Apr 15, 1981 - Thus, using th$e$ uniform continuity of $D^{r}\phi(\lambda(t)+\theta\mu(t))$ ..... $\lim_{n\rightarrow\infty}\sigma_{\Delta_{\hslash}}$ converges ...
TOKYO J. MATH. VOL. 5, No. 2, 1981
On Regular Fr\’echet-Lie Groups IV Definition and Fundamental Theorems
Hideki OMORI, Yoshiaki MAEDA, Science University of Tokyo, Keio University
Akira YOSHIOKA and Osamu KOBAYASHI Tokyo Metropolitan University, Keio University
Introduction In the previous papers [10], [11], [12], we have seen that Fourier integral operators on a compact manifold have some group theoretical characters. Indeed, one of the purposes of this series is to show that on a the group of all invertible Fourier integral operators of order $0$
compact riemannian manifold is an infinite dimensional Lie group. It should, however, be remarked that we have not given in the previous papers the definition of infinite dimensional Lie groups. It will be given in this paper, hence one may read this paper without knowing anything about the previous papers. Now, it continues to be a basic question when one may call a group $G$ an infinite dimensional Lie group. However, taking the basic properties of finite dimensional Lie groups in mind, we suggest the following $(L1)\sim(L3)$ are necessary at least, where (L1) $G$ is a infinite dimensional manifold and the tangent space at the identity has a Lie algebra structure, called the Lie algebra of $G$ . into $G$ such (L2) There exists the exponential mapping exp of that { $\exp$ tu; $t\in R$ } is a smooth one parameter subgroup of $G$ for every $C^{\infty}$
$C^{\infty}$
$\mathfrak{g}$
$\mathfrak{g}$
$u\in \mathfrak{g}$
.
Local group structures of $G$ ( $i.e.$ , a neighborhood of the identity) can be determined by its Lie algebra . Hilbert or Banach-Lie groups [1], [5] satisfy these conditions and so do strong ILB- (or strong ILH-) Lie groups defined by Omori [8], [9]. (L3)
$\mathfrak{g}$
Received April 15, 1981
366
H. OMORI, Y. MAEDA, A. YOSHIOKA AND
$0$
. KOBAYASHI
Note that strong ILB-Lie groups include Banach-Lie groups, hence finitt dimensional Lie groups. In this paper, we shall define a wider concepl of infinite dimensional Lie groups, which will be called regular Fr\’echet. Lie groups throughout this series. Roughly speaking, a regular Fr\’echet. Lie group is a manifold modeled on a locally convex Fr\’echet space having a group structure on which product integrals can be well. defined and have some smoothness properties. $C^{\infty}$
$C^{\infty}$
\S 1. Several remarks on differentiability. In this paper, we shall use the notion of differentiability defined in Let $U$ be an open subset of a Fr\’echet space , and another Fr\’echet space, where all Fr\’echet spaces in this series are assumed to be locally convex. A mapping $f:U\rightarrow F$ is a mapping, if it is con. tinuous. is called to be r-times diferentiable at $xeU$, if is on a neighborhood of and there exists a continuous symmetric r-linear [3].
$E$
$F$
$C^{0}$
$f$
$f$
$C^{r-1}$
$x$
mapping $(D^{r}f)(x):E\times\cdots\times E\rightarrow F$
$\overline{r}$
such that $F(v)=f(x+v)-f(x)-(Df)(x)(v)-\cdots-\frac{1}{r!}(D^{r}f)(x)(v, \cdots, v)$
satisfies the property that $R(t, v)=\left\{\begin{array}{ll}F(tv)/t^{r}, & t\neq 0\\0 , & t=0\end{array}\right.$
Is continuous on a neighborhood of $(0,0)$ . is called a $C$ mapping, if is r-times differentiable at each $x\in U$ and the mapping ‘
$f$
$f$
$D^{r}f:U\times E\times\cdots\times E\rightarrow F$
is continuous.
