The Lie Algebra of the General Linear Group
The construction of
\(\mathrm{Lie}(G)\)
developed for a general Lie group identifies it, as a vector space, with the tangent space at the
identity and equips it with a bracket inherited from the Lie bracket of left-invariant vector fields.
For the abelian examples treated alongside that construction, the bracket vanishes and the structure
is trivial. The general linear group
\(GL(n, \mathbb{R})\)
is the first non-abelian example in which the bracket can be made fully explicit. Its Lie algebra,
computed in either of two natural ways, is precisely
\(\mathfrak{gl}(n, \mathbb{R})\)
with the matrix commutator bracket. The two computations — one through left-invariant vector
fields on \(GL(n, \mathbb{R})\), the other through the matrix algebra structure of
\(M(n, \mathbb{R})\) — produce Lie algebras carrying independently defined brackets. The result
of this section is that the canonical vector space isomorphism between them respects both brackets:
the bracket of left-invariant vector fields is the matrix commutator under the identification at
the identity. This is the central computational identification between the two viewpoints on Lie
algebras developed across the smooth-manifold and linear-algebra tracks of the series.
The natural isomorphisms
The group \(GL(n, \mathbb{R})\) is an open subset of the vector space \(\mathfrak{gl}(n, \mathbb{R})\)
of all real \(n \times n\) matrices, defined by the open condition \(\det \neq 0\). Its tangent
space at the identity matrix \(I_n\) is therefore
canonically identified
with \(\mathfrak{gl}(n, \mathbb{R})\), and the
evaluation isomorphism
established on the preceding page identifies the Lie algebra \(\mathrm{Lie}(GL(n, \mathbb{R}))\) with
\(T_{I_n} GL(n, \mathbb{R})\). Composing the two yields a sequence of vector space isomorphisms
\[
\mathrm{Lie}\bigl( GL(n, \mathbb{R}) \bigr)
\xrightarrow{\varepsilon}
T_{I_n} GL(n, \mathbb{R})
\xrightarrow{\;\cong\;}
\mathfrak{gl}(n, \mathbb{R}) .
\tag{1}
\]
Both source and target carry Lie algebra structures: the source has the Lie bracket of left-invariant
vector fields, established on the preceding page, while the target has the matrix commutator
\([A, B] = AB - BA\) developed in the linear-algebra track. The question of this section is whether
the composition (1), already a vector space isomorphism, preserves these brackets.
The main result
Proposition (Lie Algebra of the General Linear Group)
The composition of the natural maps (1) is a Lie algebra isomorphism between
\(\mathrm{Lie}(GL(n, \mathbb{R}))\) and the matrix algebra \(\mathfrak{gl}(n, \mathbb{R})\).
The proof occupies the remainder of this section: a setup phase fixes coordinates and derives the
explicit formula (2) for the left-invariant vector field determined by a matrix \(A\), and a
computation phase evaluates the Lie bracket of two such vector fields in those coordinates,
extracting the matrix commutator at the identity.
Matrix entry coordinates
We use the matrix entries \(X^i_j\) (\(1 \leq i, j \leq n\)) as global coordinates on
\(GL(n, \mathbb{R}) \subseteq \mathfrak{gl}(n, \mathbb{R})\). Because the indices \(i, j\) play the
dual role of coordinate indices and matrix row/column indices, the usual convention that all
coordinates carry upper indices cannot be maintained here. We adopt the convention used in the
proof itself: the summation convention remains in force, and an upper index appearing "in the
denominator" — that is, as a subscript on a coordinate beneath a partial-derivative bar —
is treated as a lower index for the purposes of summation. Concretely,
\(X^i_j\, \partial/\partial X^i_j\) is a summed expression over the index pair \((i, j)\).
Under these coordinates the canonical isomorphism
\(T_{I_n} GL(n, \mathbb{R}) \leftrightarrow \mathfrak{gl}(n, \mathbb{R})\) of the previous heading
takes the explicit form
\[
A^i_j\, \frac{\partial}{\partial X^i_j}\bigg|_{I_n}
\;\longleftrightarrow\;
\bigl( A^i_j \bigr) ,
\]
sending the tangent vector at the identity with coordinate components \(A^i_j\) to the matrix with
those entries.
