

'

In order to solve (**) with these initial values, we first take a look at the upper lines of the equations. Here we see an initial value problem for yr and we know its solution: yr(t) = (p]~)-l(p(x) * tp(y)), since we may transfer the differential equation in the upper line of (**) into gl via the isomorphism P [ [ : [ - - ' g l . Since by hypothesis, • is an analytic extension of the CH-multiplication of ga, the solution 7l is defined at least on an interval ] - e , l + e [ for e > 0 sufficiently small. Moreover, from B 1 , B t _ C~ we obtain that (adT~(t))ii = cp(p(yr(t))) is exp-regular. Hence o(ad y~(t))ii is defined for all t e l - e , 1 +~[. N o w the lower lines of (**) represent an initial value problem for 7i that remains to be solved. In view of L e m m a 3.8(ii) it is a linear (inhomogeneous) differential equation (with nonconstant coefficients). Hence this initial value problem possesses a (unique) solution ?i defined on ] - e, 1 + e [, as we know from elementary calculus. Thus there is also a solution 7 : ] - e , 1 + e [ - - , g of (,) and we define x o y := 7(1). If we do this process for all pairs (x,y)~B x B, it defines a function o:BxB-,g.

42

ANSELM EGGERT

First of all, for x, y e g sufficiently small we conclude from Proposition 2.3 that x o y is equal to the CH-product x * y o f x and y. Secondly, the solution of an initial value problem of the form V'(t)= F(V(t),y), y(O)= x depends analytically on the parameter y and the initial value x, as one may find in El, §9, no. 1.8]. Thus o : B x B --, g is an analytic extension of the CH-multiplication of g and the proof of Proposition 3.4 is complete. (The final remark, that B o B _~ p-I(C1) holds, is immediate from the construction of o.) [] The entire notation from Proposition 3.4 m a y be forgotten beyond this point.

4.

T H E MAIN THEOREM

Now, the proof of our main theorem will not be difficult after we have formulated and proven two lemmas. 4.1. L E M M A . Let a, x e g be two elements of g which are contained in a solvable subalgebra. (This is true, for example, whenever a is contained in the radical of g.) Then spec(ad(a + x)) _~ spec(ad a) + spec(ad x). Proof. Let D denote a solvable subalgebra that contains a and x. Then p : x ~ ad x : b ~ Der(g) is a representation of [- Lie's theorem yields a H61der series ~ c = ~ . -~ " " ~- o~ -~ ~o = { 0 }

in the complexification gc of £, i.e. all the oi are ad Dc-invariant and we have dim Di = i. The induced one-dimensional representations P i ' [ c - * gl(o//oi-1) - C are called the weights of p. F o r y e b we have spec(ad y) = {p~(y):i = 1. . . . . n}, whence the assertion is immediate in view of the linearity of the weights. [] 4.2. L E M M A . Let j be a solvable ideal of g and assume that for every a e i the mapping ad a] j is exp-regular. Then there is a neighborhood U of 0 in g, such that each a e i + U is exp-regular. Proof As in the preceding proof we start with a H61der series ~ c = o . _ _ .-. __ ~

__ go = { 0 }

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MULTIPLICATION

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o f a d jc-invariant subspaces of t c with ,_'.imc o i = i. As above, let pi denote the corresponding weights. Let a ~ i be arbitrary. Since (ad a)(t) -~ i holds, the eigenvalues of ad a and of ad a]i are the same. Hence ad a is exp-regular, too. Thus the following is true: The restrictions p~ [ i : J ~ C of the weights do not take a value in 2niY\{0}. Hence the images &(i) are at most one-dimensional 0~-subspaces of C, i.e. straight lines that are not the imaginary axis. Let d > 0 denote the minimal distance of such a straight line from 2hi. Let I[.[[ denote a n o r m on t that satisfies ]][x,y]t] ~<[Ixl [[y[[. Now, U = { x ~ t : [[xil < d} is the neighborhood we have been looking for: Let a + x ~i + U with a E i and x E U. Then a and x are contained in the solvable subalgebra ~x + i and the preceding L e m m a 4.1 yields spee(ad(a + x)) G spec(ad a) + spee(ad x). The elements on the right-hand side are not contained in 2~riZ\{0} because the eigenvalues of ad x are by absolute value smaller than []ad x I and we have /adx]] ~< ]lxl[ < d. [] 4.3. T H E O R E M . Let g be a Lie algebra and i an ideal of t. Assume that for x ~ i the mapping ad x i i does not have an eigenvalue contained in 2~ri7/\{0}. Then there is an open ball V around 0 in 9 and an analytic extension ° : (J + V) x (i + V) ~ 9 of the CH-multiplication. Furthermore, i is solvable. Proof Firstly, i is solvable, because if not, i would contain a copy of sl(2, ~) or of so(3) and both of these algebras contain elements that are not expregular. F r o m L e m m a 4.2 we obtain a neighborhood U of 0 in 9, such that each x E i + U is exp-regular. N o w let V be a C H - n e i g h b o r h o o d in 9 that satisfies V * V _ U. As usual, let j~") denote the nth element of the c o m m u t a t o r series of i. We define in each algebra 9/j ~") two neighborhoods of 0 by B,-

