19.4 Lie groups of transformations

中文

Let $M$ be a differentiable manifold, and $G$ a Lie group. By an action of $G$ on $M$ we mean a differentiable map $\varphi :G\times M\to M$ , often denoted $\varphi (g,x)=gx$ such that

(i) $ex=x$ for all $x\in X$ , where $e$ is the identity element in $G_{\ast }$

(ii) $(gh)x=g(hx).$

This agrees with the conventions of a left action as defined in Section 2.6 and we refer to $G$ as a Lie group of transformations of $M$. We may, of course, also have right actions $(g,x)\mapsto xg$ defined in the natural way.

Exercise

For any fixed $g\in G$ show that the map ${\varphi }_g:M\to M$ defined by ${\varphi }_g(x)=\varphi (g,x)=gx$ is a diffeomorphism of $M$.

The action of $G$ on $M$ is said to be effective if $e$ leaves every point $x\in M$ fixed,

$$gx=x\text{ for all }x\in M⟹g=e.$$

As in Section 2.6 the orbit $Gx$ of a point $x\in M$ is the set $Gx=\left\{gx\mid g\in G\right\}$ , and the action of $G$ on $M$ is said to be transitive if the whole of $M$ is the orbit of some point in

$M$. In this case $M=Gy$ for all $y\in M$ and it is commonly said that $M$ is a homogeneous manifold of $G$

Example 19.9

Any Lie group $G$ acts on itself by left translation $L_g:G\to G$ , in which case the map $\varphi :G\times G\to G$ is defined by $\varphi (g,h)=gh=L_{g}h$ . The action is both effec tive and transitive. Similarly $G$ acts on itselfto the right with right translations $R_g:G\to G$ where $R_{g}h=hg$

Let $H$ be a closed subgroup of a Lie group $G,$ , and $\pi :G\to G/H$ be the natural map sending each element of $g$ to the left coset to which it belongs, $\pi (g)=gH$ . As in Section 10.8 the factor space $G/H$ is given the natural topology induced by $\pi$. Furthermore, $G/H$ has a unique manifold structure such that $\pi$ is $C^{\infty }$ and $G$ is a transitive Lie transformation group of $G/H$ under the action

$$\varphi (g,hH)=g(hH)=ghH.$$

A proofofthis non-trivial theorem may be found in $[6,\mathsf{p}.120]$ . The key result is the existence everywhere on $G$ of local sections; every coset $gH\in G/H$ has a neighbourhood $W$ and a smooth map $\alpha :W\to G$ with respect to the differentiable structure on $G/H$ such that $\pi \circ \alpha =\mathrm{id}\mathrm{on}\alpha (W)$

Every homogeneous manifold can be cast in the form of a left action on a space ofcosets. Let $G$ act transitively to the left on the manifold $M$ and for any point $x\in M$ define the map ${\varphi }_x:G\to M$ by ${\varphi }_x(g)=gx$ . This map is smooth, as it is the composition of two smooth maps ${\varphi }_x=\varphi \circ i_x$ where $i_x:G\to G\times M$ is the injection defined by $i_x(g)=(g,x)$ . The isotropy group $G_x$ of $x,$ , defined in Section 2.6 as $G_x=\{g|gx=x\}$ , is therefore a closed subgroup of $G$ since it is the inverse image of a closed singleton set, $G_x={\varphi }_x^{-1}(\{x\})$ . Let the map $\rho :G/G_x\to M$ be defined by $\rho (g{G}_x)=gx$ . This map is one-to-one, for

$$\begin{array}{c}\rho (gGx)=\rho (g^{\prime }{G}_x)⟹gx=g^{\prime }x\\ ⟹g^{-1}{g}^{\prime }\in G_x\\ ⟹g^{\prime }\in g{G}_x\\ ⟹g^{\prime }{G}_x=g{G}_x.\end{array}$$

Furthermore, with respect to the differentiable structure induced on $G/G_x$ , the map $\rho$ is $C^{\infty }$ since for any local section $\alpha :W\to G$ on a neighbourhood $W$ of a given coset $g{G}_x$ we can write $\rho =\pi \circ \alpha$ , which is a composition ofsmooth functions. Since $\rho (gK)=g(\rho (K))$ for all cosets $K=h{G}_x\in G/G_x$ , the group $G$ has the ‘same action’ on $G/G_x$ as it does on $M$.