The left-invariant field determined by a matrix
Given \(A = (A^i_j) \in \mathfrak{gl}(n, \mathbb{R})\), the corresponding left-invariant vector field
\(A^L\) on \(GL(n, \mathbb{R})\) is determined by its value at the identity through the formula
\(A^L|_X = d(L_X)_{I_n}(A)\) of the evaluation isomorphism proof, where \(L_X(Y) = XY\) is
left translation
by \(X\). The map \(L_X : \mathfrak{gl}(n, \mathbb{R}) \to \mathfrak{gl}(n, \mathbb{R})\) defined by
\(L_X(Y) = XY\) is linear in \(Y\) for fixed \(X\); its
differential
at any base point is therefore represented in coordinates by the same linear map, since on an open
subset of a vector space a linear map and its differential agree under the canonical identification
of the tangent space with the vector space itself.
Applying this differential to the tangent vector \(A^i_j\, \partial/\partial X^i_j|_{I_n}\) at the
identity uses the matrix product expansion \((XA)^i_k = X^i_j A^j_k\): the linear map sends the
column vector \((A^i_j)\) to the column vector \((XA)^i_k = X^i_j A^j_k\), with the appropriate
reinterpretation of the upper-then-lower index pair as a coordinate component. The result, valid at
every \(X \in GL(n, \mathbb{R})\), is
\[
A^L|_X
= X^i_j A^j_k\, \frac{\partial}{\partial X^i_k}\bigg|_X .
\tag{2}
\]
The formula (2) replaces the abstract construction \(A \mapsto A^L\) with a coordinate expression
in the matrix entries of \(X\) and the matrix entries of the fixed input matrix \(A\). It is the
point of departure for the bracket computation carried out in the next part of this section.
The bracket computation
With the coordinate expression (2) for left-invariant vector fields on \(GL(n, \mathbb{R})\) in
hand, the Lie bracket \([A^L, B^L]\) for two matrices \(A, B \in \mathfrak{gl}(n, \mathbb{R})\) can
be evaluated in coordinates using the
coordinate formula for the Lie bracket
of vector fields. The computation, while index-heavy, reduces by way of Kronecker contractions to
the matrix commutator.
Proof (of the Proposition):
Let \(A, B \in \mathfrak{gl}(n, \mathbb{R})\) and write \(A^L\) and \(B^L\) in matrix entry
coordinates using (2):
\[
A^L|_X = X^i_j A^j_k\, \frac{\partial}{\partial X^i_k}\bigg|_X ,
\qquad
B^L|_X = X^p_q B^q_r\, \frac{\partial}{\partial X^p_r}\bigg|_X ,
\]
where the index pairs \((i, k)\) and \((p, r)\) have been chosen distinct to avoid collision in
the bracket computation below. Applying the coordinate formula for the Lie bracket of vector
fields,
\[
\begin{align*}
[A^L, B^L]
&= X^i_j A^j_k\, \frac{\partial}{\partial X^i_k}\!
\left( X^p_q B^q_r \right) \frac{\partial}{\partial X^p_r} \\
&\quad - X^p_q B^q_r\, \frac{\partial}{\partial X^p_r}\!
\left( X^i_j A^j_k \right) \frac{\partial}{\partial X^i_k} ,
\end{align*}
\]
where each inner partial derivative is taken of the coefficient function in front of the
coordinate basis vector to the right of it. The components \(A^j_k\) and \(B^q_r\) of the fixed
matrices \(A\) and \(B\) are constants, so the partial derivatives act only on the coordinate
functions \(X^p_q\) and \(X^i_j\).
