What Is a Lie Group?
When we first studied matrix groups, we adopted a definition tailored to the tools available at
the time: a group sitting inside the invertible matrices as a closed subset. That definition was
rigorous and concrete, but it was a stopgap. We promised then that the matrix groups were merely
the most visible instances of a far more general object, one that could only be named once the
language of smooth manifolds was in place. That language is now ours, and we can give the
definition the subject deserves.
Definition (Lie Group)
A Lie group is a
smooth manifold
\(G\) (without boundary) that is also a group in the algebraic sense, such that the
multiplication map \(m : G \times G \to G\) and the inversion map \(i : G \to G\), given by
\[
m(g, h) = gh, \quad i(g) = g^{-1},
\]
are both
smooth.
The multiplication map is a map out of the product manifold \(G \times G\), whose smooth structure
was constructed when we assembled
product manifolds.
Asking that \(m\) be smooth is therefore a condition we already know how to test. A Lie group is in
particular a
topological group,
since smooth maps are continuous. The
matrix Lie groups
we defined earlier are Lie groups in this sense, because a
closed subgroup of the general linear group
is an embedded submanifold on which multiplication and inversion are smooth. We
lose nothing by having started with them, and we gain the freedom to speak of groups that are not
presented as matrices at all.
We pause for a remark on notation. The group operation is written multiplicatively by
juxtaposition, except in certain abelian groups such as \(\mathbb{R}^n\), where the operation is
addition. The identity element of a general Lie group is traditionally denoted \(e\), from the
German Einselement (unit element). We follow this convention except in specific examples
with more common notations, such as \(I_n\) for the identity matrix in a matrix group or \(0\) for
the identity in \(\mathbb{R}^n\).
Verifying smoothness of both \(m\) and \(i\) can sometimes be reduced to a single check, which is
occasionally more convenient.
Proposition (Smoothness Criterion)
If \(G\) is a smooth manifold with a group structure such that the map \(G \times G \to G\)
given by \((g, h) \mapsto g h^{-1}\) is smooth, then \(G\) is a Lie group.
Proof:
Write \(\mu(g, h) = g h^{-1}\) for the given smooth map. Inversion is recovered by holding the
first argument at the identity: \(i(g) = e \cdot g^{-1} = \mu(e, g)\), which is the composition
of the map \(g \mapsto (e, g)\) into \(G \times G\) with \(\mu\). That map is smooth by the
characteristic property of the product,
its two components being the constant map \(g \mapsto e\) and the identity of \(G\), so \(i\) is
smooth. Multiplication is then recovered as \(m(g, h) = g (h^{-1})^{-1} = \mu(g, i(h))\), the
composition of \((g, h) \mapsto (g, i(h))\) with \(\mu\). Since \(i\) is now known to be
smooth, the components of that map are a projection and the composite of \(i\) with a
projection, both smooth, so the same property applies again and the composition with \(\mu\) is
smooth. Thus both \(m\) and \(i\) are smooth and \(G\) is a Lie group.
Translations
Every element of a Lie group gives rise to two diffeomorphisms of the group onto itself, and these
maps are the source of much of what makes Lie groups tractable. The principle they encode is that a
Lie group looks the same near every point. There is a global symmetry carrying any point to any
other.
Definition (Left and Right Translation)
For an element \(g\) of a Lie group \(G\), the left translation and
right translation by \(g\) are the maps \(L_g, R_g : G \to G\) defined by
\[
L_g(h) = gh, \quad R_g(h) = hg.
\]
Left translation is smooth because it factors through multiplication. Writing \(\iota_g(h) = (g,
h)\) for the inclusion of \(G\) as the slice \(\{g\} \times G\), which is smooth by the
characteristic property of the product,
we have \(L_g = m \circ \iota_g\), a composition of smooth maps. The same map for \(g^{-1}\) is a two-sided inverse, since
\(L_{g^{-1}} \circ L_g = L_{g^{-1} g} = L_e = \mathrm{Id}_G\) and likewise on the other side.
Hence \(L_g\) is a
diffeomorphism
of \(G\), with inverse \(L_{g^{-1}}\). An identical argument shows that \(R_g\) is a
diffeomorphism. We will use these maps repeatedly.
Examples
The supply of Lie groups is abundant, and most of the examples are objects we have already met.
The point of listing them is not to introduce new constructions but to recognize that the single
abstract definition above subsumes a whole catalogue of familiar groups.
