Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 20 (Lie Groups)

For $X\in \mathfrak{g}=T_{e}G$, let $X^L$ be the left-invariant vector field with value $X$ at the identity. Its integral curve through the identity is the unique one-parameter subgroup ${\gamma }_X$ with ${\gamma }_X^{\prime }(0)=X$. It exists for all real time: local existence followed by translation and the identity ${\gamma }_X(t+s)={\gamma }_X(t){\gamma }_X(s)$ extends it indefinitely. The Exponential map of a Lie group isIt is smooth and $d_0\exp ={\mathrm{id}}_{\mathfrak{g}}$, because differentiating $t\mapsto \exp (tX)$ at zero gives $X$. The inverse function theorem therefore makes it a local diffeomorphism at zero.

If $G$ is abelian, the product $t\mapsto \exp (tX)\exp (tY)$ is a one-parameter subgroup with initial tangent $X+Y$. Uniqueness gives $\exp (t(X+Y))=\exp (tX)\exp (tY)$, henceConversely, if this additive homomorphism identity holds, all $\exp (tX)$ and $\exp (sY)$ commute. Differentiating their adjoint action shows $[X,Y]=0$. Alternatively, their commuting images contain an identity neighbourhood; that neighbourhood generates the identity component of a Lie group, so the identity component is abelian. Conversely, if the Lie algebra is abelian, its left-invariant vector fields commute, so their flows commute. Hence all one-parameter subgroups commute; their product with initial tangent $X+Y$ is the one-parameter subgroup for $X+Y$. The Lie exponential map is therefore additive, and its image generates an abelian identity component. Thus the stated equivalence holds for connected $G$. Without connectedness the exact assertion is that the Lie exponential map is a homomorphism precisely when $G^{\circ }$ is abelian. The converse implication to all of $G$ is false: in $O(2)$ the Lie algebra consists of $tJ$, whereRotations satisfy $R_{t+s}=R_t{R}_s$, but a reflection $S$ satisfies $S{R}_t{S}^{-1}=R_{-t}$, so the full orthogonal group is not abelian.

Now take a connected abelian Lie group of dimension $n$. Its Lie exponential map is a homomorphism whose image contains an identity neighbourhood, hence an open subgroup. A connected group has no proper open subgroup, so this map is onto. Its kernel $\Lambda$ is discrete, since it is injective on a neighbourhood of zero. The induced map is a bijective local diffeomorphism and therefore a Lie group isomorphismHere is the needed classification of $\Lambda$. Let $r$ be the dimension of its real span. Choose ${\lambda }_1,\dots ,{\lambda }_r\in \Lambda$ that form a real basis of that span. Their integer span ${\Lambda }_0$ has finite index in $\Lambda$: subtract integer multiples of the ${\lambda }_i$ to put each coset representative in a compact parallelepiped. Only finitely many points of $\Lambda$ lie there, since discreteness at zero gives a uniform positive separation between distinct points. Thus $\Lambda$ is finitely generated and torsion-free, and the structure theorem for finitely generated abelian groups makes it free abelian of rank $r$. Its integer basis is a real basis of the span. A linear change of coordinates consequently sends $\Lambda$ to ${\mathbb{Z}}^r\times \{0\}$ in ${\mathbb{R}}^n$.

This proves the classification of connected abelian Lie groups:The isomorphism depends on the choice of lattice basis and complementary vector space. Compactness is equivalent to $s=0$.

A complex analytic Lie group is a complex manifold whose multiplication and inversion are holomorphic maps between complex manifolds. Its identity tangent space is a complex Lie algebra, and the Adjoint representation of a Lie groupis holomorphic: differentiating the holomorphic conjugation map with respect to its second variable gives holomorphically varying matrix entries.