The relevant computation is that of the partial derivative of one matrix entry coordinate with
respect to another, which is the Kronecker pair
\[
\frac{\partial X^p_q}{\partial X^i_k} = \delta^p_i\, \delta^q_k ,
\]
equal to \(1\) when \(p = i\) and \(q = k\), and \(0\) otherwise. Applying this to the first
term,
\[
\frac{\partial}{\partial X^i_k}\!\left( X^p_q B^q_r \right)
= B^q_r\, \frac{\partial X^p_q}{\partial X^i_k}
= B^q_r\, \delta^p_i\, \delta^q_k
= \delta^p_i\, B^k_r ,
\]
where the summation over \(q\) collapses to \(q = k\) by \(\delta^q_k\). Substituting back, the
first term becomes
\[
X^i_j A^j_k \cdot \delta^p_i\, B^k_r \cdot \frac{\partial}{\partial X^p_r}
= X^i_j A^j_k B^k_r\, \frac{\partial}{\partial X^i_r} ,
\]
where the summation over \(p\) collapses to \(p = i\) by \(\delta^p_i\). Symmetrically, the
second term reduces to
\[
X^p_q B^q_r A^r_k\, \frac{\partial}{\partial X^p_k} ,
\]
and after renaming the dummy indices \((p, q, r, k) \to (i, j, k, r)\) to bring it onto the
common basis vector \(\partial/\partial X^i_r\), the second term becomes
\(X^i_j B^j_k A^k_r\, \partial/\partial X^i_r\). The Lie bracket is therefore
\[
[A^L, B^L]\bigl|_X
= X^i_j\, \bigl( A^j_k B^k_r - B^j_k A^k_r \bigr)\, \frac{\partial}{\partial X^i_r}
\bigg|_X .
\]
The bracketed expression in \(j\) and \(r\) is the \((j, r)\) entry of the matrix commutator:
\[
A^j_k B^k_r - B^j_k A^k_r
= (AB)^j_r - (BA)^j_r
= [A, B]^j_r ,
\]
where \(A^j_k B^k_r = (AB)^j_r\) is matrix multiplication summed over \(k\), and similarly for
\(BA\). Substituting,
\[
[A^L, B^L]\bigl|_X
= X^i_j\, [A, B]^j_r\, \frac{\partial}{\partial X^i_r}\bigg|_X ,
\]
which is exactly the formula (2) applied to the matrix \([A, B]\): in other words,
\([A^L, B^L]|_X = [A, B]^L|_X\) for every \(X \in GL(n, \mathbb{R})\), so
\[
[A^L, B^L] = [A, B]^L
\]
as left-invariant vector fields on \(GL(n, \mathbb{R})\).
Evaluating at the identity matrix, where \(X^i_j = \delta^i_j\), the formula above contracts to
\[
[A^L, B^L]\bigl|_{I_n}
= [A, B]^i_r\, \frac{\partial}{\partial X^i_r}\bigg|_{I_n} ,
\]
which corresponds under the canonical isomorphism
\(T_{I_n} GL(n, \mathbb{R}) \leftrightarrow \mathfrak{gl}(n, \mathbb{R})\) to the matrix
\([A, B] = AB - BA\). The composition (1) therefore sends the bracket of left-invariant
vector fields to the matrix commutator,
\[
[A^L, B^L]\bigl|_{I_n} \longleftrightarrow [A, B] ,
\]
which is the statement that the composition is bracket-preserving. Being already a vector space
isomorphism, it is a Lie algebra isomorphism between
\(\mathrm{Lie}(GL(n, \mathbb{R}))\) and \(\mathfrak{gl}(n, \mathbb{R})\).
The Central Computation of the Series
The identity \([A^L, B^L] = [A, B]^L\) just proved is the central conceptual payoff of the
development from the linear-algebra track through to this point. On the linear-algebra side,
the
matrix commutator
\([A, B] = AB - BA\) was introduced as a binary operation on \(M(n, \mathbb{R})\) with no
differential-geometric content. On the smooth-manifold side, the
Lie bracket of vector fields
\([X, Y] = XY - YX\) was introduced as the antisymmetrization of operator composition on
\(C^\infty(M)\), with no obvious connection to matrix multiplication.
The two operations, sharing nothing but a symbol, are revealed by the computation above to be
the same operation under the canonical correspondence between \(\mathrm{Lie}(GL(n, \mathbb{R}))\)
and \(\mathfrak{gl}(n, \mathbb{R})\): a left-invariant vector field on the general linear group
is determined by a matrix at the identity, and the Lie bracket of two such vector fields
evaluates at the identity to the matrix commutator of the corresponding matrices. The
linear-algebra and smooth-manifold tracks of the series describe the same Lie algebra of
\(GL(n, \mathbb{R})\) from different angles, and the computation above is the explicit bridge
between them.
