Group Actions
Lie groups matter for geometry because they act. A group is an abstract record of symmetry, but
symmetry is only realized when the group operates on something. The operation may rotate a sphere,
translate a space, or permute the solutions of an equation. The notion of an action makes this
precise. An action is a rule assigning to each group element a transformation of a manifold, in a
way that respects the group law. When the manifold and the group are smooth and the assignment is
smooth, the transformations are diffeomorphisms, and the group structure becomes available for
studying the space it acts on.
Definition (Left and Right Actions)
Let \(G\) be a group and \(M\) a set. A left action of \(G\) on \(M\) is a map
\(G \times M \to M\), written \((g, p) \mapsto g \cdot p\), satisfying
\[
g_1 \cdot (g_2 \cdot p) = (g_1 g_2) \cdot p, \quad e \cdot p = p,
\]
for all \(g_1, g_2 \in G\) and \(p \in M\). A right action is a map
\(M \times G \to M\), written \((p, g) \mapsto p \cdot g\), satisfying
\((p \cdot g_1) \cdot g_2 = p \cdot (g_1 g_2)\) and \(p \cdot e = p\).
A right action can always be turned into a left action by setting \(g \cdot p = p \cdot g^{-1}\),
and conversely, so results about one transfer to the other. We work with left actions unless the
situation favors the other convention. Writing \(\theta_g\) for the map \(p \mapsto g \cdot p\), the
two axioms read \(\theta_{g_1} \circ \theta_{g_2} = \theta_{g_1 g_2}\) and
\(\theta_e = \mathrm{Id}_M\).
Definition (Smooth Action)
If \(M\) is a topological space and \(G\) a topological group, an action is
continuous if the defining map \(G \times M \to M\) is continuous. In that case
\(M\) is a \(G\)-space. If in addition \(M\) is a smooth manifold, with or without boundary,
\(G\) is a Lie group, and the defining map is smooth, the action is a smooth
action.
For a smooth action, each \(\theta_g : M \to M\) is a diffeomorphism. The map \(\theta_g\) is smooth
as a restriction of the action map, and \(\theta_{g^{-1}}\) is a smooth inverse, since
\(\theta_g \circ \theta_{g^{-1}} = \theta_e = \mathrm{Id}_M\). Thus a smooth action realizes \(G\)
as a group of diffeomorphisms of \(M\), which is exactly the sense in which a Lie group can serve as
the symmetry group of a geometric structure.
Orbits, Isotropy, Transitivity, Freeness
Four pieces of standard terminology organize the way an action moves the points of \(M\) around. For
these definitions no continuity or smoothness is needed, and they apply to any action of a group on
a set.
Definition (Orbit)
For \(p \in M\), the orbit of \(p\) is the set of all its images under the action,
\[
G \cdot p = \{ g \cdot p : g \in G \}.
\]
Definition (Isotropy Group)
For \(p \in M\), the isotropy group, or stabilizer, of \(p\) is
the set of group elements fixing it,
\[
G_p = \{ g \in G : g \cdot p = p \},
\]
which is a subgroup of \(G\).
Definition (Transitive Action)
An action is transitive if for every pair of points \(p, q \in M\) there is a
\(g \in G\) with \(g \cdot p = q\). Equivalently, an action is transitive when the only orbit is
all of \(M\).
Definition (Free Action)
An action is free if the only element fixing any point is the identity.
Equivalently, an action is free when every isotropy group is trivial.
Examples
Examples:
(a) The trivial action \(g \cdot p = p\) of any Lie group on
any smooth manifold is smooth. Each orbit is a single point and each isotropy group is all of
\(G\).
(b) The natural action of the
general linear group
\(GL(n, \mathbb{R})\) on \(\mathbb{R}^n\) by \((A, x) \mapsto Ax\) is smooth, because the entries of
\(Ax\) are polynomials in the entries of \(A\) and \(x\). Any nonzero vector can be carried to any
other by an invertible matrix, so there are exactly two orbits: \(\{0\}\) and
\(\mathbb{R}^n \setminus \{0\}\).
(c) A Lie group acts smoothly on itself by
left translation.
Given \(g_1, g_2\), the unique left translation carrying \(g_1\) to \(g_2\) is \(L_{g_2 g_1^{-1}}\),
so this action is both free and transitive. More generally, if \(H\) is a Lie subgroup of \(G\),
the restriction to \(H\) gives a smooth free action of \(H\) on \(G\), in general no longer
transitive.