Every scalar holomorphic function on a compact connected complex manifold is constant. Indeed its modulus attains a maximum; the maximum modulus principle, applied in complex coordinate charts and along complex lines, makes it locally constant there, and connectedness propagates that constancy. Apply this to each entry of $\mathrm{Ad}$. Since ${\mathrm{Ad}}_e=I$, it follows that ${\mathrm{Ad}}_g=I$ for every $g$. Differentiation givesThus $\mathfrak{g}$ is abelian, and the connected group $G$ is abelian by the identity-neighbourhood argument in part (i). This proves that compact connected complex Lie groups are complex tori.

If $n={\dim }_{\mathbb{C}}G$, identify $\mathfrak{g}$ with ${\mathbb{C}}^n$. The Lie exponential map is holomorphic, additive, onto, and a local biholomorphism, by the holomorphic left-invariant differential equation and its identity derivative. Its kernel $B$ is a discrete subgroup of the additive space ${\mathbb{C}}^n$. The induced map yieldsas complex Lie groups, not merely as real groups. Compactness further forces $B$ to be a full-rank real lattice. If its real span $W$ were proper, projection would give a surjective continuous map ${\mathbb{C}}^n/B\to {\mathbb{R}}^{2n}/W$, contradicting compactness. The discrete-subgroup argument in part (i) therefore gives $B\cong {\mathbb{Z}}^{2n}$. The quotient is a complex torus; it need not split into one-dimensional complex tori.

Interpret $\theta$ as a continuous finite-dimensional complex group representation, so its values lie in ${\mathrm{GL}}_{\mathbb{C}}(V)$, and interpret approximation in the uniform norm. Assume the usual Hausdorff convention for the compact topological group. The following argument proves the required form of the Peter-Weyl theorem without first assuming that $G$ is a Lie group.

Start with normalized Haar measure. It is both left- and right-invariant on a compact group. Average any Hermitian inner product to make a finite-dimensional representation unitary. The left regular action $L_{a}h(x)=h(a^{-1}x)$ is already unitary on $L^2(G)$.

Choose a continuous nonnegative function $k$ supported in a small symmetric identity neighbourhood, with $k(z^{-1})=k(z)$ and $\int k=1$. Such functions are obtained from a nonzero continuous bump near the identity, symmetrizing and normalizing. DefineThe convolution operator $T_k$ commutes with left translations. Its continuous kernel is Hermitian, hence it is self-adjoint. It is compact on $L^2(G)$: approximate the continuous kernel uniformly by finite sums of separated functions of $x$ and $y$, using the Stone-Weierstrass theorem on the compact product. The corresponding operators have finite rank and converge in operator norm. It also maps into $C(G)$ and satisfiesby the Cauchy-Schwarz inequality; continuity follows from the uniform continuity of the translated kernel.

The spectral theorem for compact self-adjoint operators decomposes the orthogonal complement of its kernel into finite-dimensional eigenspaces with nonzero eigenvalues. Every such eigenspace is left-translation invariant, and all its elements are continuous, since $h=T_{k}h/\lambda$ there. If $W\subset C(G)$ is one of these invariant spaces, write $\rho (a)=L_a{|}_W$. Evaluation at the identity givesThus $h$ is a matrix coefficient of the dual representation. This identifies each finite spectral sum with finite-dimensional representation data.

To get uniform rather than just $L^2$ approximation, let $S_N$ project onto increasing finite sums of the nonzero eigenspaces. For $f\in C(G)$, $T_{k}f$ is orthogonal to the kernel, so $S_N{T}_{k}f\to T_{k}f$ in $L^2$. Applying the displayed smoothing bound givesMeanwhile, choosing the support of $k$ sufficiently small makes $T_{k}f$ uniformly close to $f$, by uniform continuity under small translations. Since $\Vert T_k{\Vert }_{C\to C}\le 1$, it also makes $T_k^{2}f$ uniformly close to $f$. Every $T_k{S}_N{T}_{k}f$ belongs to a finite sum of invariant eigenspaces, hence is a finite sum of matrix coefficients. This completes the convolution proof of uniform Peter-Weyl approximation.