Generalization to abstract vector spaces
The argument for \(GL(n, \mathbb{R})\) carries over to the Lie group \(GL(V)\) of invertible linear
transformations on any finite-dimensional real vector space \(V\), with the corresponding Lie
algebra \(\mathfrak{gl}(V)\) of all linear endomorphisms of \(V\) under the commutator bracket. The
classification of \(V\) up to isomorphism by its dimension reduces the abstract setting to the
matrix setting, but the formulation in terms of \(V\) itself is the version used downstream when
the same Lie algebra appears in representation-theoretic or operator-algebraic contexts.
The vector space \(\mathfrak{gl}(V)\) and the Lie group \(GL(V)\) sit in the same relationship as
\(\mathfrak{gl}(n, \mathbb{R})\) and \(GL(n, \mathbb{R})\): the group \(GL(V)\) is an open subset of
\(\mathfrak{gl}(V)\), defined by the non-vanishing of the determinant, and the
tangent space at the identity
\(\mathrm{Id} \in GL(V)\) is canonically identified with the ambient vector space \(\mathfrak{gl}(V)\).
Composing with the
evaluation isomorphism
yields a sequence of vector space isomorphisms
\[
\mathrm{Lie}\bigl( GL(V) \bigr)
\xrightarrow{\varepsilon}
T_{\mathrm{Id}} GL(V)
\xrightarrow{\;\cong\;}
\mathfrak{gl}(V) ,
\tag{3}
\]
parallel to the sequence (1) for \(GL(n, \mathbb{R})\).
Corollary (Lie Algebra of \(GL(V)\))
Let \(V\) be a finite-dimensional real vector space. The composition of the canonical
isomorphisms in (3) is a Lie algebra isomorphism between \(\mathrm{Lie}(GL(V))\) and
\(\mathfrak{gl}(V)\).
Proof:
Choose a basis for \(V\), and let \(n = \dim V\). The basis induces a vector space isomorphism
\(V \cong \mathbb{R}^n\), and the corresponding identifications of linear transformations with
matrices yield a Lie group isomorphism \(GL(V) \cong GL(n, \mathbb{R})\) and a Lie algebra
isomorphism \(\mathfrak{gl}(V) \cong \mathfrak{gl}(n, \mathbb{R})\); the second of these
preserves the commutator bracket because matrix multiplication and operator composition agree
under the matrix representation of linear maps. The induced Lie algebra isomorphism between
\(\mathrm{Lie}(GL(V))\) and \(\mathrm{Lie}(GL(n, \mathbb{R}))\) fits into a commutative diagram
with the canonical sequences (3) and (1), the vertical maps of which are Lie algebra
isomorphisms induced by the basis choice. Since the bottom row of the diagram is a Lie algebra
isomorphism by the proposition above, so is the top row. The conclusion is independent of the
choice of basis: a different basis induces a different but parallel commutative diagram, and
either one establishes that (3) is bracket-preserving.
The corollary closes this section. Two further generalizations — from \(GL(n, \mathbb{R})\) to
its complex analogue \(GL(n, \mathbb{C})\), and from open subsets of matrix algebras to general Lie
subgroups of \(GL(n, \mathbb{R})\) — appear in the next two sections, both as applications of
the induced Lie algebra homomorphism associated with a Lie group homomorphism.
Induced Lie Algebra Homomorphisms
The Lie algebra of a Lie group encodes much of the group's structure, and one of the constructions
that exhibits this correspondence is the assignment of a Lie algebra homomorphism to each Lie group
homomorphism. Given a smooth homomorphism \(F : G \to H\) between Lie groups, the differential at
the identity transports tangent vectors in a linear and bracket-respecting way; the
left-invariance
of the source vector field is enough to extend the construction from a single point to a vector
field on \(H\), without the usual hypothesis that \(F\) be a diffeomorphism.
Lie group homomorphisms induce Lie algebra homomorphisms
Theorem (Induced Lie Algebra Homomorphisms)
Let \(G\) and \(H\) be Lie groups, and let \(\mathfrak{g}\) and \(\mathfrak{h}\) be their Lie
algebras. Suppose \(F : G \to H\) is a
Lie group homomorphism.
For every \(X \in \mathfrak{g}\), there is a unique vector field in \(\mathfrak{h}\) that is
\(F\)-related
to \(X\). Denoting this vector field by \(F_* X\), the map
\[
F_* : \mathfrak{g} \to \mathfrak{h}
\]
is a Lie algebra homomorphism.