Exercise

Show that $\rho$ is a continuous map with respect to the factor space topology induced on $G/G_x$

Example 19.10

The orthogonal group $O(n+1)$ acts transitively on the unit $n$-sphere

$$S^n=\left\{\mathbf{x}\in {\mathbb{R}}^{n+1}\mid (x^1{)}^2+(x^2{)}^2+\cdots +(x^{n+1}{)}^2=1\right\}\subset {\mathbb{R}}^{n+1},$$

since for any $\mathbf{x}\in S^n$ there exists an orthogonal transformation $A$ such that $\mathbf{x}=\mathsf{A}\mathbf{e}$ where $\mathbf{e}=(0,0,\dots ,0,1)$ . In fact any orthogonal matrix with the last column having the same components as $x$, i.e. $A_{n+1}^i=x^i$ , will do. Such an orthogonal matrix exists by a Schmidt orthonormalization in which $x$ is transformed to the $(n+1)$ )th unit basis vector

Let $H$ be the isotropy group of the point $e$, consisting of all matrices of the form

$$\left(\begin{array}{ccc}& & 0\\ [B_j^i]& & ⋮\\ & & 0\\ \dots & 0& 1\end{array}\right)\text{ where }[B_j^i]\text{ is }n\times n\text{ orthogonal. }$$

Hence $H\cong O(n)$ . The map $O(n+1)/H\to S^n$ defined by $\mathsf{A}H\mapsto \mathsf{A}\mathbf{e}$ is clearly one-toone and continuous with respect to the induced topology on $O(n+1)/H$ . Furthermore, $O(n+1)/H$ is compact since it is obtained by identification from an equivalence relation on the compact space $O(n+1)$ (see Example 10.16). Hence the map $O(n+1)/H\to S^n$ is a homeomorphism since it is a continuous map from a compact Hausdorff space onto a compact space (see Problem 10.17). Smoothness follows from the general results outlined above. Thus $O(n+1)/O(n)$ is diffeomorphic to $S^n$ , and similarly it can be shown that $SO(n+1)/SO(n)\cong S^n$

Example 19.11

The group of matrix transformations ${\mathbf{x}}^{\prime }=\mathsf{L}\mathbf{x}$ leaving invariant the inner product $\mathbf{x}\cdot \mathbf{y}=x^1{y}^1+x^2{y}^2+x^3{y}^3-x^4{y}^4$ of Minkowski space is the Lorentz group $O(3,1)$ . The group of all transformations leaving this form invariant including translations,

$$x^{\prime i}=L_j^i{x}^j+b^j,$$

is the Poincaré group $P(4)$ (see Example 2.30 and Chapter 9). The isotropy group of the origin is clearly $O(3,1)$ . The factor space $P_4/O(3,1)$ is diffeomorphic to ${\mathbb{R}}^4$ , for two Poincaré transformations $P$ and $P^{\prime }$ belong to the same coset if and only if their translation parts are identical,

$$\begin{array}{c}{P}^{-1}{P}^{\prime }\in O(3,1)⟺{\mathsf{L}}^{-1}({\mathsf{L}}^{\prime }\mathbf{x}+{\mathbf{b}}^{\prime })-{\mathsf{L}}^{-1}\mathbf{b}=\mathsf{Kx}\text{ for }\mathsf{K}\in O(3,1)\\ ⟺{\mathbf{b}}^{\prime }=\mathbf{b}.\end{array}$$

Normal subgroups

Theorem 19.5 · Lie group quotients by closed normal subgroups

Let $H$ be a closed normal subgroup of a Lie group $G$. Then the factor group $G/H$ is a Lie group.

Proof

The map $\Psi :(aH,bH)\mapsto a{b}^{-1}H$ from $G/H\times G/H\to G/H$ is $C^{\infty }$ with respect to the natural differentiable structure on $G/H$ . For, if $(K_a,{\alpha }_a:K_{a}G)$ and $(K_b,{\alpha }_b:K_{b}G)$ are any pair of local sections at $a$ and $b$ then on $K_a\times K_b$

$$\Psi =\pi \circ \psi \circ ({\alpha }_a\times {\alpha }_b)$$

where $\psi :G\times G\to G$ is the map $\psi (a,b)=a{b}^{-1}$ . Hence is everywhere locally a composition of smooth maps, and is therefore $C^{\infty }$ -