(Cf. See [2], [13] for the various
definitions of the differen-
tiability.)
respecitively, and $G$ another Let $U,$ $V$ be open subsets of $E,$ Fr\’echet space. mapping with respect to will be called a the first variable if for each fixed $v\in V,$ is and every $Dif$ for is a continuous mapping of into $G$ , where the suffix 1 of $D$ means the derivative with respect to the first variable. The paritial differentiability for the second variable is defined by the , etc.. similar manner and the derivative is denoted by $F$
$f:U\times V\rightarrow G$
$C^{r}$
$C^{r}$
$f:U\rightarrow G$
$0\leqq s\leqq r$
$U\times V\times E\times\cdots\times E$
$D_{2}$
367
REGULAR FR\’ECHET-LIE GROUPS
mappings are used very often in this The following properties of series. (For the proof, see [3] and [4]): mapping, and mappings (A) A composition is a of $D(f\circ g)=Df\cdot Dg$ , more precisely $C$ ‘
$C^{r}$
$f\circ g$
$f,$
$C^{r}$
$g$
.
$D(f\circ g)(x)(v)=(Df)(g(x))((Dg)(x)(v))$
If
(B)
and
is
$f:U\rightarrow F$
$D_{1}^{t}Df=D^{t+}f,$
$C^{r}$
, then
$t+\epsilon\leqq r$
$D^{S}f:U\times E\times\cdots\times E\rightarrow F$
.
is
for
$C^{r-}$
$s\leqq r$
,
is with respect to both is C’ if and only if first and second variables. As Fr\’echet spaces are assumed to be locally convex, there are a lot be one of them. For a C’ of continuous linear functionals. Let -function on $[0,1]$ . mapping $h(t)=Nf(x+tv)$ is an R-valued Using the mean value theorem for , we obtain (C)
$f:U\times V\rightarrow G$
$C^{r}$
$f$
$x:F\rightarrow R$
$C^{1}$
$f:U\rightarrow F,$
$h$
(1)
$f(x+v)=f(x)+\int_{0}^{1}(Df)(x+tv)(v)dt$
Thus, using (B) successively, one
theorem: (2)
For a
$C^{r}$
function
.
can get the following Taylor’s expansion
,
$f$
$f(x+v)=f(x)+(Df)(x)(v)+\cdots+\frac{1}{(r-1)!}(D^{r-1}f)(x)(v, \cdots, v)$
$+\int_{0}^{1}\frac{(1-t)^{r-1}}{(\gamma-1)!}(D^{r}f)(x+tv)(v, \cdots, v)dt$
$=f(x)+(Df)(x)(v)+\cdots+\frac{1}{r!}(D^{r}f)(x)(v, \cdots, v)$
$+\int_{0}^{1}\frac{(1-t)^{r-1}}{(r-1)!}\{(D^{r}f)(x+tv)-(D^{r}f)(x)\}(v, \cdots, v)dt$
By this formula, one (B’)
$f:U\rightarrow F$
is
can get the converse of $C^{r}$
$Df:U\times E\times\cdots\times E\rightarrow F$
is
if and $C^{r-}$
.
only
if
$f$
.
(B), namely,
is
$C$
for some
$s(\leqq r)$
and
REMARK. The above notion of differentiability coincides with the usual one if $E,$ $G$ are finite dimensional vector spaces. However, if $G$ are infinite dimensional Banach spaces, the above notion is weaker $E,$ than the usual definition of differentiability. For instance, the above , while definition requests only the continuity of is usually requested, in Banach spaces the continuity where the last one is the Banach space of symmetric r-linear mappings with the uniform topology. The continuity of into of $F,$
$F,$
$D’ f:U\times E\times\cdots\times E\rightarrow F$
$D^{r}f:U\rightarrow L_{8}^{r_{F^{m}}}(E, F)$
$E\times\cdots\times E$
$F$
368
H. OMORI, Y. MAEDA, A. YOSHIOKA AND O. KOBAYASHI
ensures
$D^{r}f:U\times E\times\cdots\times E\rightarrow F$
bounded, namely for any positive constant $C$ such that
$x\in U$
only that
is locally of , and ,
$D^{r}f:U\rightarrow L_{sym}^{r}(E, F)$
there is a neighborhood
$W$
$x$
$||(D^{r}f)(y)(v_{1}, \cdots, v_{r})||\leqq C||v_{1}\Vert||v_{2}||\cdots||v_{r}||$
for every $y\in W,$ E. value theorem (1), we have $v_{1},$
$\cdots,$
$ v_{r}\in$
Thus, by
$D_{1}D^{r-1}f=D^{r}f$
$||(D^{r-1}f)(x+w)-(D^{r-1}f)(x)\Vert\leqq C||w||$
and the mear
.
