Apr 21, 1980 - we denote by $A(g)X$ an element of $G_{e}^{R}$ ... on $G_{e}^{R}$ , which will be $ca!led$ ... Ad $(\exp tu)g_{x}=\mathfrak{g}_{\phi_{t}(x)}$.
J. Math. Soc.
Japan
Vol. 33, No. 4, 1981
A remark on non-enlargable Lie algebras By Hideki OMORI (Received April 21, 1980)
be a connected, non-compact, separable, manifold of finite dimension, and let be the Lie algebra of all vector fields on $N$. In this short note, we shall remark the following: THEOREM. There is no “infinite dimensional Lie group” with the Lie algebra
Let
$N$
$C^{\infty}$
$\Gamma(T_{N})$
$\Gamma(T_{N})$
$C^{\infty}$
.
The above result shows that Lie’s third theorem does not hold in a sense in case of infinite dimensional Lie algebras, but of course it depends how we define the concept of infinite dimensional Lie groups. (See also \S 3 below.) Thus to give a precise statement of the above theorem we have to fix at first the meaning of ”infinite dimensional Lie groups”. However, since the result that we want to obtain is a negative one, we shall fix here the definition as wide as possible. \S 1. Definition of infinite dimensional Lie groups. be an abstract group. As usual, Let denotes the group of all mappings of $R$ into , where the group operations are defined pointwisely. By we denote the subgroup consisting of all $X\in G^{R}$ such that $X(O)=e$ , the identity. For each $g\in G,$ we denote by $A(g)X$ an element of defined by $(A(g)X)(t)=gX(t)g^{-1}$ . $A$ is an action of , which will be $ca!led$ the on adjoint action. A structure of an infinite dimensional Lie group on is a triple such that if $g(t)\in S$ then $g(t+s)g(s)^{-1}$ of an adjoint invariant subgroup of for any , an infinite dimensional topological Lie algebra and a homomorphism of onto the underlying additive group of satisfying the following: (a) For every $g\in G$ , there is an automorphism Ad $(g)$ of such that $G^{R}$
$G$
$G$
$G_{e}^{R}$
$X\in G_{e}^{R}$
$G_{e}^{R}$
$G$
$G_{e}^{R}$
$G$
$S$
$\in S$
$G_{e}^{R}$
$s$
$\pi$
$\{S, \mathfrak{g}, \pi\}$
$\mathfrak{g}$
$S$
$\mathfrak{g}$
$\mathfrak{g}$
$\pi(A(g)X)=Ad(g)\pi(X)$
For every
.
, the mapping is of class such $[, ]$ $d/dt|_{t=0}$ $u=\pi(X)$ $(X(t))v=[u, v]$ , where that Ad and is the Lie bracket (See [2], [3] for the definition of differentiability.) product defined on (b)
$X\in S$
$\mathfrak{g}$
and
$ t-\rightarrow Ad(X(t))\nu$
$v\in \mathfrak{g}$
$C^{\infty}$
.
such that for every There is a mapping $exp$ : $t\in R$ is an element of } is a one parameter subgroup of { $\exp$ tu;
(c)
$\mathfrak{g}\rightarrow G$
$S,$
$u\in \mathfrak{g},$
$ X(t)=\exp$ $G$
and
tu
$\pi(X)$
708 $=u$
H. OMORI
.
An element of will be called a smooth curve in , and will be called the Lie algebra of . A group with a structure stated above will be called an infinite dimensional Lie group. $G$
$S$
$\mathfrak{g}$
$G$
\S 2. Proof of Theorem. Assume for a while that there is an infinite dimensional Lie group having . As $N$ is non-compact, there is $u\in\Gamma(T_{N})$ which is not the Lie algebra a complete vector field on $N$. Nevertheless by assumption (c), exp $tu$ is a smooth is a one one parameter subgroup of , and hence Ad ( $\exp$ tu): parameter automorphism group. at $x\in N$. Then, by Theorem 3 Let be the isotropy subalgebra of is characterized by a maximal finite codimensional subalgebra of of [1], , and by Theorem 2 of [1] there is a one parameter family of diffeomorphisms of $N$ onto itself such that $G$
$\Gamma(T_{N})$
$G$
$\Gamma(T_{N})\rightarrow\Gamma(T_{N})$
$\Gamma(T_{N})$
$\mathfrak{g}_{x}$
$\mathfrak{g}_{x}$
$\Gamma(T_{N})$
Ad $(\exp
(1)
where Ad fined by
$C^{\infty}$
$\phi_{t}$
$(\phi_{t})v$
is defined by
(Ad
Ad
(2)
tu)v=Ad(\phi_{t})v$
$(\phi_{t})v$
,
) $(x)=d\phi_{t}v(\phi_{t}^{-1}(x))$ .
$(\exp tu)g_{x}=\mathfrak{g}_{\phi_{t}(x)}$
$\phi_{t}$
$\frac{d}{dt}$
Ad $(\exp
tu)v=$ [ $u$ ,
Ad $(\exp
Using (1) and the assumption (b), we see that Ad (4)
$\frac{d}{dt}$
Ad
$(\phi_{t})v=$
[ , Ad $u$
$\phi_{t}$
is de-
.
By (2) we get that is a one parameter subgroup of onto itself. By the assumed property (b), we see easily (3)
Recall that
$C^{\infty}$
diffeomorphisms of
$N$
tu)v$ ].
$(\phi_{t})v$
$(\phi_{t})v$
is
in
$C^{\infty}$
$t$
such that
]
for every . Remark that the above equality makes sense on every open subset of $N$. For a relatively compact open subset $U$ of $N$, we denote by a local one parameter group on $U$ generated by . We assume is dePned for such , Ad that . For every is well-dePned as a local vector field on $U$ , and it is easy to see that $v\in\Gamma(T_{N})$
$\psi_{t}$
$u$
$|t|0$
$v\in\Gamma(T_{N})$
$\frac{d}{dt}$
Ad
$(\psi_{t}^{-1})v=-$
[ , Ad $u$
$t$
$\psi_{t}$
$(\psi_{t})v$
$(\psi_{t}^{-1})v$
],
$|t|