The
general linear groups
\(GL(n, \mathbb{R})\) and \(GL(n, \mathbb{C})\) are Lie groups. Each is an open submanifold of the
vector space of matrices, multiplication is smooth because the entries of a product are
polynomials in the entries of the factors, and inversion is smooth by Cramer's rule. More
generally, for any finite-dimensional real or complex vector space \(V\), the group \(GL(V)\) of
invertible linear maps of \(V\) is a Lie group. A choice of basis identifies it with \(GL(n,
\mathbb{R})\) or \(GL(n, \mathbb{C})\). The change-of-basis map between two such identifications
has the form \(A \mapsto BAB^{-1}\), which is smooth, so the smooth structure on \(GL(V)\) is
independent of the basis.
The additive groups \(\mathbb{R}^n\) and \(\mathbb{C}^n\) are Lie groups, since the coordinates of
a difference are linear functions of the coordinates of the summands. The nonzero reals
\(\mathbb{R}^*\) and nonzero complex numbers \(\mathbb{C}^*\) are Lie groups under multiplication,
the former identifiable with \(GL(1, \mathbb{R})\) and the latter with \(GL(1, \mathbb{C})\). The
circle \(\mathbb{S}^1 \subseteq \mathbb{C}^*\), whose
smooth structure we built alongside the
sphere, is a Lie group under complex multiplication, called the circle group. Its
multiplication and inversion read \((\theta_1, \theta_2) \mapsto \theta_1 + \theta_2\) and \(\theta
\mapsto -\theta\) in angle coordinates.
New Lie groups arise from old ones by taking
products.
Given Lie groups \(G_1, \dots, G_k\), the product manifold \(G_1 \times \cdots \times G_k\) is a
Lie group under componentwise multiplication. The \(n\)-torus \(\mathbb{T}^n = \mathbb{S}^1 \times
\cdots \times \mathbb{S}^1\) is the prototypical example, an \(n\)-dimensional abelian Lie group.
Any group equipped with the discrete topology is a topological group, and if it is countable it is
a zero-dimensional discrete Lie group. At the opposite extreme from these
zero-dimensional examples lie the classical matrix groups
\(O(n)\), \(SO(n)\), \(U(n)\), \(SU(n)\), and the
special Euclidean group
\(SE(n)\), each of positive dimension. These too are Lie groups, a fact we will recover with new
tools once group actions are available.
Homomorphisms and the Constant-Rank Property
Maps between groups that respect the algebraic structure are the natural morphisms of the theory,
and when the groups are Lie groups we ask the maps to respect the smooth structure as well. The
algebra alone, combined with the homogeneity supplied by translations, forces a strong
differential-topological constraint. Such a map can never change its rank from one point to
another. This single fact organizes much of what follows.
Definition (Lie Group Homomorphism)
If \(G\) and \(H\) are Lie groups, a Lie group homomorphism from \(G\) to
\(H\) is a smooth map \(F : G \to H\) that is also a group homomorphism. It is called a
Lie group isomorphism if it is also a diffeomorphism. In that case its inverse
is automatically a Lie group homomorphism, and we say that \(G\) and \(H\) are
isomorphic.
We have already encountered such maps in the matrix setting, where a
homomorphism between matrix Lie groups
appeared as a continuous group homomorphism. Here we recognize it as a smooth map between abstract
Lie groups, and the definition specializes to the earlier one.
The examples are again familiar, beginning with the inclusion \(\mathbb{S}^1 \hookrightarrow
\mathbb{C}^*\). The exponential \(\exp : \mathbb{R} \to \mathbb{R}^*\), \(t \mapsto e^t\), is a
homomorphism onto the positive reals, with smooth inverse \(\log\), hence a Lie group isomorphism
of \(\mathbb{R}\) with \(\mathbb{R}^+\). A third example is \(\varepsilon : \mathbb{R} \to
\mathbb{S}^1\), \(\varepsilon(t) = e^{2\pi i t}\), a homomorphism with kernel \(\mathbb{Z}\), and
more generally \(\varepsilon^n : \mathbb{R}^n \to \mathbb{T}^n\) is a homomorphism with kernel
\(\mathbb{Z}^n\). The determinant \(\det : GL(n, \mathbb{R}) \to \mathbb{R}^*\) is a Lie group
homomorphism, since the determinant is a polynomial in the matrix entries and \(\det(AB) = (\det
A)(\det B)\). The same holds for \(\det : GL(n, \mathbb{C}) \to \mathbb{C}^*\).
One further example is important enough to name.
Definition (Conjugation and Normal Subgroups)
For \(g\) in a Lie group \(G\), the conjugation by \(g\) is the Lie group
isomorphism \(C_g : G \to G\), \(C_g(h) = ghg^{-1}\). A subgroup \(H \subseteq G\) is
normal if \(C_g(H) = H\) for all \(g \in G\).
Conjugation is smooth because multiplication and inversion are, and a direct computation shows
that it is a group homomorphism. It is an isomorphism because \(C_{g^{-1}}\) is a two-sided
inverse.