Therefore, if is in the above sense, then $f$ is in $tht$ usual sense in Banach spaces. Thus, if we concem only $C^{\infty}mappings_{1}$ the above two notions of differentiability make no difference. It should be remarked, however, that above two notions make a difference for the partial differentiability. In our definition, we requesl only the continuity of for . For a fixed $v\in V,$ is continuous by the above remark, but this does not mean the continuity of $Di^{-1}f:U\times V\rightarrow L_{sym}^{*-1}(E, G)$ . Suppose $E,$ $F$ are Fr\’echet spaces and $U,$ $V$ are open subsets of $E,$ respectively. Let $f:U\rightarrow V$ be a $C$ ‘ mapping $(r\geqq 1)$ . We define the differential map $df:U\times E\rightarrow V\times F$ by $C^{r}$
$f:U\rightarrow F$
$C^{r-1}$
$D_{1}^{g}f:U\times V\times E\times\cdots\times E\rightarrow G$
$s\leqq r$
$D_{1}^{*-1}f:U\rightarrow L_{sym}^{\iota-1}(E, G)$
$F$
$(df)(x, v)=(f(x), (Df)(x)(v))$ $Df(x)(v)$
is denoted sometimes by $(df)_{x}v,$ $(Tf)_{x}v$ or $f_{*g}v$ . Obviously, is a mapping by $(B’)$ . If $f$ is a ‘-diffeomorphism exists and ), then $df$ is a diffeomorphism.
$df:U\times E\rightarrow V\times F$
(i.e.,
$f^{-1}$
$C^{r-1}$
$C$
$C^{r}$
LEMMA 1.1. If mapping, then is if
$C^{r-1}$
$f:U\rightarrow V$
$f^{-1}$
$C^{1}$
is invertible and mapping. is
$f^{-1}$
By using the composition rule is given by $(Df^{-1})(y)=(Df(x))^{-1}$
Therefore, if
$y,$
$C$ ‘
mapping
$(r\geqq 1)$
and
$C^{r}$
PROOF.
$y$
.
$(A)$
, the derivative of
$f^{-\iota}$
at
, $y=f(x)eV$ .
$y+w\in V$
$(Df^{-1})(y+w)-(Df^{-1})(y)$
$=-(Df^{-1})(y)((Df)(f^{-1}(y+w))-(Df)(f^{-1}(y)))(Df^{-1}(y+w))$
.
Hence by the continuity of $(Df^{-1})(y)w$ and th smoothness of , we see that $(Df^{-1})(y)w$ is , and hence is . The desired regularity follows inductively by this manner. $e$
$C^{1}$
$f^{-1}$
$f$
$C^{2}$
$\square $
369
REGULAR FR\’ECHET-LIE GROUPS
Fr\’echet manifold $M$ modeled on we define the notion of $i.e.,$ $M$ admits an admissible atlas which is a collection of pairs as coordinate transformations. A tangent of local charts with where , at $x\in M$, is an equivalence class of triples vector $M$ is a vector of ; two and at is an arbitrary chart of triples are equivalent if Now, usual,
$E$
$C^{\infty}$
$C^{\infty}$
$(U, \phi)$
$(U_{i}, \phi_{i}, X_{i})$
$X_{x}$
$E$
$X_{i}$
$x$
$(U_{1}, \phi_{i})$
$X_{j}=D(\phi_{j^{\circ}}\phi_{i}^{-1})(\phi_{i}(x))X_{i}$
.