(d) A Lie group acts smoothly on itself by
conjugation,
\(g \cdot h = ghg^{-1}\).
(e) An action of a discrete group is smooth precisely when each individual map
\(p \mapsto g \cdot p\) is smooth. For instance \(\mathbb{Z}^n\) acts smoothly and freely on
\(\mathbb{R}^n\) by translation,
\[
(m^1, \dots, m^n) \cdot (x^1, \dots, x^n) = (m^1 + x^1, \dots, m^n + x^n).
\]
Actions arise wherever a space carries structure worth preserving. If \(M\) is a vector space, a
metric space, or a manifold with some geometric structure, the diffeomorphisms preserving that
structure typically form a Lie group acting smoothly on \(M\). The left-translation action of (c) is
the seed of much of what follows. Homomorphisms turn out to be equivariant against it, and it is the
mechanism by which one orbit's local geometry is the same as another's.
Actions from Covering Maps
Covering maps supply a class of actions that connects this page back to the topology of the
fundamental group. The deck transformations of a cover are the self-maps that permute the sheets
while leaving the projection unchanged, and they form a group acting on the total space. When the
cover is smooth this group is a zero-dimensional Lie group acting freely. This is the same machinery
that, in the special case of a simply connected cover, produced the
universal covering group.
Here the lifting properties of covers reappear as the engine driving an action.
The transitivity of the automorphism action on fibers is itself a named condition, and it is the
hypothesis under which deck transformations detect global structure such as orientability.
Definition (Normal Covering)
A covering map \(\pi : E \to M\) is a normal covering, also called a
regular covering, if the automorphism group \(\mathrm{Aut}_\pi(E)\) acts
transitively on each fiber of \(\pi\). Equivalently, \(\pi\) is normal when, for every
\(q \in E\), the image \(\pi_*\bigl(\pi_1(E, q)\bigr)\) is a
normal subgroup
of \(\pi_1\bigl(M, \pi(q)\bigr)\).
For a smooth cover, the automorphism group is small in a strong sense. It is discrete, and its
action is free.
Proof:
Each automorphism \(\varphi\) is a lift of \(\pi\) through \(\pi\) itself, in the sense that
\(\pi \circ \varphi = \pi\). If \(\varphi\) fixes a single point \(p \in E\), then \(\varphi\) and
the identity map are two lifts of \(\pi\) agreeing at \(p\), so by the
uniqueness of lifts
they coincide: \(\varphi = \mathrm{Id}_E\). Hence no nonidentity automorphism fixes any point, and
the action is free.
Countability of \(\mathrm{Aut}_\pi(E)\) is what makes the discrete topology a zero-dimensional
manifold structure. To prove it, fix \(q \in E\) and set \(p = \pi(q)\). Choose an evenly
covered neighborhood \(U\) of \(p\). Since \(E\) is
second countable,
the preimage \(\pi^{-1}(U)\) has at most countably many components, and each component contains
exactly one point of the fiber \(\pi^{-1}(p)\), so \(\pi^{-1}(p)\) is countable.
The evaluation map \(\varphi \mapsto \varphi(q)\) sends \(\mathrm{Aut}_\pi(E)\) into
\(\pi^{-1}(p)\), and it is injective because the action is free. Two automorphisms agreeing at
\(q\) are equal. Thus \(\mathrm{Aut}_\pi(E)\) is countable, and with the discrete topology it is
a zero-dimensional smooth manifold.
Smoothness of the action is the remaining point. Because \(\mathrm{Aut}_\pi(E)\) is discrete,
the action map out of \(\mathrm{Aut}_\pi(E) \times E\) is smooth precisely when every
automorphism \(\varphi : E \to E\) is itself smooth. Fix an automorphism \(\varphi\) and a point
\(x \in E\). Let \(U\) be an evenly covered neighborhood of \(\pi(x)\) and let \(W\) be the
component of \(\pi^{-1}(U)\) containing \(\varphi(x)\), so that \(\pi\) restricts to a
diffeomorphism of \(W\) onto \(U\). On the neighborhood \(\varphi^{-1}(W)\) of \(x\) the
identity \(\pi \circ \varphi = \pi\) reads \(\varphi = (\pi|_W)^{-1} \circ \pi\), a composition
of smooth maps. Since \(x\) and \(\varphi\) were arbitrary, every automorphism is smooth, and
the action is smooth.