Now suppose $\phi :G\to H$ is any Lie group homomorphism, and let $N={\phi }^{-1}(\{e^{\prime }\})$ ) be the kernel of the homomorphism, where $e^{\prime }$ is the identity of the Lie group $H$. It is clearly a closed subgroup of $G.$ . The tangent map ${\phi }_{\ast }:\mathcal{GH}$ induced by the map $\phi$ is a Lie algebra homomorphism. Its kernel $\mathcal{N}=({\phi }_{\ast }{)}^{-1}(0)$ is an ideal of $\mathcal{G}$ and is the Lie algebra corresponding to the Lie subgroup $N$, since

$$\begin{array}{rl}Z\in \mathcal{N}& ⟺{\phi }_{\ast }Z=0\\ & ⟺\exp t{\phi }_{\ast }Z=e^{\prime }\\ & ⟺\phi (\exp tZ)=e^{\prime }\text{ by Eq. (19.10) }\\ & ⟺\exp tZ\in N.\end{array}$$

Thus, if $N$ is any closed normal subgroup of $G$ then it is the kernel of the homomorphism $\pi :G\to H=G/N_{}$ , and its Lie algebra is the kernel of the Lie algebra homomorphism ${\pi }_{\ast }:\mathcal{GH}.$ . That is, $\mathcal{H}\cong \mathcal{G}/\mathcal{N}$

Example 19.12

Let $G$ be the additive abelian group $G={\mathbb{R}}^n$ , and $H$ the discrete subgroup consisting of all points with integral coordinates. Evidently $H$ is a closed normal subgroup of $G_{:}$ , and its factor group

$$T^n={\mathbb{R}}^n/H$$

is the $n$-dimensional torus (see Example 10.14). In the torus group two vectors are identified if they differ by integral coordinates, $[\mathbf{x}]=[\mathbf{y}]$ in $T^n$ if and only if $\mathbf{x}-\mathbf{y}=(k_1,k_2,\dots ,k_n)$ where $k_i$ are integers. The one-dimensional torus $T=T^1$ is diffeomorphic to the unit circle in ${\mathbb{R}}^2$ , and the $n$-dimensional torus group is the product of one-dimensional groups $T^n=T\times I\times \cdot \cdot \times T$ . It is a compact group.

Induced vector fields

Let $G$ be a Lie group of transformations of a manifold $M$ with action to the right defined by a map $\rho :G\times M\to M$ . We set $\rho (p,g)=pg$ , with the stipulations $pe=p$ and $p(gh)=(pg)h$ . Every left-invariant vector field $X$ on $G$ induces a vector field $\tilde{X}$ on $M$ by setting

$${\tilde{X}}_{p}f=\frac{\mathrm{d}}{\mathrm{d}t}f(p\exp tX){|}_{t=0}(p\in M)\text{\hspace{1em}(19.12)}$$

for any smooth function $f:M\to \mathbb{R}$ . This is called the vector field induced by the left invariant vector field $X$.

Exercise

Show that $\tilde{X}$ is a vector field on $M$ by verifying linearity and the Leibnitz rule at each point $p\in M_{:}$ , Eqs. (15.3) and (15.4).

Theorem 19.6 · Induced vector fields preserve Lie brackets

Lie brackets of a pair ofinduced vectorfields correspond to the Lie prod ucts of the corresponding left-invariant vectorfields,

$$\widetilde{[X,Y]}=[\tilde{X},\tilde{Y}].\text{\hspace{1em}(19.13)}$$

Proof

Before proceeding with the main part of the proof, we need an expression for $[X,Y{]}_e$ . Let ${\sigma }_y$ be a local one-parameter group of transformations on $G$ generated by the vector field $X$. By Eq. (19.6),

$${\sigma }_t(g)={\sigma }_t(ge)=g{\sigma }_t(e)=g\exp tX=R_{\exp tX}(g)$$

where the operation $R_h$ is right translation by $h$. Hence

$$[X,Y{]}_e=\underset{t\to 0}{\lim }\frac{1}{t}[Y_e-\left({\sigma }_{t\ast }Y{)}_e\right]=\underset{t\to 0}{\lim }\frac{1}{t}[Y_e-(R_{\exp (tX)\ast }Y{)}_e].\text{\hspace{1em}(19.14)}$$

