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the following conditions:
(i) 1 2 S and SS \ U S
(ii) L(S) = w .
(For a proof see e. g. [4].)
De nition 1.6. A set S G satisfying (i) and (ii) is called a local subsemi-
group of G with respect to U .
The following conjecture is consistent with what we knowtoday:
Conjecture 1.6. For every pair (g w) consisting of a Lie algebra and a Lie
wedge w generating g as a Lie algebra there are
(i) a (simply connected) Lie group G with L(G) = g,
(ii) a pathwise connected (and simply connected) topological semigroup S ,
and
(iii) a continuous morphism of semigroups with identity p: S ! G and
(iv) open identity neighborhoods U and V of S and G , respectively, such
that
(a) pj U : U ! p(U ) is a homeomorphism and p(U ) is a local sub-
semigroup of G with respect to V .
;p(U
(b) L ) = w .
Hofmann 39
Proposition 1.7. (W. Weiss) If w is pointed then Conjecture 1.6 is correct.
Very little is known beyond this general result. Better progress was
made regarding the question when for a given Lie group G , and for a given Lie
wedge w in g = L(G) there exists a semigroup S in G such that L(S) = w .
Such wedges w g we call global with respect to G and simply global if G is
simply connected. First results are to be found in [4], and better insights are due
to Neeb. As a rst orientation we can formulate the following necessary and
su cient condition:
Proposition 1.8. A Lie wedge w L(G) is global with respect to a connected
Lie group G if and only if the analytic subgroup H with L(H) = h(w) is closed
and there is a smooth function : G ! R such that for all X 2 w n h(W) and
all g 2 G the number hd (g) d (1)(X)i is positive.
g
Such functions are also called strictly positive . Their geometric signi -
cance is as follows: One may say that the homogeneous space M = G=H carries
a causal structure de ned by the transport via the left action of G on M of the
pointed cone w=h(W) to each tangent space T(M)gH , yielding a unique pointed
cone (gH) in this tangent space. The fact that w is a Lie wedge is exactly
adequate for this formalism to work. The prescription of strictly positive func-
tions on G as speci ed above and that of strictly positive functions on M are
one and the same thing on M one could call such a function global time since
;a
every \time like" curve ! x(t): [t0 ! M with _ 2 x(t) then can be
7
;t ;x(tT] R ;x(x( t)
t
assigned an eigentime x(t) = ) + hd s) x(s)i ds which
t0
;_ ;x(T)allows a
reparametrisation the curve to an equivalent curve : [ x(t0) ] ! M
;of ;
given by (r) = x ( x); 1 (r) with (r) = r .
Special Lie wedges are always global since half-space Lie wedges are
global and the property of globality is compatible with the formation of inter-
sections. Invariant wedges are not always global. Neeb's lecture will contain
more on this. The issue of globality of invariant cones will settle the question of
globality of Lie semialgebras.
An observation on Lie' s Third Theorem
Concerning the consideration of pairs (g w) one should observe that the
statement that w generates g sometimes imposes undesireable restrictions. Let
us discuss this aspect brie y, illustrated in the context of split wedges w = c + h
with g = q + h and c = w \ q , h = h(w).
We form the semidirect sum g h with ad: h ! Der(g) given by
ad
ad(X)(Y ) = [X Y] . In other words, on g h we consider the componentwise
vector spaces structure and the bracket given by
0 0
(1) [(X Y) (X0 Y0)] =([X X0 ] +ad(Y )(X0 ) ; ad(Y )(X) [Y Y ]):
The we have a surjective morphism of Lie algebras : g h ! g with kernel
ad
ker( ) = f(;Y Y) : Y 2 hg h:
=
40 Hofmann
If Y 2 h then ad(0 Y) on g h is (ad Y ad Y) by (1) and thus ead(0 Y ) =
ad
def
(ead Y ead Y ) . It follows that g# = q h is a subalgebra of g such that
h
j g#: g# ! g is an isomorphism on account of ker( ) \ (q h) = f(0 0)g . Fur-
thermore, the wedge w# def c h g h is a Lie wedge. The restriction
=
ad
and corestriction j w#: w# ! w is an isomorphism, so that (g# w#) (g w) .
=
However, Weiss' Theorem given us a topological semigroup T and a homomor-
phism p: T ! G such that for suitable open identity neighborhoods f(U ) is a
;p(U
local subsemigroup of G with respect to V and that L ) = c . If Weiss'
construction allows us to choose T in such a fashion that H acts continuously
on T as a group of continuous automorphisms via : H ! Aut(T) such that
;
p (h)(t) = hp(t)h; 1 |which is certainly the case if h is compactly embedded
into g, i. e., if head h i is relatively compact in Aut(g) |then S = T is a
;p(t)H
semigroup such that P: S ! G H , I(h)(g) = hgh; 1 , P(t h) = h is
I
a homomorphism mapping U H homeomorphically onto its image p(U ) H .
0 0
such that P(U U ) is a local subsemigroup of G H with respect to V U
I
for a suitable identity neighborhood U of H . Thus the Conjecture 1.6 holds for
w# g h whereas we have at present no way of asserting that it holds for
ad
w# g# , that is, for w g.
The main open problem in the domain of Lie's fundamental theorems
for semigroups is a general proof or refutation of Conjecture 1.6.
References
[1] Bourbaki, N., Int egration, Chapitres 7 et 8, Hermann, Paris, 1963.
[2] |, Groupes et alg ebres de Lie, Chapitres 2 et 3, Hermann Paris, 1972.
[3] |, Groupes et alg ebres de Lie, Chapitres 7 et 8, Hermann Paris, 1975.
[4] Hilgert, J., K. H. Hofmann, and J. D. Lawson, Lie groups, Convex Cones,
and Semigroups, Oxford University Press 1989.
[5] Hofmann, K. H., Zur G eschichte des Halbgruppenbegri s, Historia Math-
ematica, 1991, to appear.
[6] |, A memo on the singularities of the e xponential function, Seminar
Notes 1990.
[6] Hofmann, K. H., J. D. Lawson, and J. S.Pym, Editors, The Analytical
and Topological Theory of Semigroups, de Gruyter Verlag, Berlin, 1990.
Fachbereich Mathematik
Technische Hochschule Darmstadt
Schlossgartenstr. 7
D-6100 Darmstadt
e-mail XMATDA4L@DDATHD21
Received January 17, 1991 [ Pobierz całość w formacie PDF ]