The covering arc, continued
The thread running from the fundamental group through covering manifolds to the universal
covering group continues here. Lifting first gave covers their power to compute fundamental
groups. Then, over a connected Lie group, the lifted multiplication turned the universal cover
into a group. Now the same lifting uniqueness shows that the symmetries of any smooth cover
organize themselves into a discrete group acting freely. That group is the deck group, and its
size measures how many sheets the cover has. The freeness is not incidental. It is precisely the
rigidity that a lift is determined by its value at one point.
Equivariant Maps and the Equivariant Rank Theorem
A map between two spaces carrying group actions behaves well when it respects those actions.
Respecting the actions means that moving a point by a group element and then applying the map gives
the same result as applying the map first and then moving by the same element. Such maps are called
equivariant, and they are the structure-preserving maps of the category of \(G\)-spaces.
One fact about them is that equivariance with respect to a transitive action forces constant rank,
which converts a soft symmetry hypothesis into the hard conclusion that a map is a submersion, an
immersion, or a diffeomorphism. This generalizes the constant-rank property already established for
homomorphisms, which are exactly the maps equivariant for left translation.
Definition (Equivariant Map)
Let \(G\) be a Lie group and let \(M, N\) be smooth manifolds, with or without boundary,
carrying smooth left \(G\)-actions \(\theta\) and \(\varphi\). A smooth map \(F : M \to N\) is
equivariant if
\[
F(g \cdot p) = g \cdot F(p) \quad \forall g \in G,\ p \in M,
\]
equivalently if \(\varphi_g \circ F = F \circ \theta_g\) for every \(g\). One says that \(F\)
intertwines the two actions. The corresponding condition for right actions is
\(F(p \cdot g) = F(p) \cdot g\).
Example:
The covering homomorphism \(\varepsilon^n : \mathbb{R}^n \to \mathbb{T}^n\) seen earlier is
equivariant for the translation action of \(\mathbb{R}^n\) on itself and the action on
\(\mathbb{T}^n\) obtained by pushing it forward. Translating in \(\mathbb{R}^n\) and then
projecting agrees with projecting and then translating on the torus.
The link to rank rests on a single observation. If the action on the source is transitive, then any
two points are related by a group element, and equivariance ties the behavior of the map at one
point to its behavior at the other through diffeomorphisms, and diffeomorphisms cannot change rank.
Theorem (Equivariant Rank Theorem)
Let \(M\) and \(N\) be smooth manifolds (without boundary) and \(G\) a Lie group. Suppose
\(F : M \to N\) is a smooth map that is equivariant with respect to a transitive smooth
\(G\)-action on \(M\) and any smooth \(G\)-action on \(N\). Then \(F\) has constant rank.
Consequently, a surjective \(F\) is a
smooth submersion,
an injective \(F\) is a
smooth immersion,
and a bijective \(F\) is a diffeomorphism.
Proof:
Let \(\theta\) and \(\varphi\) denote the actions on \(M\) and \(N\), and let \(p, q\) be any
two points of \(M\). Since the action on \(M\) is transitive, there is a \(g \in G\) with
\(\theta_g(p) = q\). Equivariance gives \(\varphi_g \circ F = F \circ \theta_g\).
Differentiating this identity at \(p\) yields
\[
d(\varphi_g)_{F(p)} \circ dF_p = dF_q \circ d(\theta_g)_p.
\]
Both \(\theta_g\) and \(\varphi_g\) are diffeomorphisms, so their differentials
\(d(\theta_g)_p\) and \(d(\varphi_g)_{F(p)}\) are linear isomorphisms. Composing a linear map
with isomorphisms on either side leaves its rank unchanged, so \(dF_p\) and \(dF_q\) have the
same rank. As \(p\) and \(q\) were arbitrary, \(F\) has
constant rank.
The consequences for surjective, injective, and bijective \(F\) then follow from the
global rank theorem.
Homomorphisms as a special case
The constant-rank property of
Lie group homomorphisms
is this theorem in disguise. A homomorphism \(F : G \to H\) intertwines the left-translation
action of \(G\) on itself with the action on \(H\) defined by \(g \cdot h = F(g) h\), and the
left-translation action is transitive. The theorem above therefore applies and returns the
earlier result.