Define the maps ${\rho }_p:G\to M$ and ${\rho }_g:M\to M$ for any $p\in M,g\in G$ by

$${\rho }_p(g)={\rho }_g(p)=\rho (p,g)=pg.$$

Then

$${\tilde{X}}_p={\rho }_{p\ast }{X}_e\text{\hspace{1em}(19.15)}$$

for if $f$ is any smooth function on $M$ then, on making use of Eq. (19.8), we have

$$\begin{array}{c}{\rho }_{p\ast }{X}_{e}f=X_e(f\circ {\rho }_p)=\frac{\mathrm{d}}{\mathrm{d}t}(f\circ {\rho }_p(\exp tX)){|}_{t=0}\\ =\frac{\mathrm{d}}{\mathrm{d}t}(f(p\exp tX)){|}_{t=0}={\tilde{X}}_{p}f.\end{array}$$

The maps ${\tilde{\sigma }}_t:M\to M$ defined by

$${\tilde{\sigma }}_t(p)=p\exp tX={\rho }_{\exp tX}(p)$$

form a one-parameter group of transformations of $M$ since, using Eq. (19.9),

$${\tilde{\sigma }}_{t+s}(p)={\tilde{\sigma }}_t\circ {\tilde{\sigma }}_t(p)$$

for all $t$ and $s$. By Eq. (19.12) they induce the vector field $\tilde{X}.$ , whence

$$[\tilde{X},\tilde{Y}{]}_p=\underset{t\to 0}{\lim }\frac{1}{t}[{\tilde{Y}}_p-({\tilde{\sigma }}_{t\ast }\tilde{Y}{)}_p].\text{\hspace{1em}(19.16)}$$

Applying the definition of ${\tilde{\sigma }}_t$ we have

$$\begin{array}{rl}& {\left({\tilde{\sigma }}_{t\ast }\tilde{Y}\right)}_p={\left({\rho }_{\exp (tX)\ast }\tilde{Y}\right)}_p\\ & ={\rho }_{\exp (tX)\ast }{\rho }_p\exp (-tX)\ast Y_e\\ & ={\left({\rho }_{\exp tX}\circ {\rho }_p\exp (-tX)\right)}_{\ast }{Y}_e.\end{array}$$

The map in the brackets can be written

$${\rho }_{\exp tX}\circ {\rho }_{p\exp (-tX)}={\rho }_p\circ R_{\exp tX}\circ L_{\exp (-tX)}$$

since

$${\rho }_{\exp tX}\circ {\rho }_{p\exp (-tX)}(g)=p\exp (-tX)g\exp tX={\rho }_p\circ R_{\exp tX}\circ L_{\exp (-tX)}(g).$$

Hence, since $Y$ is left-invariant

$$\begin{array}{rl}& {\left({\tilde{\sigma }}_{t\ast }\tilde{Y}\right)}_p={\rho }_{p\ast }\circ R_{\exp (tX)\ast }\circ L_{\exp (-tX)\ast }{Y}_e\\ & ={\rho }_{p\ast }\circ R_{\exp (tX)\ast }\left(Y_{\exp (-tX)}\right)\\ & ={\rho }_{p\ast }(R_{\exp (tX)\ast }{Y}_{\exp (-tX)}{)}_e.\end{array}$$

Since ${\tilde{Y}}_p={\rho }_{p\ast }{Y}_e$ by Eq. (19.15), substitution in Eq. (19.16) and using Eq. (19.14) gives

$$\begin{array}{rl}& [\tilde{X},\tilde{Y}{]}_p={\rho }_{p\ast }({\lim }_{t\to 0}\frac{1}{t}[Y_e-(R_{\exp (tX)\ast }Y{)}_e\left]\right)\\ & ={\rho }_{p\ast }[X,Y{]}_e={\widetilde{[X,Y]}}_p,\end{array}$$

which proves Eq. (19.13).

Problems

Problem 19.11

Show that a group $G$ acts effectively on $G/H$ if and only if $H$ contains no norma subgroup of $G$. [Hint: The set of elements leaving all points of $G/H$ fixed is $\bigcap {}_{a\in G}aH{a}^{-1}$ .]