The Constant-Rank Theorem
The key structural fact about homomorphisms is the following. Its proof is a paradigm for arguments
in this subject. A property is established at the identity, and translations spread it to every
point.
Theorem (Homomorphisms Have Constant Rank)
Every Lie group homomorphism has
constant rank.
Proof:
Let \(F : G \to H\) be a Lie group homomorphism, and let \(e\) and \(\tilde{e}\) denote the
identity elements of \(G\) and \(H\). Fix an arbitrary \(g_0 \in G\). We show that
\(dF_{g_0}\) has the same rank as \(dF_e\). The homomorphism property gives, for every \(g \in G\),
\[
\begin{align*}
F\bigl(L_{g_0}(g)\bigr) &= F(g_0 g) \\\\
&= F(g_0) F(g) \\\\
&= L_{F(g_0)}\bigl(F(g)\bigr),
\end{align*}
\]
which is to say \(F \circ L_{g_0} = L_{F(g_0)} \circ F\). Taking differentials at \(e\) and
applying the chain rule,
\[
dF_{g_0} \circ d(L_{g_0})_e = d\bigl(L_{F(g_0)}\bigr)_{\tilde{e}} \circ dF_e .
\]
The translations \(L_{g_0}\) and \(L_{F(g_0)}\) are diffeomorphisms, so their differentials are
linear isomorphisms. Composing a linear map with an isomorphism on either side does not change
its rank. Hence \(dF_{g_0}\) and \(dF_e\) have the same rank. Since \(g_0\) was arbitrary, \(F\)
has constant rank.
The payoff is immediate. A bijective homomorphism is as good as an isomorphism. No separate
verification of smoothness of the inverse is needed.
Corollary (Bijective Homomorphisms Are Isomorphisms)
A Lie group homomorphism is a Lie group isomorphism if and only if it is bijective.
Proof:
A Lie group isomorphism is bijective by definition. Conversely, a bijective Lie group
homomorphism has constant rank by the theorem, and a bijective constant-rank map is a
diffeomorphism by the
global rank theorem.
Hence it is a Lie group isomorphism.
The Universal Covering Group
Every connected Lie group sits underneath a simply connected one, related to it by a covering map that
is simultaneously a homomorphism. This is the point at which the topology developed for covering spaces
feeds back into the algebraic theory. The
universal covering manifold
of a connected Lie group inherits a multiplication that turns the bare manifold into a group, and the
covering projection turns out to respect it. The construction is purely a matter of lifting the
multiplication and inversion maps through the cover and checking that the lifted operations satisfy the
group axioms.
Existence
Theorem (Existence of a Universal Covering Group)
Let \(G\) be a connected Lie group. There exists a simply connected Lie group \(\widetilde{G}\),
called the universal covering group of \(G\), together with a
smooth covering map
\(\pi : \widetilde{G} \to G\) that is also a Lie group homomorphism.
Proof:
Let \(\widetilde{G}\) be the universal covering manifold of \(G\) and let \(\pi : \widetilde{G} \to G\)
be the corresponding smooth covering map. Both exist and are determined up to isomorphism by the
existence and uniqueness of the
universal covering manifold.
A finite product of smooth covering maps is again a smooth covering map, so
\(\pi \times \pi : \widetilde{G} \times \widetilde{G} \to G \times G\) is a smooth covering map.
Write \(m : G \times G \to G\) and \(i : G \to G\) for the multiplication and inversion of \(G\), and
fix an element \(\widetilde{e}\) in the fiber \(\pi^{-1}(e)\) over the identity. Consider the smooth map
\(m \circ (\pi \times \pi) : \widetilde{G} \times \widetilde{G} \to G\). Because \(\widetilde{G} \times \widetilde{G}\)
is simply connected, the standard lifting criterion of covering-space theory produces a continuous map
\[
\begin{align*}
&\widetilde{m} : \widetilde{G} \times \widetilde{G} \to \widetilde{G}, \\\\
&\widetilde{m}(\widetilde{e}, \widetilde{e}) = \widetilde{e}, \\\\
&\pi \circ \widetilde{m} = m \circ (\pi \times \pi),
\end{align*}
\]
and the
lifting properties of covering maps
make this lift unique once its value at the basepoint is fixed. The existence of such a lift is supplied
by the covering-space theory already in place, which requires only the
simple connectivity
of the domain. Uniqueness is the tool we use actively below. Since \(\pi\) is a local diffeomorphism and
\(\pi \circ \widetilde{m} = m \circ (\pi \times \pi)\) is smooth, \(\widetilde{m}\) is smooth.
The same construction applied to \(i \circ \pi : \widetilde{G} \to G\) yields a smooth lift
\(\widetilde{i} : \widetilde{G} \to \widetilde{G}\) with \(\widetilde{i}(\widetilde{e}) = \widetilde{e}\)
and \(\pi \circ \widetilde{i} = i \circ \pi\).