What was proved by hand for homomorphisms is the prototype of a single principle: transitivity
upstream plus equivariance forces uniform behavior of the differential everywhere. This is also
the property that makes equivariance the right hypothesis in geometric learning, where a map
between feature spaces is required to commute with the symmetry group of the data.
Orbits, Isotropy, and the Classical Groups
The equivariant rank theorem applies directly to the internal structure of an action. Fixing a point
and letting the group sweep it through its orbit defines a map from the group to the manifold. This
orbit map is equivariant for left translation, so it has constant rank, and its level set over the
chosen point is the isotropy group. Constant rank then makes the isotropy group a properly embedded
Lie subgroup.
Applied to the action of the general linear group on matrices, the same mechanism realizes the
orthogonal, special orthogonal, unitary, and special unitary groups as embedded submanifolds. The
classical groups, defined earlier as matrix groups, thereby acquire their manifold structure from a
uniform argument.
The Orbit Map
Proposition (Properties of the Orbit Map)
Let \(\theta\) be a smooth left action of a Lie group \(G\) on a smooth manifold \(M\) (without
boundary). For each \(p \in M\), the orbit map \(\theta^{(p)} : G \to M\),
\(\theta^{(p)}(g) = g \cdot p\), is smooth and has constant rank, so the
isotropy group
\(G_p = (\theta^{(p)})^{-1}(p)\) is a
properly embedded
Lie subgroup of \(G\). If \(G_p\) is trivial, then \(\theta^{(p)}\) is an injective smooth
immersion, and the orbit \(G \cdot p\) is an immersed submanifold of \(M\).
Proof:
The orbit map is smooth as the composition
\(G \cong G \times \{p\} \hookrightarrow G \times M \xrightarrow{\theta} M\). It is equivariant
with respect to the left-translation action of \(G\) on itself and the given action on \(M\):
for \(g' \in G\),
\[
\begin{align*}
\theta^{(p)}(g' g) &= (g' g) \cdot p \\\\
&= g' \cdot (g \cdot p) \\\\
&= g' \cdot \theta^{(p)}(g).
\end{align*}
\]
Because left translation is transitive, the
equivariant rank theorem
shows \(\theta^{(p)}\) has constant rank. Its level set \(G_p = (\theta^{(p)})^{-1}(p)\) is then
a properly embedded submanifold by the
constant-rank level set theorem,
and a subgroup, hence a
Lie subgroup.
If \(G_p\) is trivial, then \(g' \cdot p = g \cdot p\) forces \(g^{-1} g' \in G_p = \{e\}\), so
\(\theta^{(p)}\) is injective. By the equivariant rank theorem it is an immersion, and the
image of an injective immersion
is an immersed submanifold.
Every orbit is a submanifold
The proposition handles orbits with trivial isotropy, but in fact every orbit of a smooth action
is an immersed submanifold, regardless of its isotropy group. The general statement rests on
passing to the quotient \(G / G_p\), which requires the theory of quotient manifolds developed
later. The version here is the part reachable with the equivariant rank theorem alone.
The Classical Groups by Equivariant Rank
The same level-set technique identifies the classical subgroups of the general linear group. In each
case a smooth map out of \(GL\) is shown to be equivariant for a suitable action, hence of constant
rank, so that the group, a level set of that map, is properly embedded. The group itself, its
dimension, and its compactness were recorded when these groups were first introduced as matrix
groups. What the present method contributes is the rigorous manifold structure, obtained without
separate coordinate computations.
The orthogonal group.
The
orthogonal group
\(O(n)\) consists of the real matrices preserving the Euclidean dot product, equivalently those
with \(A^\top A = I_n\). Define \(\Phi : GL(n, \mathbb{R}) \to M(n, \mathbb{R})\) by
\(\Phi(A) = A^\top A\), so that \(O(n) = \Phi^{-1}(I_n)\). Let \(GL(n, \mathbb{R})\) act on
itself by right multiplication and on \(M(n, \mathbb{R})\) by \(X \cdot B = B^\top X B\). Then
\(\Phi\) is equivariant,
\[
\begin{align*}
\Phi(AB) &= (AB)^\top (AB) \\\\
&= B^\top A^\top A B \\\\
&= B^\top \Phi(A) B \\\\
&= \Phi(A) \cdot B,
\end{align*}
\]
and the action on \(GL(n, \mathbb{R})\) is transitive, so \(\Phi\) has constant rank by the
equivariant rank theorem.