A coefficient $\ell (\theta (g)v)$ equals $\mathrm{tr}(\alpha \theta (g))$ for the rank-one endomorphism $\alpha =v\otimes \ell$. Finite sums of coefficients can be combined using a direct-sum representation and a block-diagonal $\alpha$. Therefore the conclusion has exactly the requested form:

For a class function $f$, apply conjugation averaging on a compact group,It fixes $f$ and is a contraction in the uniform norm, so averaging an approximation preserves its error bound. A trace coefficient averages to $\mathrm{tr}(\overline{\alpha }\theta (g))$, whereThis endomorphism commutes with the representation. A finite-dimensional unitary representation splits into irreducibles by repeatedly taking invariant orthogonal complements. On each irreducible diagonal block, Schur lemma makes its average a scalar multiple of the identity, namely its trace divided by the block dimension. Off-diagonal blocks do not contribute to the total trace. Thus the averaged coefficient is a finite linear combination of irreducible characters. Consequently irreducible complex characters span a uniformly dense subspace of the continuous class functions.

By part (i), finite-dimensional continuous unitary representations separate points of the compact group. More explicitly, if every representation sent some $g\ne e$ to the identity, all their matrix coefficients would take the same values at $g$ and $e$. Uniform density would force every continuous scalar function to do so, contradicting separation of points on a compact Hausdorff space. HenceEach representation kernel is a closed normal subgroup. We claim that the intersection can be reduced to finitely many kernels. Begin with $K_0=G$. If $K_j\ne \{e\}$, choose $g_j\in K_j\setminus \{e\}$ and a representation ${\theta }_{j+1}$ with ${\theta }_{j+1}(g_j)\ne I$. ThenAn indefinite continuation would violate the descending chain condition for closed subgroups. Thus after finitely many steps, $K_N=\{e\}$.

The direct sum $\rho ={\theta }_1\oplus \cdots \oplus {\theta }_N$ is consequently a finite-dimensional faithful representation, with continuous injective homomorphismCompactness of $G$ makes this a homeomorphism onto a compact, hence closed, image. The closed-subgroup theorem makes that image an embedded Lie subgroup of the finite-dimensional unitary group. Transporting its smooth structure to $G$ provesThis is the mechanism behind the assertion that compact groups with a descending chain condition are Lie groups: the chain condition turns an arbitrarily large separating family into one finite-dimensional faithful representation.

The Lie group–Lie algebra correspondence separates local differential structure from global topology. For a Lie group $G$, define $LG=T_{e}G$ as a real vector space. Extend each $X\in T_{e}G$ to the left-invariant vector fieldThe commutator of two such vector fields is again left-invariant. SetThe vector-field commutator is bilinear, antisymmetric and satisfies the Jacobi identity, so this makes $T_{e}G$ a Lie algebra. For a matrix group this bracket is $XY-YX$.

For a smooth Lie group homomorphism $\varphi :G_1\to G_2$, its differential at the identity is the linear mapThe identity $\varphi \circ L_g=L_{\varphi (g)}\circ \varphi$ shows that the corresponding left-invariant fields are $\varphi$-related. Brackets of related vector fields are related, henceThe chain rule also gives $L(\psi \circ \varphi )=L\psi \circ L\varphi$ and $L(\mathrm{id})=\mathrm{id}$. Thus differentiation is a functor from Lie groups and smooth homomorphisms to real Lie algebras and bracket-preserving linear maps.

To see what information the differential retains, apply $\varphi$ to a one-parameter subgroup. Its initial tangent is $L\varphi (X)$, so uniqueness yieldsIf $G_1$ is connected, a small exponential neighbourhood generates it. Every $g\in G_1$ is a finite product $\exp X_1\cdots \exp X_k$, and thereforeFor an already existing homomorphism this expression is independent of the chosen product because it equals $\varphi (g)$. Two homomorphisms with the same differential agree on that neighbourhood and hence everywhere. This proves that a Lie group homomorphism determined by its differential requires a connected source.

On a disconnected source only the restriction to $G_1^{\circ }$ is determined. For instance a nontrivial finite discrete group has zero Lie algebra, so its identity and trivial endomorphisms have the same zero differential but are different maps. The discussion of all components must therefore include additional global data. The remaining parts explain the corresponding existence and topology conditions.