plays the same in the triple The representative $X\in E$ of role as the components in a local coordinate system. The space of vectors tangent at the point , together with its natural vector space structure is the tangent vector space $T.M$. It can be easily verified that th set of vectors tangent to $M,$ $TM=\bigcup_{xeM}T_{x}M$ has a $structu\grave{r}e$ of a by the family of charts differentiable manifold modeled on and $\{(U, \phi)\}$ $\{(\bigcup_{xeM}T_{x}M, T\phi)\}$ is an atlas of $M$ modeled on where defined by the homeomorphism of $\bigcup_{xeU}T_{x}M$ on $(U, \phi_{i}, X_{i})$
$X_{x}$
$x$
$e$
$C^{\infty}$
$E\times E$
$E$
$ T\phi$
$U\times E$
$T\phi(x, X_{x})=(\phi(x), X)$
in the map $(U, \phi)$ . Moreover, the being the representative of space is given a fibre bundle structure with the natural projection mapping and is called the Tangent bundle. Thus, the of
$X$
$X_{x}$
$T_{M}$
$\pi:T_{M}\rightarrow M$
$C^{\infty}$
. sections of vector fields are defined as usual as , for it However, we do not define the structure of vector bundle on topological $GL(E)$ as a is not easy to define a reasonable topology for $\pi:T_{H}\rightarrow M$
$C^{k}$
$C^{k}(k\leqq\infty)$
$T_{M}$
group. Frechet submanifold, if there A subset $N$ of $M$ will be called a local and for every $x\in N$ there is a of is a closed subspace maps $U\cap F$ homeosuch that of $M$ at coordinate system morphically onto the arcwise connected component of of a neighborhood in $N$ under the relative topology. of if the following will be called a subbundle of A subset ST of $C^{\infty}$
$C^{\infty}$
$E$
$F$
$\phi:U\rightarrow M$
$x$
$\phi$
$x$
$x$
$T_{M}$
$T_{M}$
conditions are satisfied: is a closed linear sub(SB1) is surjective, and by and call it space of $T_{x}M$ for every $x\in M$. We denote at . the fiber of of each $x\in M$, and a (SB2) There exists an open neighborhood such that onto diffeomorphism of , and is linear with respect to the second variable. . is , then (SB3) If and a unit interval $I=[0,1]$ , let $C^{k}(I, E)$ Given a Fr\’echet space $\pi^{-1}(x)\cap\Psi$
$\pi:\ovalbox{\tt\small REJECT}\rightarrow M$
$\pi^{-1}(x)\cap\ovalbox{\tt\small REJECT}^{\prime}$
$\Psi^{-}$
$F_{x}$
$x$
$V_{x}$
$C^{\infty}$
$\psi_{\dot{x}}$
$y,$
$y\in V_{x}$
$V_{x}\times F_{x}$
$\pi^{-1}(V_{x})\cap\chi’\subset T_{M}$
$\pi\psi_{x}(y, v)\equiv$
$\psi_{x}$
$C^{\infty}$
$\psi_{x}^{-1}\psi_{y}:(V_{x}\cap V_{\nu})\times F_{y}\rightarrow(V_{l}\cap V_{y})\times F_{x}$
$ V_{x}\cap V_{y}\neq\emptyset$
$E$
370
H. OMORI,
$Y$
. MAEDA,
A. YOSHIOKA AND O. KOBAYASHI
be the set of all mappings from into $E(k=0,1,2, \cdots)$ . The set $C^{k}(I, E)$ has the structure of vector space in the obvious way. Consider $C^{k}$
$I$
the evaluation map
$ev^{k}:I\times C^{k}(I, E)\rightarrow\frac{E\times\cdots\times E}{k+1}$
given by $ev^{k}(t, c)=(c(t), (Dc)(t),$
$\cdots,$
$(D^{k}c)(t))$
.
Putting a Fr\’echet structure into , we obtain the uniform topology of $C^{k}(I, E)$ . Let $M$ be a Fr\’echet manifold modeled on $E$ and let $C^{k}(I, M)$ , be a space of all $k=0,1,2,$ mappings from into $M$. Then, we have the following. $E\times\cdots\times E$
$C^{\infty}$
$C^{k}$
$\cdots$
LEMMA 1.2. $C^{k}(I, M)$ is a . For a for each $k=0,1,2,$ $\cdots$
$I$
Fr\’echet manifold modeled on fixed point $peM$,
$C^{\infty}$
$C^{k}(I, E)$
$C_{p}^{k}(I, M)=\{c\in C^{k}(I, M);c(0)=p\}$
is a
$C^{\infty}$
Fr\’echet
submanifold of $C^{k}(I, M)$ .
