The Lie Group–Lie Algebra Correspondence
Over the past three pages, we have built a dictionary between Lie groups and Lie
algebras. A
matrix Lie group
\(G\) determines a
Lie algebra
\(\mathfrak{g} = T_I G\) — a vector space equipped with a
Lie bracket
\([X, Y] = XY - YX\). The
matrix exponential
\(\exp : \mathfrak{g} \to G\) sends the algebra to the group, converting linear
(infinitesimal) data into nonlinear (finite) group elements. We now address the
central question: to what extent does the Lie algebra determine the Lie group?
The Exponential Map as a Local Diffeomorphism
The exponential map \(\exp : \mathfrak{g} \to G\) is, in general, neither injective
nor surjective globally. For instance, in \(SO(2)\), the map
\(\theta \mapsto \exp(\theta J)\) wraps \(\mathbb{R}\) infinitely many times around
the circle, so it is not injective. However, near the origin, the exponential map is
well-behaved:
Theorem: Local Diffeomorphism
Let \(G\) be a matrix Lie group with Lie algebra \(\mathfrak{g}\). The exponential
map \(\exp : \mathfrak{g} \to G\) is a diffeomorphism from a
neighborhood of \(0 \in \mathfrak{g}\) onto a neighborhood of \(I \in G\). That
is, near the identity, every group element has a unique logarithm in the Lie
algebra.
The proof uses the inverse function theorem on manifolds: the derivative of \(\exp\) at
\(0\) is the identity map \(\mathfrak{g} \to \mathfrak{g}\) (since
\(\frac{d}{dt}\big|_{t=0} \exp(tA) = A\)), which is invertible, so \(\exp\) is a local
diffeomorphism by the inverse function theorem. The full details require the machinery of
smooth manifolds, which will be developed in a future page. For now, we state the theorem
and draw its consequences.
The local diffeomorphism property means that the Lie algebra \(\mathfrak{g}\) faithfully
encodes the local structure of \(G\) — the structure in a neighborhood
of the identity. Two Lie groups with isomorphic Lie algebras are
locally isomorphic: they look identical near their respective identities. They
may, however, differ globally — in their topology. We will see a dramatic
example of this phenomenon in the next section.
Lie Algebra Homomorphisms
The correspondence between groups and algebras extends to their maps. Just as a
group homomorphism
preserves the group operation, a Lie algebra homomorphism preserves the bracket.
Definition: Lie Algebra Homomorphism
A Lie algebra homomorphism is a linear map
\(\varphi : \mathfrak{g} \to \mathfrak{h}\) between Lie algebras that preserves
the bracket:
\[
\varphi([X, Y]) = [\varphi(X), \varphi(Y)]
\quad \text{for all } X, Y \in \mathfrak{g}.
\]
A bijective Lie algebra homomorphism is a Lie algebra isomorphism.
The key theorem of the Lie correspondence is that group homomorphisms automatically
induce Lie algebra homomorphisms — the passage from group to algebra is
functorial.
Theorem: Group Homomorphisms Induce Lie Algebra Homomorphisms
Let \(\Phi : G \to H\) be a Lie group homomorphism (i.e., a continuous group
homomorphism between matrix Lie groups). Then the derivative at the
identity,
\[
d\Phi_I : \mathfrak{g} \to \mathfrak{h}, \qquad
d\Phi_I(A) = \left.\frac{d}{dt}\right|_{t=0} \Phi(\exp(tA)),
\]
is a Lie algebra homomorphism:
\(d\Phi_I([A, B]) = [d\Phi_I(A),\, d\Phi_I(B)]\).
Proof sketch:
Since \(\Phi\) is a group homomorphism, it intertwines the exponential maps:
\[
\Phi(\exp(tA)) = \exp(t \cdot d\Phi_I(A))
\]
for all \(A \in \mathfrak{g}\) and \(t \in \mathbb{R}\). (This follows from the
fact that \(t \mapsto \Phi(\exp(tA))\) is a one-parameter subgroup of \(H\) with
initial velocity \(d\Phi_I(A)\), so by the
one-parameter subgroup theorem,
it equals \(\exp(t \cdot d\Phi_I(A))\).)