Define multiplication and inversion on \(\widetilde{G}\) by \(xy = \widetilde{m}(x, y)\) and
\(x^{-1} = \widetilde{i}(x)\). The two lifting identities then read
\[
\begin{align*}
\pi(xy) &= \pi(x)\,\pi(y), \\\\
\pi(x^{-1}) &= \pi(x)^{-1},
\end{align*}
\]
for all \(x, y \in \widetilde{G}\). These say precisely that \(\pi\) will be a homomorphism once
\(\widetilde{G}\) is known to be a group, and they are the only properties of \(\widetilde{m}\) and
\(\widetilde{i}\) used in what follows.
It remains to verify the group axioms, and each one is settled by the uniqueness of lifts. For the
identity, the map \(x \mapsto \widetilde{e}\,x\) is a lift of \(x \mapsto \pi(x)\) agreeing with the
identity map of \(\widetilde{G}\) at \(\widetilde{e}\), since both send \(\widetilde{e}\) to
\(\widetilde{e}\). By uniqueness \(\widetilde{e}\,x = x\), and similarly \(x\,\widetilde{e} = x\). For
associativity, the two maps
\[
\begin{align*}
\alpha_L(x, y, z) &= (xy)z, \\\\
\alpha_R(x, y, z) &= x(yz)
\end{align*}
\]
are both lifts of the single map \((x, y, z) \mapsto \pi(x)\pi(y)\pi(z)\), and they agree at
\((\widetilde{e}, \widetilde{e}, \widetilde{e})\). Uniqueness forces \(\alpha_L = \alpha_R\). The same
pattern shows \(x^{-1}x = xx^{-1} = \widetilde{e}\). Hence \(\widetilde{G}\) is a group, \(\widetilde{m}\)
and \(\widetilde{i}\) are smooth, and \(\pi\) is a smooth covering map and a homomorphism.
Why the cover carries the algebra
Nothing in this argument computes a product by hand. The entire group structure on \(\widetilde{G}\) is
transported from \(G\) by the demand that \(\pi\) commute with multiplication, and every axiom reduces
to the statement that two maps which project to the same map downstairs and agree at one point must
coincide. Simple connectivity supplies the maps. The uniqueness of lifts supplies the equations between
them. This is the mechanism by which the topological hypothesis that loops in \(\widetilde{G}\) contract
becomes an algebraic conclusion about how products associate.
Uniqueness
The covering group is as unique as the covering manifold from which it is built.
Theorem (Uniqueness of the Universal Covering Group)
Let \(G\) be a connected Lie group. If \(\widetilde{G}\) and \(\widetilde{G}'\) are simply connected Lie
groups admitting smooth covering maps \(\pi : \widetilde{G} \to G\) and \(\pi' : \widetilde{G}' \to G\)
that are also Lie group homomorphisms, then there is a Lie group isomorphism
\(\Phi : \widetilde{G} \to \widetilde{G}'\) with \(\pi' \circ \Phi = \pi\).
Proof Sketch:
Two simply connected covers of the same connected manifold are related by a diffeomorphism commuting
with the projections, obtained by lifting one projection through the other and using uniqueness of
lifts exactly as above. This is the manifold-level uniqueness already recorded for the
universal covering manifold.
The resulting \(\Phi\) carries products to products by the same lift-uniqueness argument that built the
multiplication, so it is a bijective Lie group homomorphism, and a
bijective homomorphism is an isomorphism.
Examples
The universal covers of the torus and of the punctured complex plane are both exponential maps. In
each case the covering homomorphism is a map of a familiar shape, and the only new information is
that its domain is simply connected.
Examples:
(a) For each \(n\), the map \(\varepsilon^n : \mathbb{R}^n \to \mathbb{T}^n\),
\[
\varepsilon^n(x^1, \dots, x^n) = \left( e^{2\pi i x^1}, \dots, e^{2\pi i x^n} \right),
\]
is a Lie group homomorphism and a smooth covering map. Since the domain is simply connected, the
additive group \(\mathbb{R}^n\) is the universal covering group of the
\(n\)-torus \(\mathbb{T}^n\).
(b) The exponential homomorphism \(\exp : \mathbb{C} \to \mathbb{C}^*\),
\(\exp(z) = e^z\), is a smooth covering map. As \(\mathbb{C}\) is simply connected, it is the universal
covering group of the multiplicative group \(\mathbb{C}^*\). The single circle factor inside
\(\mathbb{C}^*\) accounts for the noncontractible loops that \(\mathbb{C}\) unwinds, just as the
circle is unwound by \(\mathbb{R}\) in the
one-dimensional case.