$V$ be open PROOF. Let 8ubsets of and let morphism. Define a mapping $f:C^{0}(I, U)\rightarrow C^{0}(I, V)$ by Obviously, th crucial part of the proof is to show that that purpose, set $U,$
$E$
$\phi:U\rightarrow V$
$e$
a
$C^{\infty}$
diffeo-
$f(\lambda)(t)=\phi(\lambda(t))$
$f$
is of
$C^{\infty}$
.
.
For
$ F(\mu)(t)=\phi((x+\mu)(t))-\phi(\lambda(t))-(D\phi)(\lambda(t))(\mu(t))-\cdots$
$-\frac{1}{r!}(D^{r}\phi)(\lambda(t))(\mu(t), \cdots, \mu(t))$
.
By Taylor’s theorem, we have $ F(\mu)(t)=\int_{0}^{1}\frac{(1-\theta)^{-1}}{(r-1)!}\{D^{r}\phi(x(t)+\theta\mu(t))-D^{r}\phi(\lambda(t))\}(\mu(t), \cdots, \mu(t))d\theta$
Thus, using th
$e$
uniform continuity of
$D^{r}\phi(\lambda(t)+\theta\mu(t))$
.
we see that
$R(s, \mu)=\left\{\begin{array}{ll}F(\epsilon\mu)/s^{f} & s\neq 0\\0 & s=0\end{array}\right.$
is continuous on a neighborhood of is continuous with respect to
$(0,0)$
.
Since
$\lambda\in C^{0}(I, U),$
$\mu_{1},$
$D^{r}\phi(\lambda(t))(\mu_{1}(t), \cdots, \mu_{r}(t))$
$\cdots,$
$\mu_{r}eC^{0}(I, E)$
, for all
371
REGULAR FR\’ECHET-LIE GROUPS
, we have that
$r\geqq 0$
$f:C^{0}(I, U)\rightarrow C^{0}(I, V)$
$D^{r}\phi(\lambda(t))(\mu_{1}(t), \cdots\mu_{r}(t))$
.
is
$C$ ‘
and
$D^{r}f(x)(\mu_{1}, \cdots, \mu_{r})(t)=$
To prove the differentiability for $k>0$ , we have only to use instead of . $\square $
$d^{k}\phi;U\times E\times\cdots\times E\rightarrow V\times E\times\cdots\times E$
$\phi$
\S 2. Several remarks on FL-groups. An FL-group is the combined concept of a topological group and a . Therefore, Fr\’echet manifold such that the group operations are one may call an FL-group a Fr\’echet-Lie group. However, we hesitate to use the name “Lie”, for a general FL-group may not have the properties $(L1)\sim(L3)$ mentioned in the introduction. of $G$ at the Now, let $G$ be an FL-group. The tangent space identity is naturally identified with its model space, and hence is diffeomorphism of an a Fr\’echet space. By definition, there is a in $G$ of open neighborhood $U$ of in onto an open neighborhood will be called a local coordinate such that $\xi(0)=e,$ $(d\xi)_{0}=Id$ . is called a “local coordinate system”. system at . (Usually, We use th above definition in accordance with the exponential map of the group.) As satisfies the first countability axiom, so also does $G$ , and hence $G$ has a right-invariant metric (cf [6] p. 34), where a metric on $G$ is called a right-invariant metric if $\rho(xa, ya)=\rho(x, y)$ for every $C^{\infty}$
$C^{\infty}$
$\mathfrak{g}$
$\mathfrak{g}$
$e$
$C^{\infty}$
$\xi$
$\tilde{U}$
$0$
$e$
$\mathfrak{g}$
$\xi:U\rightarrow\tilde{U}$
$\xi^{-1}:\tilde{U}\rightarrow U$
$e$
$e$
$\mathfrak{g}$
$\rho$
$x,$
$y,$
a $eG$ .
As $G$ is a topological group, there is an open neighborhood $V$ of Therefore, by the local coordinate $\xi(V)^{-1}=\xi(V)$ . in such system at , the group operations of $G$ are represented by $0$
$\xi(V)^{2}\subset\tilde{U},$
$\mathfrak{g}$
$\xi$
$e$
$\eta(u, v)=\xi^{-1}(\xi(u)\xi(v))$
For every that
$g\in G$
.