Its level set \(O(n)\) is therefore a properly embedded Lie subgroup.
Computing the differential at the identity along the curve \(t \mapsto I_n + tY\) gives
\[
\begin{align*}
d\Phi_{I_n}(Y) &= \left.\frac{d}{dt}\right|_{t=0} (I_n + tY)^\top (I_n + tY) \\\\
&= Y^\top + Y,
\end{align*}
\]
whose image is the space of symmetric matrices. This fixes the rank of \(\Phi\) and hence the
codimension of \(O(n)\), in agreement with the
dimension recorded earlier.
Special orthogonal, unitary, special unitary.
The
special orthogonal group
\(SO(n) = O(n) \cap SL(n, \mathbb{R})\) is the subset of \(O(n)\) of determinant \(+1\). Since
every \(A \in O(n)\) satisfies \((\det A)^2 = \det(A^\top A) = 1\), the determinant takes only
the values \(\pm 1\) on \(O(n)\), so \(SO(n)\) is the open subgroup where it equals \(+1\). The
subgroup \(SO(n)\) is therefore an embedded Lie subgroup, compact as a closed subset of the
compact \(O(n)\).
The
unitary group
\(U(n)\), the complex matrices with \(A^* A = I_n\), is handled by the same equivariant
level-set argument applied to \(A \mapsto A^* A\), giving a properly embedded Lie subgroup of
\(GL(n, \mathbb{C})\). Finally the
special unitary group
\(SU(n) = U(n) \cap SL(n, \mathbb{C})\) is the level set \(\det^{-1}(1)\) of the determinant on
\(U(n)\). The determinant restricts to a Lie group homomorphism \(U(n) \to \mathbb{C}^*\), so it
has
constant rank,
and \(SU(n)\) is a properly embedded submanifold of \(U(n)\) by the
constant-rank level set theorem,
hence a
Lie subgroup.
Since the composition of embeddings
\(SU(n) \hookrightarrow U(n) \hookrightarrow GL(n, \mathbb{C})\) is again an embedding,
\(SU(n)\) is embedded in \(GL(n, \mathbb{C})\) as well.
Semidirect Products
Group actions also build new Lie groups out of old ones. When a group acts on another by
automorphisms, the two can be combined into a single group whose underlying manifold is the product
but whose multiplication twists one factor by the action of the other. The construction captures the
structure of groups that split into a normal piece and a complementary piece. The example to keep in
mind is Euclidean space with its rigid motions, where translations form the normal piece and
rotations act on them.
An action \(\theta : H \times N \to N\) of one Lie group on another is an action by
automorphisms if each \(\theta_h : N \to N\) is a group automorphism of \(N\). Given such an
action, the semidirect product \(N \rtimes_\theta H\) is the manifold \(N \times H\)
with multiplication twisting the first factor by the action.
Definition (Semidirect Product)
Let \(N\) and \(H\) be Lie groups and \(\theta : H \times N \to N\) a smooth action by
automorphisms. The semidirect product \(N \rtimes_\theta H\) is the smooth manifold
\(N \times H\) with the group multiplication
\[
(n, h)(n', h') = \big(n\, \theta_h(n'),\, h h'\big),
\]
identity \((e, e)\), and inversion \((n, h)^{-1} = \big(\theta_{h^{-1}}(n^{-1}), h^{-1}\big)\). When
the action is understood, it is written \(N \rtimes H\).
Example (the Euclidean group).
Take \(\mathbb{R}^n\) under addition and let \(O(n)\) act on it in the natural way. Each
orthogonal map is an automorphism of the additive group, so this is an action by automorphisms.
The resulting semidirect product \(E(n) = \mathbb{R}^n \rtimes O(n)\) is the Euclidean
group, with multiplication \((b, A)(b', A') = (b + Ab', AA')\). It acts on
\(\mathbb{R}^n\) by \((b, A) \cdot x = b + Ax\), and this action preserves lines, distances, and
angles, exactly the relations of Euclidean geometry. Restricting the rotational factor to
\(SO(n)\) gives the orientation-preserving subgroup \(\mathbb{R}^n \rtimes SO(n)\), which is the
group of
rigid body motions
underlying robotic kinematics.