Now, using the closure proof from
Lie Algebras and the Lie Bracket,
recall that
\([A, B] = \frac{d}{dt}\big|_{t=0} \exp(tA)\,B\,\exp(-tA)\). Applying
\(d\Phi_I\) and using the intertwining property:
\[
\begin{align*}
d\Phi_I([A, B])
&= \left.\frac{d}{dt}\right|_{t=0}
\Phi(\exp(tA))\, d\Phi_I(B)\, \Phi(\exp(-tA)) \\
&= \left.\frac{d}{dt}\right|_{t=0}
\exp(t\,d\Phi_I(A))\, d\Phi_I(B)\, \exp(-t\,d\Phi_I(A)) \\
&= [d\Phi_I(A),\, d\Phi_I(B)]. \qquad \square
\end{align*}
\]
Example: The Determinant and the Trace
The determinant
\(\det : GL(n, \mathbb{R}) \to \mathbb{R}^*\) is a Lie group homomorphism. Its
derivative at the identity is:
\[
d(\det)_I(A) = \left.\frac{d}{dt}\right|_{t=0} \det(\exp(tA))
= \left.\frac{d}{dt}\right|_{t=0} e^{t\,\mathrm{tr}(A)}
= \mathrm{tr}(A).
\]
So the induced Lie algebra homomorphism
\(\mathrm{tr} : \mathfrak{gl}(n, \mathbb{R}) \to \mathbb{R}\) is the trace.
Since \(\mathbb{R}\) is abelian (its Lie bracket is zero), the homomorphism
condition \(\mathrm{tr}([A, B]) = [\mathrm{tr}(A), \mathrm{tr}(B)] = 0\)
reduces to \(\mathrm{tr}(AB - BA) = 0\), which indeed holds because
\(\mathrm{tr}(AB) = \mathrm{tr}(BA)\).
The
kernel
of \(\det\) is \(SL(n)\); correspondingly, the kernel of \(\mathrm{tr}\) is
\(\mathfrak{sl}(n)\). Group-level and algebra-level kernels correspond perfectly.
The Baker-Campbell-Hausdorff Formula
We now arrive at the deepest result of the Lie correspondence: the group multiplication
near the identity is entirely determined by the Lie bracket. This is
formalized by the Baker-Campbell-Hausdorff (BCH) formula.
The full series involves increasingly complex nested brackets and is determined by a
universal recursive formula (the Dynkin formula). We do not prove the BCH formula — the
proof requires analysis of formal power series in non-commuting variables. Instead, we
focus on its meaning and consequences.
The key message: The group multiplication
\((g, h) \mapsto gh\) is a nonlinear operation on a curved manifold. The BCH
formula shows that, near the identity, this nonlinear operation is entirely encoded
by the Lie bracket — a bilinear operation on a vector space. Every coefficient
in the BCH series is built from iterated brackets and nothing else. This is why the Lie
algebra, with its bracket, determines the local structure of the group.
Special cases. The BCH formula illuminates the results of the
preceding pages:
(a) Commuting case. If \([X, Y] = 0\), then all bracket terms in the
BCH series vanish, and \(Z = X + Y\). This recovers
\(\exp(X)\exp(Y) = \exp(X + Y)\) —
property (b)
of the matrix exponential.
(b) First-order approximation. Keeping only the first bracket term:
\[
\exp(X)\exp(Y) \approx \exp\!\left(X + Y + \tfrac{1}{2}[X, Y]\right)
\]
for small \(X, Y\). The failure of \(\exp\) to be a homomorphism is measured, to
leading order, by \(\frac{1}{2}[X, Y]\). This makes precise the statement from
The Matrix Exponential
that the commutator is the "first correction term."
(c) Abelian groups. If \(\mathfrak{g}\) is abelian
(\([X, Y] = 0\) for all \(X, Y\)), then the BCH formula reduces to
\(Z = X + Y\), i.e., \(\exp(X)\exp(Y) = \exp(X + Y)\) always. The exponential
map is a group homomorphism from \((\mathfrak{g}, +)\) to \(G\). This is exactly
what happens for \(SO(2)\): the exponential
\(\theta \mapsto \exp(\theta J)\) is a homomorphism because
\(\mathfrak{so}(2) \cong \mathbb{R}\) is abelian.
The Double Cover: \(SU(2)\) and \(SO(3)\)
The BCH formula tells us that isomorphic Lie algebras produce locally isomorphic Lie
groups. But "locally isomorphic" does not mean "isomorphic" — two groups can share
the same infinitesimal structure while differing in their global topology. The most
important example of this phenomenon is the relationship between
\(SU(2)\)
and
\(SO(3)\).