$g\xi(W)g^{-1}\subset\tilde{U}$
,
$c(u)=\xi^{-1}(\xi(u)^{-1})$
, there is an open neighborhood We set $\tilde{A}_{g}(u)=\xi^{-1}(g\xi(u)g^{-1})$
. $W$
of
$0$
in
$\mathfrak{g}$
such
.
on the domain on which they are are and As $G$ is FL-group, defined. , we see $(D_{1}\eta)_{(0,0)}=(D_{2}\eta)_{(0,0)}=Id$ (the idenSince $\eta(u, O)\equiv u,$ curves in $G$ such that $a(O)=b(O)=e$ . Then, tity). Let $a(t),$ $b(t)$ be eg, where $\dot{a}(0)=(d/dt)|_{t=0}a(t)$ . The product $c(t)=a(t)hb(t)$ is a curve in $G$ with $c(O)=h\in G$ . The following is easy to prove. $\tilde{A}_{g}$
$\eta,$
$C^{\infty}$
$\iota$
$\eta(0, v)\equiv v$ $C^{1}$
$\dot{a}(0),\dot{b}(0)$
LEMMA 2.1.
$C^{1}$
372
H. OMORI, Y. MAEDA, A. YOSHIOKA AND O. KOBAYASHI $\frac{d}{dt}|_{t=0}a(t)hb(t)=dR_{h}\dot{a}(0)+dL_{h}\dot{b}(0)$
$\frac{d}{dt}|_{t=0}a(t)^{-1}=-\dot{a}(0)$
.
where $R_{h}(resp. L_{g})$ denotes the right- (resp. left-) translation. Let $G$ be an FL-group. For every $g\in G$ , define the map by $A_{g}h=ghg^{-1}$ . Then the map $A:G\times G\rightarrow G$ , (3)
$A_{g}$
on
$G$
$A(g, h)=A_{g}h$
is . We often use the notation $A(g)h$ instead of $A,h$ . Given , there is a curve $c(t)$ in $G$ such that $c(O)=e$ , and $\dot{c}(0)=u$ (for instance, $c(t)=\xi(tu)$ satisfies this). Define the adjoint map $Ad(g),$ $g\in G$ on by $C^{\infty}$
$u\in \mathfrak{g}$
$C^{\infty}$
$\mathfrak{g}$
(4)
$Ad(g)u=\frac{d}{dt}|_{t=0}gc(t)g^{-1}(=\frac{d}{dt}|_{t=0}A_{g}c(t))$
The adjoint map
.
can be expressed in terms of the left and right If we denote the left- (resp. right-) translation
$Ad(g)$
auxiliary functions. by
$L_{g}:h\rightarrow gh$
(resp.
$R_{g}:h\rightarrow hg$
),
$g,$
$h\in G$
,
then we get $Ad(g)u=(dR_{g})^{-1}(dL_{g})u$ . Therefore, the definition (4) does not depend on the choice of the curve $c(t)$ . Also, we put the following map of to by $G\times \mathfrak{g}$
$\mathfrak{g}$
$Ad(g, u)=Ad(g)u$
,
$geG,$
$ue\mathfrak{g}$
.
be the tangent bundle of FL-group $G$ . Since , we see that is diffeomorphic to . The differential map of the product operation on $G$ gives the group structure on . Namely, under the identification between , we obtain and the product by Let
$(dR_{g})_{e}\mathfrak{g}),$
$T_{a}$
$T_{g}G=\mathfrak{g}\cdot g(=$
$g\in G$
$C^{\infty}$
$T_{a}$
$\mathfrak{g}\times G$
$T_{\theta}$
$T_{a}$
$\mathfrak{g}\times G$
$*on\mathfrak{g}\times G$
(5)
$(u, g)*(v, h)=(dR_{gh})(u+Ad(g)v)(=(u+Ad(g)v)\cdot gh)$
.
LEMMA 2.2. By the product $*of(5),$ turns out to be an FLgroup with $(0, e)$ as the identity and can be identified with as FLgroup. Moreover, is a normal abelian subgroup of . The above FL-group will be denoted by Let $g(t)$ be a curve in such that $g(O)=e,\dot{g}(0)=u$ . We define the bracket $[u, v]$ on by $\mathfrak{g}\times G$
$T_{a}$
$\mathfrak{g}\times\{e\}$
$\mathfrak{g}\times G$
$\mathfrak{g}*G$
$G$
$C^{\infty}$
$\mathfrak{g}$
373
REGULAR FR\’ECHET-LIE GROUPS
(6)
$[u, v]=\frac{d}{dt}|_{t=0}Ad(g(t))v$
,
$v\in \mathfrak{g}$
.