An isomorphism of Lie algebras identifies the local multiplication laws of the identity components. In exponential coordinates the local product is governed by the Baker--Campbell--Hausdorff formula,A bracket-preserving linear isomorphism carries this law to the corresponding law in the other Lie algebra. Thus the identity components are locally isomorphic. This statement is local; the exponential coordinates need not cover the whole group.

The stronger global description uses a universal covering Lie group. The universal cover of a connected Lie group acquires a group law by lifting multiplication with a chosen identity; uniqueness of lifts supplies associativity and inversion. The covering map is a local diffeomorphism and a Lie group homomorphism, with discrete kernel. That kernel is central: conjugation of one of its elements gives a continuous map from the connected covering group into a discrete set, so it is constantly that element.

By integration of the given Lie algebra isomorphism and its inverse on simply connected covers, these covers are isomorphic. Their two compositions have identity differential and therefore are identity maps by the preceding uniqueness argument. Consequently, for some connected simply connected Lie group $\tilde{G}$,where ${\Gamma }_1,{\Gamma }_2$ are discrete central subgroups. Different quotients need not be isomorphic. The additive group $\mathbb{R}$ and the circle group $\mathbb{R}/2\pi \mathbb{Z}$ have the same one-dimensional Lie algebra but different compactness and fundamental groups. Another familiar pair is the SU(2) group and the SO(3) group, with the latter the quotient of the former by $\{I,-I\}$.

Thus isomorphic Lie algebras give isomorphic connected simply connected groups, and identify arbitrary identity components up to discrete central quotients. For disconnected groups the Lie algebra also omits the group of components and its conjugation and extension data. For example $\mathbb{R}\times F$, for any finite group $F$, always has Lie algebra $\mathbb{R}$, while its component group is $F$.

For a finite-dimensional real Lie algebra $\mathfrak{h}$, the Lie third theorem asserts the existence of a connected simply connected Lie group whose Lie algebra is $\mathfrak{h}$. There are two complementary views of the construction.

Locally, the Baker--Campbell--Hausdorff formula defines multiplication on a sufficiently small neighbourhood of zero in $\mathfrak{h}$. Its leading terms are $X+Y+\frac{1}{2}[X,Y]$, its identity is zero, and inversion is $X\mapsto -X$. The convergent BCH series in finite dimensions and the Jacobi identity provide the local associative law. Continuing and globalizing this local group gives its simply connected integration; the globalization is a substantive step, rather than a claim that one exponential chart gives all of the group.

A matrix construction makes that global step more concrete. The Ado theorem gives a faithful finite-dimensional representationConsider the left-invariant distribution $D_A=A\iota (\mathfrak{h})$ on ${\mathrm{GL}}_N(\mathbb{R})$. It is involutive because $\iota (\mathfrak{h})$ is closed under commutators. The Frobenius theorem integrates it to its connected leaf through the identity. This leaf carries the intrinsic immersed-submanifold structure of the connected Lie subgroup generated by the matrix exponentials $e^{\iota (X)}$. Its tangent Lie algebra is exactly $\iota (\mathfrak{h})$; multiplication and inversion preserve the leaf and give its Lie group operations. Call this immersed group $H_0$.

Taking its universal covering Lie group givesAdo's faithfulness matters: the adjoint representation alone loses central elements. Also, the subgroup generated by the exponentials need not be closed in the ambient matrix group. For an irrational $a$, the map $t\mapsto (R_t,R_{at})$ gives a dense one-dimensional immersed subgroup of the two-dimensional torus. Its intrinsic group is $\mathbb{R}$, whereas its closure has a two-dimensional Lie algebra. Taking an ambient closure in the construction would therefore be incorrect.

Finally, any other connected Lie group with Lie algebra $\mathfrak{h}$ is obtained from this simply connected integration by a discrete central quotient, as in part (i). The simply connected integration is unique up to isomorphism, but an arbitrary corresponding group is not.