Proof:
Throughout the proof we identify \(\mathfrak{g}\) with the space of smooth left-invariant
vector fields on \(G\), and similarly for \(\mathfrak{h}\), via the evaluation isomorphism. Let
\(X \in \mathfrak{g}\) be given.
Step 1: the unique candidate. Any \(Y \in \mathfrak{h}\) that is \(F\)-related to
\(X\) must satisfy \(Y_{F(p)} = dF_p(X_p)\) for every \(p \in G\); evaluating at the identity
\(p = e_G\), and using that \(F\) is a Lie group homomorphism and hence \(F(e_G) = e_H\),
\[
Y_{e_H} = dF_{e_G}( X_{e_G} ) .
\]
Since \(Y \in \mathfrak{h}\) is left-invariant, its value at \(e_H\) uniquely determines \(Y\)
across all of \(H\), so any \(F\)-related \(Y\) must equal the left-invariant vector field on
\(H\) with value \(dF_{e_G}(X_{e_G})\) at the identity, namely
\[
Y := \bigl( dF_{e_G}(X_{e_G}) \bigr)^L .
\]
It remains to verify that this \(Y\) is in fact \(F\)-related to \(X\), and that the assignment
\(X \mapsto Y\) is a Lie algebra homomorphism.
Step 2: the candidate is \(F\)-related to \(X\). The Lie group homomorphism identity
\(F(gh) = F(g) F(h)\), applied with \(h\) varying over \(G\), gives the equality of smooth maps
\(G \to H\)
\[
F \circ L_g = L_{F(g)} \circ F ,
\]
since for every \(h \in G\), \((F \circ L_g)(h) = F(gh) = F(g) F(h) = L_{F(g)}(F(h)) =
(L_{F(g)} \circ F)(h)\). Taking the differential at \(e_G\) and applying the
chain rule
yields
\[
dF_g \circ d(L_g)_{e_G} = d(L_{F(g)})_{e_H} \circ dF_{e_G}
\]
as linear maps \(T_{e_G}G \to T_{F(g)}H\). Applying both sides to \(X_{e_G}\) and using the
defining formulas for \(X_g = d(L_g)_{e_G}(X_{e_G})\) (left-invariance of \(X\)) and
\(Y_{F(g)} = d(L_{F(g)})_{e_H}( dF_{e_G}(X_{e_G}) )\) (the construction of \(Y\)),
\[
dF_g( X_g ) = dF_g \circ d(L_g)_{e_G}( X_{e_G} )
= d(L_{F(g)})_{e_H} \circ dF_{e_G}( X_{e_G} )
= Y_{F(g)} ,
\]
which is exactly the condition for \(Y\) to be \(F\)-related to \(X\). Combining with the
uniqueness from Step 1, there is a unique \(F\)-related vector field in \(\mathfrak{h}\), and
we denote it \(F_*X\).
Step 3: \(F_*\) preserves brackets. For \(X_1, X_2 \in \mathfrak{g}\), the
naturality of the Lie bracket
applied to the pairs \((X_1, F_*X_1)\) and \((X_2, F_*X_2)\) of \(F\)-related vector fields
gives that \([X_1, X_2]\) is \(F\)-related to \([F_*X_1, F_*X_2]\). Since the unique
\(F\)-related vector field in \(\mathfrak{h}\) associated with \([X_1, X_2]\) is by definition
\(F_*[X_1, X_2]\), we conclude
\[
F_*[X_1, X_2] = [F_*X_1, F_*X_2] ,
\]
which is the bracket-preservation condition for a Lie algebra homomorphism. Linearity of
\(F_*\) follows from linearity of the differential \(dF_{e_G}\), since \(F_*X = (dF_{e_G}(X_e))^L\)
and the assignment \(v \mapsto v^L\) is linear by construction. Hence \(F_*\) is a Lie algebra
homomorphism.
Pushforward Without Diffeomorphism
The
pushforward
of a vector field by a smooth map was originally defined only when the map is a
diffeomorphism: without surjectivity the construction misses points in the target, and without
injectivity it produces a multivalued assignment. The induced Lie algebra homomorphism
\(F_*\) of the theorem above uses the same symbol \(F_*\) but is defined under a strictly
weaker hypothesis: the map \(F\) need only be a Lie group homomorphism, not a diffeomorphism.