Proposition (Properties of the Semidirect Product)
Let \(N, H\) be Lie groups, \(\theta\) a smooth action by automorphisms, and
\(G = N \rtimes_\theta H\). Then the subsets \(\widetilde{N} = N \times \{e\}\) and
\(\widetilde{H} = \{e\} \times H\) are closed Lie subgroups isomorphic to \(N\) and \(H\). The
subgroup \(\widetilde{N}\) is
normal in
\(G\), and \(\widetilde{N} \cap \widetilde{H} = \{(e,e)\}\) while
\(\widetilde{N}\,\widetilde{H} = G\).
Proof Sketch:
Each of \(\widetilde{N}\) and \(\widetilde{H}\) is the image of a smooth injective homomorphism
(inclusion of a factor), hence a
Lie subgroup
isomorphic to \(N\) or \(H\). Both are closed as preimages of \(\{e\}\) under the smooth
projections to the other factor, and a
closed Lie subgroup is embedded.
Normality of \(\widetilde{N}\) is a direct computation from the multiplication law. Conjugating
\((n', e)\) by \((n, h)\) returns an element of \(\widetilde{N}\), since the second coordinate
of the product stays at \(e\). The intersection and product statements are immediate from the
definitions.
Recognizing a Semidirect Product
The construction has a converse: a group that contains a normal subgroup and a complementary subgroup,
meeting only at the identity and together spanning the whole group, is automatically a semidirect
product of the two. This is the form in which semidirect products are usually recognized in practice.
Theorem (Characterization of Semidirect Products)
Let \(G\) be a Lie group, and let \(N, H \subseteq G\) be
closed Lie subgroups
with \(N\) normal, \(N \cap H = \{e\}\), and \(NH = G\). Then the map \((n, h) \mapsto nh\) is a Lie
group isomorphism \(N \rtimes_\theta H \to G\), where \(\theta\) is the conjugation action
\(\theta_h(n) = hnh^{-1}\). One says that \(G\) is the internal semidirect product
of \(N\) and \(H\).
Proof Sketch:
Since \(N\) is normal, conjugation by an element of \(H\) carries \(N\) to itself, so
\(\theta_h(n) = hnh^{-1}\) defines a smooth action of \(H\) on \(N\) by automorphisms. The map
\(\Phi(n, h) = nh\) is a homomorphism for this action, because
\(\Phi\big((n,h)(n',h')\big) = n\,(hn'h^{-1})\,hh' = nh \cdot n'h' = \Phi(n,h)\Phi(n',h')\). It is
bijective: surjective because \(NH = G\), and injective because \(N \cap H = \{e\}\) forces a
product \(nh\) to determine \(n\) and \(h\). A
bijective Lie group homomorphism is an isomorphism,
so \(\Phi\) realizes \(G\) as \(N \rtimes_\theta H\).
Rigid motions as a semidirect product
The Euclidean and special Euclidean groups are the prototypes a roboticist meets first. A rigid
motion is a rotation followed by a translation, and composing two such motions twists the second
translation by the first rotation, precisely the semidirect multiplication
\((b, A)(b', A') = (b + Ab', AA')\). Recognizing the rigid-motion group as
\(\mathbb{R}^n \rtimes SO(n)\) is what lets one separate the rotational and translational
degrees of freedom while keeping track of how they interact, the starting point for describing
the configuration space of an articulated mechanism.
Representations
Most of the Lie groups encountered so far can be realized as subgroups of a general linear group,
and it is natural to ask whether every Lie group is of this form. The question is answered by the
theory of representations: linear actions of a group on a vector space, or equivalently
homomorphisms into a general linear group. A representation turns abstract group elements into
matrices, and a faithful one realizes the group as a matrix group when such a representation exists.
The catch is that not every Lie group admits one, and the obstruction connects directly back to the
universal covering group.
Definition (Representation)
Let \(G\) be a Lie group and \(V\) a finite-dimensional real or complex vector space, so that
\(GL(V)\), the group of invertible linear maps of \(V\), is a Lie group isomorphic to a
general linear group.
A (finite-dimensional) representation of \(G\) is a Lie group homomorphism
\(\rho : G \to GL(V)\). It is faithful if it is injective.