To obtain that (6) satisfies the properties of the Lie bracket, we give by the rightthe other description of (6). Denote for every $i.e.$ $G$ , such that $u^{*}(e)=u,$ vector field on invariant $u^{*}$
$u\in \mathfrak{g}$
$C^{\infty}$
(7)
.
$u^{*}(g)=u\cdot g(=(dR_{g})_{e}u)$
Then,
we have the following by
$\dot{c}(0)=v$
:
taking
a
curve such that
$C^{\infty}$
$c(O)=e$ ,
$\frac{d}{dt}|_{t=0}Ad(g(t))v=\frac{\partial^{2}}{\partial t\partial s}|_{t=0 ,l--0}g(t)c(s)g(t)^{-1}$
$=\frac{d}{ds}|_{=0}\{dR_{c(*)}u-dL_{a(\epsilon)}u\}$
(Lemma 2.1)
$=\partial_{v}u^{*}-\frac{\partial^{2}}{\partial s\partial t}|_{t=0 ,\epsilon=0}c(s)g(t)$
$=\partial_{v}u^{*}-\frac{d}{dt}|_{t=0}dR_{g(t)}v$
$=\partial_{v}u^{*}-\partial_{u}v^{*}$
,
where in the above computations, differentials are computed by taking depend on the choice a local coordinate expression. Though does not depend on it. Moreof a local coordinate system, over, we see that $(d/dt)|_{t=0}Ad(g(t))v$ does not depend on the choice of $g(t)$ . Hence, we get $\partial_{u}v^{*}$
$\partial_{v}u^{*},$
$\partial_{v}u^{*}-\partial_{u}v^{*}$
(8)
$[u, v]=\partial_{v}u^{*}-\partial_{u}v^{*}$
.
, the above bracket product Since is a Lie algebra such that satisfies the Jacobi identity, and hence $[, ]$ : is a continuous bi-linear mapping, $i.e.,$ is a Fr\’echet-Lie algebra. will be called the Lie algebra of $G$ . $(d^{2}u^{*})_{\epsilon}(v, w)=(d^{2}u^{*})_{e}(w, v),$
$u,$
$v,$
$w\in \mathfrak{g}$
$\mathfrak{g}$
$\mathfrak{g}$
$\mathfrak{g}\times \mathfrak{g}\rightarrow \mathfrak{g}$
$\mathfrak{g}$
LEMMA 2.3.
Let
$g(t)$
be a
$C^{1}$
curve in G.
$\frac{d}{dt}Ad(g(t))v=[u(t), Ad(g(t))v]$
Then,
,
$\frac{d}{dt}Ad(g(t)^{-1})v=-Ad(g(t)^{-1})[u(t), v]$
where
$u(t)=(dg(t)/dt)\cdot g(t)^{-1}\in \mathfrak{g}$
.
,
374
H. OMORI, Y. MAEDA, A. YOSHIOKA AND O. KOBAYASHI
PROOF. Remark at first that $Ad(g(t))v$ is C’ with respect to . Note that $Ad(g)Ad(h)=Ad(gh)$ and $(dg/dt)\cdot g(t)^{-1}=d\xi(su(t))/ds|_{\iota=0}$ . We compute as follows: $t$
$\frac{d}{dt}Ad(g(t))v=\frac{d}{ds}|_{=0}Ad(g(t+s))v$
$=\frac{d}{ds}|_{=0}Ad(\xi(su(t))\cdot g(t))v$
$=\frac{d}{ds}|_{=0}Ad(\xi(\epsilon u(t)))Ad(g(t))v$
$=[u(t), Ad(g(t))v]$
(Remark
2.1 and
that
$\xi(su(t))$
is
w.r.t.
$C^{\infty}$
$s$
.
and use (6)).