The Lie Algebra Isomorphism \(\mathfrak{su}(2) \cong \mathfrak{so}(3)\)
The Lie algebra \(\mathfrak{su}(2)\) consists of \(2 \times 2\) traceless
skew-Hermitian matrices. A standard basis is:
\[
F_1 = \begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix}, \quad
F_2 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \quad
F_3 = \begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix}.
\]
(These are \(-i\) times the Pauli matrices \(\sigma_1, \sigma_2, \sigma_3\).) One
verifies that each \(F_k\) is skew-Hermitian (\(F_k^* = -F_k\)) and traceless, and
that
\[
[F_1, F_2] = 2F_3, \qquad [F_2, F_3] = 2F_1, \qquad [F_3, F_1] = 2F_2.
\]
Setting \(\tilde{E}_k = \frac{1}{2}F_k\) for \(k = 1, 2, 3\) gives a basis satisfying
\[
[\tilde{E}_1, \tilde{E}_2] = \tilde{E}_3, \qquad
[\tilde{E}_2, \tilde{E}_3] = \tilde{E}_1, \qquad
[\tilde{E}_3, \tilde{E}_1] = \tilde{E}_2
\]
— exactly the same bracket relations as the basis \(\{E_1, E_2, E_3\}\) of
\(\mathfrak{so}(3)\) computed in
Lie Algebras and the Lie Bracket.
The linear map \(\tilde{E}_k \mapsto E_k\) is therefore a Lie algebra isomorphism:
\[
\mathfrak{su}(2) \cong \mathfrak{so}(3).
\]
Both are 3-dimensional real Lie algebras with structure constants given by the
Levi-Civita symbol. At the Lie algebra level, they are indistinguishable.
The Groups are Not Isomorphic
Despite their identical Lie algebras, \(SU(2)\) and \(SO(3)\) are not
isomorphic as groups. The difference is topological:
\(SU(2)\) is homeomorphic to \(S^3\). Every element of \(SU(2)\) has
the form
\(\begin{pmatrix} \alpha & -\bar{\beta} \\ \beta & \bar{\alpha} \end{pmatrix}\) with
\(|\alpha|^2 + |\beta|^2 = 1\), which identifies \(SU(2)\) with the unit sphere in
\(\mathbb{C}^2 \cong \mathbb{R}^4\). In particular, \(SU(2)\) is
simply connected — every closed loop can be continuously shrunk to a
point.
\(SO(3)\) is homeomorphic to \(\mathbb{R}P^3\). The real projective
space \(\mathbb{R}P^3 = S^3 / \{x \sim -x\}\) is obtained from \(S^3\) by identifying
antipodal points. This space is not simply connected: it has
fundamental group \(\pi_1(SO(3)) \cong \mathbb{Z}/2\mathbb{Z}\). There exist loops in
\(SO(3)\) (a rotation by \(2\pi\) about any axis) that cannot be continuously deformed
to the identity, but traversing such a loop twice (rotation by \(4\pi\))
yields a contractible loop.
The Double Cover Homomorphism
The Lie algebra isomorphism \(\mathfrak{su}(2) \cong \mathfrak{so}(3)\) lifts to a
group homomorphism — but not an isomorphism. There is a surjective 2:1 map
\(\Phi : SU(2) \to SO(3)\), called the double cover.
Construction (following Stillwell):
The group \(SU(2)\) acts on its own Lie algebra \(\mathfrak{su}(2)\) by
conjugation: for \(U \in SU(2)\) and \(X \in \mathfrak{su}(2)\), define
\[
\Phi_U(X) = U X U^*.
\]
Since conjugation preserves skew-Hermiticity and tracelessness, \(\Phi_U\) maps
\(\mathfrak{su}(2)\) to itself. Moreover, \(\Phi_U\) preserves the inner product
\(\langle X, Y \rangle = -\frac{1}{2}\,\mathrm{tr}(XY)\) on \(\mathfrak{su}(2)\):
\[
\langle \Phi_U(X), \Phi_U(Y) \rangle
= -\tfrac{1}{2}\,\mathrm{tr}(UXU^* \cdot UYU^*)
= -\tfrac{1}{2}\,\mathrm{tr}(XY) = \langle X, Y \rangle.
\]
Using the isomorphism \(\mathfrak{su}(2) \cong \mathbb{R}^3\) (via the basis
\(\{\tilde{E}_1, \tilde{E}_2, \tilde{E}_3\}\)), the map \(\Phi_U\) becomes an
isometry of \(\mathbb{R}^3\) preserving orientation (since \(\det(\Phi_U)\) is
continuous, equals 1 at \(U = I\), and \(SU(2)\) is connected). Hence
\(\Phi_U \in SO(3)\).