A Lie algebra homomorphism $\theta :{\mathfrak{g}}_1\to {\mathfrak{g}}_2$ always defines a local group homomorphism near the identities by the exponential charts:for sufficiently small $X$. Preservation of the BCH multiplication law makes this a local homomorphism. The issue is extending it consistently around all paths and periods in the source.

Suppose first that $G_1$ is connected and simply connected. For a piecewise smooth path $\gamma :[0,1]\to G_1$ from $e_1$ to $g$, letbe its left logarithmic velocity. Solve in $G_2$ the differential equationThe equation exists throughout the finite path interval: bounded smooth body velocity gives a uniform local existence interval at the identity, and left translation repeatedly continues the solution. Define the proposed map by the endpoint $\Phi (g)=\eta (1)$.

Bracket preservation makes this endpoint invariant under fixed-endpoint homotopy. To see the mechanism, for a two-parameter path write $A={\gamma }^{-1}{\partial }_t\gamma$ and $B={\gamma }^{-1}{\partial }_s\gamma$, using the Maurer-Cartan form rather than literal multiplication for nonmatrix groups. The Maurer-Cartan equation givesAfter applying $\theta$, the same identity holds for $a=\theta A$ and $b=\theta B$. If $c={\eta }^{-1}{\partial }_s\eta$, the transport equation impliesBoth have zero initial value because the path starts at a fixed identity. Uniqueness gives $c=b$. At the fixed endpoint, $B(s,1)=0$, hence ${\partial }_s\eta (s,1)=0$. The endpoint is therefore unchanged by the homotopy. Simple connectedness then makes it independent of the chosen path.

For paths to $g$ and $h$, concatenate the first with the left translate by $g$ of the second. Its transported endpoint is $\Phi (g)\Phi (h)$, because left logarithmic velocity is unchanged by left translation. It follows that $\Phi (gh)=\Phi (g)\Phi (h)$. Near the identity, choose $\gamma (t)=\exp (tX)$; transport yields $\Phi (\exp X)=\exp (\theta X)$. Thus $\Phi$ is smooth near the identity and, by translation, everywhere, and its differential is $\theta$. The uniqueness argument from the root solution givesThis is integration of a Lie algebra homomorphism.

For a connected source that is not simply connected, first integrate $\theta$ on its universal cover $p:{\tilde{G}}_1\to G_1$. The resulting map $\tilde{\Phi }:{\tilde{G}}_1\to G_2$ descends exactly whenIndeed two lifts of one element differ by a kernel element, so this condition is both necessary and sufficient for their images to agree. It is the period obstruction to integration of a Lie algebra homomorphism. For $G_1=G_2=S^1$, identify the Lie algebras with $\mathbb{R}$ and take $\theta (t)=ct$. The integrated map on the cover is $t\mapsto e^{ict}$, which respects the source period $2\pi$ precisely when $c\in \mathbb{Z}$. Therefore half-scaling is a valid Lie algebra homomorphism but does not integrate to a circle homomorphism. If both groups are written as simply connected covers modulo central subgroups, the equivalent descent condition is that the integrated cover map send ${\Gamma }_1$ into ${\Gamma }_2$.

For disconnected $G_1$, first solve this problem on $G_1^{\circ }$, obtaining ${\varphi }_0$. One must then assign images $a_{\gamma }$ to representatives $s_{\gamma }$ of the component group. These assignments must satisfy the conjugation and multiplication relations. Explicitly, with $s_e=e$ and $s_{\gamma }{s}_{\delta }=s_{\gamma \delta }{c}_{\gamma ,\delta }$ for $c_{\gamma ,\delta }\in G_1^{\circ }$, the conditions areWhen these can be solved, $\varphi (s_{\gamma }h)=a_{\gamma }{\varphi }_0(h)$ is a smooth global homomorphism. They may prevent existence or allow several extensions with the same differential. Thus a bracket-preserving linear map gives local data; global integration requires the source's periods and, when present, its component relations to be respected.