Problem 19.12

Show that the special orthogonal group $SO(n)$ , the pseudo-orthogonal group $O(p,q)$ and the symplectic group $Sp(n)$ are all closed subgroups of $GL(n,\mathbb{R})$

(a) Show that the complex groups $SL(n,\mathbb{C}),O(n,\mathbb{C}),U(n),SU(n)$ are closed subgroups of $GL(n,\mathbb{C})$

(b) Show that the unitary groups $U(n)$ and $SU(n)$ are compact groups.

Problem 19.13

Show that the centre $Z$ of a Lie group $G$, consisting of all elements that commute with every element $g\in G,$ , is a closed normal subgroup of $G$.

Show that the general complex linear group $GL(n+1,\mathbb{C})$ acts transitively but not effectively on complex projective $n$-space $C{P}^n$ defined in Problem 15.4. Show that the centre of $GL(n+1,\mathbb{C})$ is isomorphic to $GL(1,\mathbb{C})$ and $GL(n+1,\mathbb{C})/GL(1,\mathbb{C})$ is a Lie group that acts effectively and transitively on $C{P}^n$

Problem 19.14

Show that $SU(n+1)$ ) acts transitively on $C{P}^n$ and the isotropy group of a typical point, taken for convenience to be the point whose equivalence class contains $(0,0,\dots ,0,1)$ , is $U(n)$ Hence show that the factor space $SU(n+1)/U(n)$ is homeomorphic to $C{P}^n$ . Show similarly, that

(a) $SO(n+1)/O(n)$ is homeomorphic to real projective space $P^n$

(b) $U(n+1)/U(n)\cong SU(n+1)/SU(n)$ is homeomorphic to $S^{2n+1}$

Problem 19.15

As in Problem 9.2 every Lorentz transformation $\mathsf{L}=[L^i{}_j]$ $I$ has $\det$ $L$ = ±1 and either $L_4^4\ge 1\mathrm{or}{L}_4^4\le -1$ . Hence show that the Lorentz group $G=O(3,1)$ ) has four connected components,

$$G_0=G^{++}:\det \mathsf{L}=1,L_4^4\ge 1G^{+-}:\det \mathsf{L}=1,L_4^4\le -1$$ $$G^{-+}:\det \mathsf{L}=-1,L_4^4\ge 1G^{--}:\det \mathsf{L}=-1,L_4^4\le -1.$$

Show that the group of components $G/G_0$ is isomorphic with the discrete abelian group $Z_2\times Z_2$

Problem 19.16

Show that the component of the identity $G_0$ of a locally connected group $G$ is generated by any connected neighbourhood of the identity $e$: that is, every element of $G_0$ can be written as a product of elements from such a neighbourhood.

Hence show that every discrete normal subgroup $N$ of a connected group $G$ is contained in the centre $Z$ of $G$.

Find an example of a discrete normal subgroup of the disconnected group $O(3)$ ) that is not in the centre of $O(3)$ .

Problem 19.17

Let $\mathcal{A}$ be a Lie algebra, and $X$ any element of $\mathcal{A}$.

(a) Show that the linear operator ${\mathrm{ad}}_X:\mathcal{A}\to \mathcal{A}$ defined by ${\mathbf{ad}}_X(Y)=[X,Y]$ is a Lie algebra homomorphism of $\mathcal{A}$ into $\mathcal{G}\mathcal{L}(\mathcal{A})$ (called the adjoint representation).

(b) For any Lie group $G$ show that each inner automorphism $C_g:G\to G$ defined by $C_g(a)=ga{g}^{-1}$ (see Section 2.4) is a Lie group automorphism, and the map $\mathrm{Ad}:G\to GL(\mathcal{G})$ defined by $\mathrm{Ad}(g)=C_{g\ast }$ is a Lie group homomorphism.

(c) Show that ${\mathbf{Ad}}_{\ast }=\mathbf{ad}.$

Problem 19.18

(a) Show that the group of all Lie algebra automorphisms of a Lie algebra $\mathcal{A}$ form a Lie subgroup of $\mathrm{Aut}(\mathcal{A})\subseteq GL(\mathcal{A})$

(b) A linear operator $D:\mathcal{A}\to \mathcal{A}$ is called a derivation on $\mathcal{A}$ if $D[X,Y]=[DX,Y]+[X,DY]$ Prove that the set of all derivations of $\mathcal{A}$ form a Lie algebra, $\partial (\mathcal{A})$, which is the Lie algebra of $\mathrm{Aut}(\mathcal{A})$.