A faithful representation does more than label group elements by matrices. It embeds the group among
them. By the proposition on
images of injective homomorphisms,
the image \(\rho(G)\) is a Lie subgroup of \(GL(V)\) and \(\rho\) is an isomorphism onto it. Thus a
Lie group admits a faithful representation if and only if it is isomorphic to a Lie subgroup of some
general linear group. Not every Lie group passes this test. The
universal covering group
of \(SL(2, \mathbb{R})\) admits no faithful representation, and so is a Lie group that is not
isomorphic to any matrix group. The proof of that fact belongs to a later stage of the theory.
Examples
Examples:
(a)
If \(G\) is any Lie subgroup of \(GL(n, \mathbb{R})\), the inclusion
\(G \hookrightarrow GL(n, \mathbb{R}) = GL(\mathbb{R}^n)\) is a faithful representation, called
the defining representation of \(G\). The complex case is identical.
(b)
The inclusion of the circle group into \(\mathbb{C}^* \cong GL(1, \mathbb{C})\)
is a faithful representation. More generally, sending a point of the \(n\)-torus to the diagonal
matrix with its coordinates on the diagonal gives a faithful representation
\(\mathbb{T}^n \to GL(n, \mathbb{C})\).
(c)
The additive group \(\mathbb{R}^n\) has a faithful representation into
\(GL(n+1, \mathbb{R})\) sending \(x\) to the block matrix
\(\left(\begin{smallmatrix} I_n & x \\\\ 0 & 1 \end{smallmatrix}\right)\), and another into
\(GL(n, \mathbb{R})\) sending \((x^1, \dots, x^n)\) to the diagonal matrix with entries
\(e^{x^1}, \dots, e^{x^n}\). The diagonal map into \(GL(n, \mathbb{C})\) with entries
\(e^{2\pi i x^j}\) is a representation but not faithful, its kernel being \(\mathbb{Z}^n\).
(d)
The Euclidean group \(E(n) = \mathbb{R}^n \rtimes O(n)\), a
semidirect product, has a
faithful representation into \(GL(n+1, \mathbb{R})\) sending \((b, A)\) to the block matrix
\(\left(\begin{smallmatrix} A & b \\\\ 0 & 1 \end{smallmatrix}\right)\), which encodes the
affine action \(x \mapsto Ax + b\) as a linear map in one higher dimension.
Linear Actions Are Representations
A representation has two descriptions, one as a homomorphism into \(GL(V)\) and one as a linear
action on \(V\). The two are equivalent. A smooth action of \(G\) on a vector space \(V\) is
linear if for each \(g\) the map \(x \mapsto g \cdot x\) is linear. Every
representation induces such an action by \(g \cdot x = \rho(g) x\), and conversely.
Proposition (Linear Actions and Representations)
Let \(G\) be a Lie group and \(V\) a finite-dimensional vector space. A smooth left action of \(G\)
on \(V\) is linear if and only if it is of the form \(g \cdot x = \rho(g) x\) for some
representation \(\rho\) of \(G\).
Proof:
An action arising from a representation is linear by construction. Conversely, suppose the
action is linear. For each \(g\) the map \(x \mapsto g \cdot x\) is an invertible linear map, so
it defines an element \(\rho(g) \in GL(V)\). The action axioms give
\(\rho(g_1 g_2) = \rho(g_1)\rho(g_2)\), so \(\rho\) is a group homomorphism.
To see that \(\rho\) is smooth, fix a basis \((E_i)\) of \(V\) with coordinate projections
\(\pi^i\). The matrix entries of \(\rho(g)\) in this basis are
\(\rho^i_j(g) = \pi^i(g \cdot E_j)\), each a composition of the smooth action with smooth linear
maps, hence smooth functions of \(g\). Since the matrix entries form global smooth coordinates
on \(GL(V)\), the map \(\rho\) is smooth, and therefore a representation.
Where representations lead
Representation theory threads through differential geometry, differential equations, harmonic
analysis, number theory, quantum physics, and the symmetry-based architectures of modern machine
learning. The adjoint representation already met in the matrix setting is the action of a group
on its own
Lie algebra by conjugation.
That action is the first nontrivial example, and the bridge from a group's global structure to
the linear algebra of its infinitesimal generators.
For geometric and categorical approaches to learning, a representation is the precise device
that says how a symmetry group acts on a layer of features, so that equivariant maps between
layers can be required to commute with it. The definitions assembled here are action, orbit,
equivariance, and representation. They are the vocabulary in which those theories are written.