The map \(\Phi : SU(2) \to SO(3)\), \(U \mapsto \Phi_U\), is a group homomorphism
(since conjugation respects composition). It is surjective (this can be verified
by checking that the induced Lie algebra map is the isomorphism
\(\mathfrak{su}(2) \xrightarrow{\sim} \mathfrak{so}(3)\), and applying the
connectedness of \(SO(3)\)).
The kernel. We have \(\Phi_U = \mathrm{Id}\) if and only if
\(UXU^* = X\) for all \(X \in \mathfrak{su}(2)\), i.e., \(U\) commutes with the
three basis matrices \(F_1, F_2, F_3\). Since these together with \(iI\) span
\(M_2(\mathbb{C})\) over \(\mathbb{R}\), the matrix \(U\) commutes with every
\(2 \times 2\) complex matrix, so \(U = \lambda I\) for some scalar \(\lambda\).
The conditions \(U^* U = I\) and \(\det(U) = 1\) force \(\lambda = \pm 1\), so
\[
\ker(\Phi) = \{I, -I\} \cong \mathbb{Z}/2\mathbb{Z}.
\]
By the
First Isomorphism Theorem:
\[
SU(2) / \{I, -I\} \;\cong\; SO(3).
\]
The group \(SO(3)\) is obtained from \(SU(2) \cong S^3\) by identifying each element
with its negative — precisely the operation that produces the projective space
\(\mathbb{R}P^3\) from \(S^3\).
Quaternions and 3D Graphics
The double cover \(SU(2) \to SO(3)\) is the mathematical foundation of
quaternion rotations in computer graphics and game engines. The
group \(SU(2)\) is isomorphic to the group of unit quaternions
\(\{q \in \mathbb{H} : |q| = 1\}\), and the double cover corresponds to the
action of a unit quaternion \(q\) on \(\mathbb{R}^3\) by
\(\mathbf{v} \mapsto q \mathbf{v} \bar{q}\), where \(\mathbf{v}\) is identified
with a purely imaginary quaternion. The sign ambiguity — \(q\) and \(-q\) yield
the same rotation — is precisely the \(\mathbb{Z}/2\mathbb{Z}\) kernel.
Quaternion representations of rotations are preferred over rotation matrices in
many applications because they avoid gimbal lock (a degeneracy in
Euler angle parameterizations) and admit smooth interpolation via
SLERP (Spherical Linear Interpolation) on \(S^3\). The
mathematical reason is that \(SU(2) \cong S^3\) is a smooth, simply connected
manifold — interpolation on the sphere is geometrically natural and singularity-free.
The Adjoint Representations
The double cover construction above used a specific instance of a general operation:
the group \(G\) acting on its own Lie algebra \(\mathfrak{g}\) by conjugation. This
operation — the adjoint representation — is the most natural example
of a group representation and the bridge to the systematic study of representation
theory.
The Adjoint Representation of the Group
For each \(g \in G\), the conjugation map
\(C_g : G \to G\), \(C_g(h) = ghg^{-1}\), is a group automorphism. Recall that
conjugation appeared in the definition of
normal subgroups:
a subgroup \(H\) is normal if \(gHg^{-1} = H\) for all \(g \in G\). The adjoint
representation is the infinitesimal version: instead of conjugating subgroups,
we conjugate Lie algebra elements.
Definition: The Adjoint Representation \(\mathrm{Ad}\)
Let \(G\) be a matrix Lie group with Lie algebra \(\mathfrak{g}\). For each
\(g \in G\), define the linear map
\[
\mathrm{Ad}(g) : \mathfrak{g} \to \mathfrak{g}, \qquad
\mathrm{Ad}(g)(X) = gXg^{-1}.
\]
The map \(\mathrm{Ad} : G \to GL(\mathfrak{g})\), \(g \mapsto \mathrm{Ad}(g)\),
is a group homomorphism — the adjoint representation of \(G\).
Verification:
\(\mathrm{Ad}(g)\) maps \(\mathfrak{g}\) to \(\mathfrak{g}\):
If \(X \in \mathfrak{g}\), then \(\exp(t \cdot gXg^{-1}) = g\exp(tX)g^{-1} \in G\)
for all \(t\) (since \(\exp(tX) \in G\) and \(G\) is closed under conjugation).
Hence \(gXg^{-1} \in \mathfrak{g}\).
