Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 20 / 3

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.