The reason the construction succeeds in the weaker setting is that the target vector field is
not built pointwise on \(H\) by transporting values from preimages in \(G\) — an
operation that requires \(F\) to be invertible — but rather by first pushing the single
tangent vector \(X_{e_G}\) forward through \(dF_{e_G}\) to a single tangent vector at \(e_H\),
and then propagating that single vector across all of \(H\) by left-invariance. The
construction therefore costs nothing in surjectivity (left-invariance defines the field on all
of \(H\) from one tangent vector at \(e_H\)) and nothing in injectivity (the target is
determined by the action of the differential at a single point, not by inverting \(F\)). The
same symbol \(F_*\) does double duty for the two operations — pushforward of an arbitrary
vector field, valid for diffeomorphisms, and the induced Lie algebra homomorphism, valid for
Lie group homomorphisms — with the second operation an extension of the first to a
weaker hypothesis on \(F\), at the cost of restricting the class of vector fields involved to
the left-invariant ones.
Functorial properties of the induced homomorphism
The construction \(F \mapsto F_*\) of the previous theorem behaves well with respect to identity
maps and compositions of Lie group homomorphisms, in the sense that it preserves both. As a
consequence, Lie group isomorphisms induce Lie algebra isomorphisms, formalizing the principle that
the Lie algebra is an invariant of the Lie group up to isomorphism.
Proposition (Properties of Induced Homomorphisms)
Let \(G\), \(H\), and \(K\) be Lie groups. The induced Lie algebra homomorphism construction
satisfies:
(a) Identity: The induced homomorphism of the identity map
\(\mathrm{Id}_G : G \to G\) is the identity of \(\mathrm{Lie}(G)\):
\[
(\mathrm{Id}_G)_* = \mathrm{Id}_{\mathrm{Lie}(G)} .
\]
(b) Composition: If \(F_1 : G \to H\) and \(F_2 : H \to K\) are Lie group
homomorphisms, then
\[
(F_2 \circ F_1)_* = (F_2)_* \circ (F_1)_*
\;:\; \mathrm{Lie}(G) \to \mathrm{Lie}(K) .
\]
(c) Isomorphism preservation: Isomorphic Lie groups have isomorphic Lie
algebras: if \(F : G \to H\) is a Lie group isomorphism, then
\(F_* : \mathrm{Lie}(G) \to \mathrm{Lie}(H)\) is a Lie algebra isomorphism.
Proof:
Throughout the proof we use the formula \(F_*X = (dF_{e_G}(X_{e_G}))^L\) established in Step 1
of the previous theorem, which shows that the induced homomorphism at the level of Lie algebras
is determined by the differential of \(F\) at the identity. The properties of induced
homomorphisms reduce to the corresponding properties of differentials at the identity.
(a) The
differential of the identity
is the identity: \(d(\mathrm{Id}_G)_{e_G} = \mathrm{Id}_{T_{e_G}G}\). For
\(X \in \mathrm{Lie}(G)\),
\[
(\mathrm{Id}_G)_*X = \bigl( d(\mathrm{Id}_G)_{e_G}(X_{e_G}) \bigr)^L = (X_{e_G})^L = X ,
\]
where the last equality uses that \(X\) is itself the unique left-invariant vector field with
value \(X_{e_G}\) at the identity. Hence \((\mathrm{Id}_G)_* = \mathrm{Id}_{\mathrm{Lie}(G)}\).
(b) The chain rule for differentials gives
\(d(F_2 \circ F_1)_{e_G} = d(F_2)_{e_H} \circ d(F_1)_{e_G}\), where \(e_H = F_1(e_G)\) by the
Lie group homomorphism property of \(F_1\). For \(X \in \mathrm{Lie}(G)\),
\[
\begin{align*}
(F_2 \circ F_1)_*X
&= \bigl( d(F_2 \circ F_1)_{e_G}( X_{e_G} ) \bigr)^L
= \bigl( d(F_2)_{e_H} \circ d(F_1)_{e_G}(X_{e_G}) \bigr)^L \\
&= \bigl( d(F_2)_{e_H}\bigl( (F_1)_*X|_{e_H} \bigr) \bigr)^L
= (F_2)_*\bigl( (F_1)_*X \bigr)
= \bigl( (F_2)_* \circ (F_1)_* \bigr) X ,
\end{align*}
\]
where the middle equalities use that \((F_1)_*X = (d(F_1)_{e_G}(X_{e_G}))^L\) and so
\((F_1)_*X|_{e_H} = d(F_1)_{e_G}(X_{e_G})\).