\(\mathrm{Ad}(g)\) is linear:
\(\mathrm{Ad}(g)(aX + bY) = g(aX + bY)g^{-1} = a\,gXg^{-1} + b\,gYg^{-1} = a\,\mathrm{Ad}(g)(X) + b\,\mathrm{Ad}(g)(Y)\).
\(\mathrm{Ad}\) is a group homomorphism:
\(\mathrm{Ad}(g_1 g_2)(X) = (g_1 g_2)X(g_1 g_2)^{-1} = g_1(g_2 X g_2^{-1})g_1^{-1} = \mathrm{Ad}(g_1)(\mathrm{Ad}(g_2)(X))\),
so \(\mathrm{Ad}(g_1 g_2) = \mathrm{Ad}(g_1) \circ \mathrm{Ad}(g_2)\). Also,
\(\mathrm{Ad}(I)(X) = X\), so \(\mathrm{Ad}(I) = \mathrm{Id}\). \(\square\)
The double cover \(\Phi : SU(2) \to SO(3)\) constructed in the previous section is
precisely \(\mathrm{Ad} : SU(2) \to GL(\mathfrak{su}(2)) \cong GL(3, \mathbb{R})\),
restricted to the image \(SO(3)\). This illustrates the adjoint representation's
geometric content: it describes how the group rotates its own infinitesimal generators.
The Adjoint Representation of the Lie Algebra
Applying the general principle of the Lie correspondence — differentiating a group
homomorphism to obtain a Lie algebra homomorphism — to
\(\mathrm{Ad} : G \to GL(\mathfrak{g})\) gives a Lie algebra homomorphism from
\(\mathfrak{g}\) to \(\mathfrak{gl}(\mathfrak{g})\).
Definition: The Adjoint Representation \(\mathrm{ad}\)
The derivative of \(\mathrm{Ad}\) at the identity defines the
adjoint representation of the Lie algebra:
\[
\mathrm{ad} : \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}), \qquad
\mathrm{ad}(X)(Y) = \left.\frac{d}{dt}\right|_{t=0}
\mathrm{Ad}(\exp(tX))(Y) = \left.\frac{d}{dt}\right|_{t=0}
\exp(tX)\,Y\,\exp(-tX).
\]
Computing the derivative (as in the proof of bracket closure in
Lie Algebras and the Lie Bracket):
\[
\mathrm{ad}(X)(Y) = XY - YX = [X, Y].
\]
The adjoint representation of the Lie algebra is simply the Lie bracket itself,
viewed as a linear map \(Y \mapsto [X, Y]\) for each fixed \(X\).
Theorem: \(\mathrm{ad}\) is a Lie Algebra Homomorphism
The map \(\mathrm{ad} : \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})\) is a
Lie algebra homomorphism:
\[
\mathrm{ad}([X, Y]) = [\mathrm{ad}(X),\, \mathrm{ad}(Y)]
= \mathrm{ad}(X) \circ \mathrm{ad}(Y) - \mathrm{ad}(Y) \circ \mathrm{ad}(X).
\]
Proof:
We must show that for all \(Z \in \mathfrak{g}\):
\[
\mathrm{ad}([X, Y])(Z) = \mathrm{ad}(X)(\mathrm{ad}(Y)(Z))
- \mathrm{ad}(Y)(\mathrm{ad}(X)(Z)).
\]
The left-hand side is \([[X, Y], Z]\). The right-hand side is
\([X, [Y, Z]] - [Y, [X, Z]]\). The claim is therefore:
\[
[[X, Y], Z] = [X, [Y, Z]] - [Y, [X, Z]].
\]
This is precisely the
Jacobi identity
rearranged: the standard form
\([X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0\) can be rewritten as
\(-[Z, [X, Y]] = [X, [Y, Z]] + [Y, [Z, X]]\), i.e.,
\([[X, Y], Z] = [X, [Y, Z]] - [Y, [X, Z]]\) (using antisymmetry
\([Z, X] = -[X, Z]\)). \(\square\)
This result reveals a deep connection: the statement "\(\mathrm{ad}\) is a Lie algebra
homomorphism" is equivalent to the Jacobi identity. Conversely, the Jacobi
identity can be understood as the statement that the map \(X \mapsto [X, \,\cdot\,]\)
is a derivation — it satisfies a "product rule" with respect to the bracket.