v+i i(,)

and

C, -

u+i i~,)

According to L e m m a 3.3 the algebra t/i {"+1) is an abelian extension of g/j{"). Let p,: t / j ~"+ 1 ) ~ 9/i(,) denote the canonical projection. Then we have p;a(B,) = B.+I

and

p,-l(C,) = C.+1.

Moreover, by hypothesis B o = (V + J)/i is a C H - n e i g h b o r h o o d of g/j satisfying B o * B o ___Co.

44

ANSELM EGGERT

N o w we are in a position to apply Proposition 3.4. The choice of U guarantees that the requirements on exp-regularity are satisfied. The proposition yields inductively analytic extensions of the CH-multiplication on B, _ 9/i ~") for n = 1,2 . . . . . F o r n sufficiently large i t") = {0} holds, hence B, = V + i. (If we allow ourselves the liberty of identifying 9 and g/{0}!) [] F r o m the t h e o r e m we are n o w able to obtain am old result by Dixmier [3, Th. 3] as a corollary. 4.4. T H E O R E M . Let fl be a real Lie algebra. Then the following are equivalent: (i) g is exponential. (ii) I f G denotes the simply connected group with Lie algebra g, then exp : g ~ G is an isomorphism of analytic manifolds. (iii) N o x e g possesses an eigenvalue in 2rciZ\{0}. Furthermore, every exponential Lie algebra is solvable. P r o o f (i)=*-(ii) Let o:g x g ~ g be the analytic extension of the C H multiplication. Then (g, o) is a simply connected Lie g r o u p with Lie algebra g; the exponential function being the identity. Hence there is an i s o m o r p h i s m of analytic groups from (g, o) onto G. (ii) => (i) Given the exponential function exp, we m a y pull back the group multiplication of G onto g. This is the wanted analytic extension of the C H multiplication. (i) =~ (iii) Let o denote the analytic extension of the CH-multiplication and choose x E g arbitrarily. The derivative of the function y ~ x o y at the point y = 0 is g(ad x). In particular, 0(ad x) is defined and x is exp-regular. (iii) =~ (i) This implication as well as the solvability of g is an immediate consequence of T h e o r e m 4.3 if we put i = g. []

REFERENCES 1. Bourbaki, N., Variktds diffdrentielles et analytiques, paragraphes8 ~ 15, Hermann, Paris, 1971. 2. Bourbaki, N., Groupes et Algdbres de Lie, Chapitres 1 d 8, Hermann, Paris, 1975. 3. Dixmier, J., 'L'application exponentielle dans les groupes de Lie r6solubles', Bull. Soc. Math. France 85 (1957), 113-121. 4. Eggert, A., 'Uber Lieische Semialgebren', Mitteilungen aus dem Mathematischen Seminar GieBen 204 (1991), iv+91 pp. 5. Hochschild, G., The Structure of Lie Groups, Holden-Day, San Francisco, London, Amsterdam, 1965. 6. Hofmann, K. H., 'A memo on the exponential function and regular points', Preprint No. 1349, TH Darmstadt, 1991, 8 pp.

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Author's address: Anselm Eggert, Fachbereich Mathematik, Technische Hochschule Darmstadt, Schlol3gartenstral3e 7,

6100 Darmstadt, Germany. (Received,February 24, 1992;revised version, March 20, 1992)