(c) If \(F : G \to H\) is a Lie group isomorphism, its inverse \(F^{-1} : H \to G\) is
also a Lie group homomorphism, and \(F \circ F^{-1} = \mathrm{Id}_H\),
\(F^{-1} \circ F = \mathrm{Id}_G\). Applying (a) and (b),
\[
F_* \circ (F^{-1})_*
= (F \circ F^{-1})_*
= (\mathrm{Id}_H)_*
= \mathrm{Id}_{\mathrm{Lie}(H)} ,
\]
and symmetrically \((F^{-1})_* \circ F_* = \mathrm{Id}_{\mathrm{Lie}(G)}\). Hence \(F_*\) is a
Lie algebra isomorphism with inverse \((F^{-1})_*\).
A categorical perspective
Properties (a) and (b) of the proposition together state that the assignment
\(G \mapsto \mathrm{Lie}(G)\) on objects and \(F \mapsto F_*\) on morphisms is functorial in the
sense of category theory: it sends identity morphisms to identity morphisms and respects
composition. In categorical language, this defines a covariant functor from the category of Lie
groups and Lie group homomorphisms to the category of finite-dimensional real Lie algebras and
Lie algebra homomorphisms. We use the descriptive name and the property it captures — that
the assignment is compatible with both kinds of structure preservation — without developing
the categorical formalism further on this page; the systematic study of functorial constructions in
the smooth-manifold setting is taken up in the categorical track of the series.
The complex general linear group
The complex analogue of the result for \(GL(n, \mathbb{R})\) holds with the same structure of proof.
Throughout this section we treat \(GL(n, \mathbb{C})\) and the complex matrix algebra
\(\mathfrak{gl}(n, \mathbb{C})\) as real manifolds and real vector spaces respectively, of real
dimension \(2n^2\), in keeping with the convention adopted in this development that all Lie algebras
are real.
Just as in the real case, the group \(GL(n, \mathbb{C})\) is an open subset of \(\mathfrak{gl}(n,
\mathbb{C})\), defined by the non-vanishing of the complex determinant. Its
tangent space at the identity
is therefore canonically identified with \(\mathfrak{gl}(n, \mathbb{C})\) as a real vector space, and
composing with the
evaluation isomorphism
yields a sequence of vector space isomorphisms
\[
\mathrm{Lie}\bigl( GL(n, \mathbb{C}) \bigr)
\xrightarrow{\varepsilon}
T_{I_n} GL(n, \mathbb{C})
\xrightarrow{\;\varphi\;}
\mathfrak{gl}(n, \mathbb{C}) ,
\tag{4}
\]
parallel to (1) for \(GL(n, \mathbb{R})\) and (3) for \(GL(V)\).
Proposition (Lie Algebra of \(GL(n, \mathbb{C})\))
The composition of the maps in (4) is a Lie algebra isomorphism between
\(\mathrm{Lie}(GL(n, \mathbb{C}))\) and the matrix algebra \(\mathfrak{gl}(n, \mathbb{C})\).
Proof:
The strategy is to compare (4) with the corresponding sequence (1) for
\(GL(2n, \mathbb{R})\) via the standard complex-to-real Lie group homomorphism
\[
\beta : GL(n, \mathbb{C}) \to GL(2n, \mathbb{R})
\]
sending a complex \(n \times n\) matrix to its real \(2n \times 2n\) representation under the
identification \(\mathbb{C}^n \cong \mathbb{R}^{2n}\); this map is a Lie group homomorphism by
construction. By the
induced homomorphism theorem,
\(\beta\) induces a Lie algebra homomorphism
\(\beta_* : \mathrm{Lie}(GL(n, \mathbb{C})) \to \mathrm{Lie}(GL(2n, \mathbb{R}))\). Composing
\(\beta_*\) with the canonical isomorphisms (4) at the top and (1) at the bottom yields a
commutative diagram of vector spaces and linear maps:
\[
\begin{array}{ccccc}
\mathrm{Lie}(GL(n, \mathbb{C}))
& \xrightarrow{\varepsilon}
& T_{I_n} GL(n, \mathbb{C})
& \xrightarrow{\varphi}
& \mathfrak{gl}(n, \mathbb{C}) \\
{\scriptstyle \beta_*}\downarrow
& & {\scriptstyle d\beta_{I_n}}\downarrow
& & {\scriptstyle \alpha}\downarrow \\
\mathrm{Lie}(GL(2n, \mathbb{R}))
& \xrightarrow{\varepsilon}
& T_{I_{2n}} GL(2n, \mathbb{R})
& \xrightarrow{\varphi}
& \mathfrak{gl}(2n, \mathbb{R}) ,
\end{array}
\tag{5}
\]
where \(\alpha = \varphi \circ d\beta_{I_n} \circ \varphi^{-1}\) is the linear map of matrix
algebras induced by \(d\beta_{I_n}\) under the canonical identifications.