Similarly, by Lemma
(6), $\frac{d}{dt}$
Ad $(g(t)^{-1})v=\frac{d}{ds}|_{=0}Ad(g(t)^{-1})Ad(\xi(su(t))^{-1})v$ $=Ad(g(t))^{-1}\frac{d}{ds}|_{=0}Ad(\xi(su(t))^{-1})v$
$=-Ad(g(t)^{-1})[u(t), v]$
LEMMA 2.4.
Ad:
$G\times \mathfrak{g}\rightarrow \mathfrak{g},$
$Ad(g):\mathfrak{g}\rightarrow \mathfrak{g}$
.
$\square $
is an automorphism of the Lie algebra and $g\in G,$ , is a mapping.
$Ad(g, u)=Ad(g)u,$
$C^{\infty}$
$ue\mathfrak{g}$
$(g, h)\mapsto ghg^{-1}$ PROOF. As is obvious by definition. For every $g\in G,$ isomorphism. Thus, we have only to show This is given as follows: $G\times G\rightarrow G,$
$C^{\infty}$
, the second assertion is is obviously a linear
$Ad(g):\mathfrak{g}\rightarrow \mathfrak{g}$
$Ad(g)[u, v]=[Ad(g)u, Ad(g)v]$
.
$Ad(g)[u, v]=Ad(g)\frac{d}{dt}|_{t=0}Ad(c(t))v$
$=\frac{d}{dt}|_{t=0}Ad(gc(t))v$
$=\frac{d}{dt}|_{=0}Ad(gc(t)g^{-1})Ad(g)v$
$=[Ad(g)u, Ad(g)v]$
,
is a curve in $G$ such that $c(O)=e$ and $\dot{c}(0)=u$ . equality follows from (6), for $(d/dt)gc(t)g^{-1}|_{t=0}=Ad(g)u$ . where
$c(t)$
$C^{\infty}$
The last
Remark that it is not known whether there is the exponential mapping on every FL-group. Namely one might not be able $exp:\mathfrak{g}\rightarrow G$
375
REGULAR FR\’ECHET-LIE GROUPS
to solve the equation
$(d/dt)g(t)=u(t)\cdot g(t)$
.
However, one can get the
following uniqueness theorem:
LEMMA 2.5. Let $u(t)$ be a -valued continuous function on Then, the uniqueness holds for the equation $\mathfrak{g}$
$\frac{d}{dt}g(t)=u(t)\cdot g(t)$
PROOF.
Let
,
$g(O)=e$
$h(O)=e$
.
Then, by
.
$\frac{d}{dt}h(t)^{-1}g(t)=-dL_{htt)}-1dR_{g(t)}u(t)+dL_{h(t)}-1dR_{g(t)}u(t)\equiv 0$
Hence by the mean value theorem (1), we have Let
$g(t)$
be a
$(dg(t)/dt)\cdot g(t)^{-1}\in \mathfrak{g}$
$C^{1}$
.
curve in
LEMMA 2.6. Let $w(t)$ be a -valued above $u(t)$ , the differential equation
$C^{0}$
$\mathfrak{g}$
$\frac{d}{dt}v(t)-[u(t), v(t)]=w(t)$
has a unique solution
PROOF.
$h(t)^{-1}g(t)\equiv e$
defined on
$G$
$\lceil 0,1$
].
function on ,
.
.
be another solution such that
$h(t)$
$[0,1]$
.
$\square $
We set $[0,1]$
.
$u(t)=$
For the
$v(O)=0$
$Ad(g(t))\int_{0}^{t}Ad(g(s)^{-1})w(s)ds$
.
By using Lemma 2.3, we see easily that $Ad(g(t))\int_{0}^{t}Ad(g(s)^{-1})w(s)ds$
is a solution. Suppose there is another solution . Then satisfies $(d/dt)(v(t)-\overline{v}(t))=[u(t), v(t)-\overline{v}(t)]$ and $v(O)-\overline{v}(0)=0$ . Compute $(d/dt)Ad(g(t)^{-1})(v(t)-\overline{v}(t))$ , by using Lemma 2.3, and we 8ee easily that it is identically . Hence by the mean value theorem (1), we have $\overline{v}(t)$
$v(t)-\overline{v}(t)$
$0$
$v(t)=\overline{v}(t)$
.
$\square $
\S 3. Product integrals and the definition of regular
Fr\’echet-Lie
groups.
we start with considering a division $\Delta:a=t_{0}