Example: \(\mathrm{ad}\) for \(\mathfrak{so}(3)\)
Using the basis \(\{E_1, E_2, E_3\}\) of \(\mathfrak{so}(3)\) and the bracket
relations from
Lie Algebras and the Lie Bracket,
we can write \(\mathrm{ad}(E_i)\) as a \(3 \times 3\) matrix acting on the basis
\(\{E_1, E_2, E_3\}\).
For \(\mathrm{ad}(E_1)\): we have \(\mathrm{ad}(E_1)(E_1) = [E_1, E_1] = 0\),
\(\mathrm{ad}(E_1)(E_2) = [E_1, E_2] = E_3\), and
\(\mathrm{ad}(E_1)(E_3) = [E_1, E_3] = -E_2\). So:
\[
\mathrm{ad}(E_1) = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix} = E_1.
\]
Similarly:
\[
\mathrm{ad}(E_2) = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix} = E_2, \qquad
\mathrm{ad}(E_3) = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} = E_3.
\]
The matrices of \(\mathrm{ad}(E_1)\), \(\mathrm{ad}(E_2)\), \(\mathrm{ad}(E_3)\) in
the basis \(\{E_1, E_2, E_3\}\) are exactly the basis elements themselves. In
other words, the map
\(\mathrm{ad} : \mathfrak{so}(3) \to \mathfrak{gl}(\mathfrak{so}(3))\) is, in this
basis, the identity map. This remarkable fact reflects the exceptional
isomorphism \(\mathfrak{so}(3) \cong (\mathbb{R}^3, \times)\): the Lie algebra acts on
itself by the cross product, and the matrix of the cross product map
\(\mathbf{v} \mapsto \boldsymbol{\omega} \times \mathbf{v}\) is precisely the
skew-symmetric matrix \(\hat{\boldsymbol{\omega}}_\times\) — the hat map.
Connections and Outlook
Lie Algebras in Deep Learning
In equivariant neural networks, layers must commute with the
action of a symmetry group \(G\) on the input data — a condition called
equivariance. For instance, a network processing 3D point clouds
should produce the same output regardless of how the input is rotated (equivariance
under \(SO(3)\)).
Enforcing equivariance at the group level requires checking the constraint
for every group element — an uncountable family of conditions. The Lie
correspondence provides a shortcut: for connected groups, equivariance under
\(G\) is equivalent to infinitesimal equivariance under the Lie
algebra \(\mathfrak{g}\). This reduces the problem to finitely many linear
constraints (one for each basis element of \(\mathfrak{g}\)), which can be
incorporated directly into the network architecture. This is why modern equivariant
architectures often work with \(\mathfrak{g}\)-valued features and the Lie bracket
rather than explicit group elements.
Summary of the Four-Page Arc
We have completed the development of Lie groups and Lie algebras for matrix groups.
Let us trace the logical arc:
Matrix Lie Groups defined the classical
groups as closed subgroups of \(GL(n)\), with Cartan's theorem guaranteeing smooth
manifold structure.
The Matrix Exponential provided
the bridge between linear data and nonlinear group elements, establishing one-parameter
subgroups as the "straight lines" in the group.
Lie Algebras and the Lie Bracket
formalized the tangent space at the identity as a Lie algebra — a vector space with a
bracket encoding non-commutativity.
The present page established the Lie correspondence: the algebra
determines the group locally (BCH formula), group homomorphisms induce algebra
homomorphisms, and the adjoint representations describe the group's action on its own
infinitesimal generators.
Where We Go from Here
Representation Theory (next page): The adjoint representations
\(\mathrm{Ad}\) and \(\mathrm{ad}\) are examples of
group and algebra representations — homomorphisms from \(G\)
(or \(\mathfrak{g}\)) to \(GL(V)\) for some vector space \(V\). The next page studies
representations systematically: which vector spaces can \(G\) act on? When can a
representation be decomposed into simpler pieces (irreducible representations)?
Schur's lemma and character theory will provide the answers.
Smooth Manifolds (Section II): The tangent space \(T_I G\), defined
here concretely as velocity vectors of curves in \(M_n(\mathbb{C})\), will be
generalized to the tangent space \(T_p M\) at any point of an abstract smooth manifold.
The exponential map on a Lie group will become a special case of the exponential map of
a Riemannian manifold.
Equivariant Neural Networks (Section V): Networks whose architecture
encodes the symmetry groups \(SO(3)\), \(SE(3)\), and the Lie algebras
\(\mathfrak{so}(3)\), \(\mathfrak{se}(3)\) defined across these four pages. The
adjoint representation and the BCH formula will appear as tools for constructing
equivariant layers.