The bottom row is a Lie algebra isomorphism. The composition along the bottom row of
(5) is exactly the sequence (1) for \(GL(2n, \mathbb{R})\), and the
proposition for the real general linear group
established that this composition is a Lie algebra isomorphism between
\(\mathrm{Lie}(GL(2n, \mathbb{R}))\) and \(\mathfrak{gl}(2n, \mathbb{R})\).
The map \(\alpha\) preserves matrix commutators. Since \(\beta\) is a Lie group
homomorphism on matrix groups, where the group operation is matrix multiplication, the map
\(\beta\) preserves matrix products: \(\beta(AB) = \beta(A)\beta(B)\) for all
\(A, B \in GL(n, \mathbb{C})\); the restriction of \(\beta\) to the open subset
\(GL(n, \mathbb{C}) \subseteq \mathfrak{gl}(n, \mathbb{C})\) extends to a linear map
\(\mathfrak{gl}(n, \mathbb{C}) \to \mathfrak{gl}(2n, \mathbb{R})\) with the same coordinate
expression, since the embedding into block real matrices is itself linear. This linear extension
is the map \(\alpha\), and the product-preservation property of \(\beta\) descends to
\(\alpha(AB) = \alpha(A)\alpha(B)\) for all \(A, B \in \mathfrak{gl}(n, \mathbb{C})\). Applying
this to the commutator,
\[
\alpha[A, B]
= \alpha(AB - BA)
= \alpha(A)\alpha(B) - \alpha(B)\alpha(A)
= [\alpha(A), \alpha(B)] ,
\]
where the middle equality uses linearity of \(\alpha\). Hence \(\alpha\) is an injective Lie
algebra homomorphism \(\mathfrak{gl}(n, \mathbb{C}) \to \mathfrak{gl}(2n, \mathbb{R})\) when
both are regarded as Lie algebras under the matrix commutator.
Diagram chase. Let \(\Phi := \varphi \circ \varepsilon\) denote the top row composition
for \(GL(n, \mathbb{C})\) and \(\Psi := \varphi \circ \varepsilon\) the bottom row composition
for \(GL(2n, \mathbb{R})\); both are vector space isomorphisms. We show \(\Phi\) is
bracket-preserving by chasing the diagram. Fix \(X_1, X_2 \in \mathrm{Lie}(GL(n, \mathbb{C}))\)
and compute \(\alpha\bigl( \Phi[X_1, X_2] \bigr)\) along the two paths in (5) connecting the
top-left and bottom-right corners.
Going right then down,
\[
\alpha\bigl( \Phi[X_1, X_2] \bigr) = \alpha \circ \Phi\, [X_1, X_2] .
\]
Going down then right and using commutativity \(\alpha \circ \Phi = \Psi \circ \beta_*\),
\[
\alpha \circ \Phi\, [X_1, X_2]
= \Psi\bigl( \beta_*[X_1, X_2] \bigr)
= \Psi\bigl( [\beta_*X_1, \beta_*X_2] \bigr) ,
\]
where the last equality uses that \(\beta_*\) is a Lie algebra homomorphism by the induced
homomorphism theorem. Since the bottom row \(\Psi\) is a Lie algebra isomorphism and \(\alpha\)
preserves brackets,
\[
\Psi\bigl( [\beta_*X_1, \beta_*X_2] \bigr)
= [\Psi \beta_*X_1, \Psi \beta_*X_2]
= [\alpha \Phi X_1, \alpha \Phi X_2]
= \alpha\bigl( [\Phi X_1, \Phi X_2] \bigr) ,
\]
applying commutativity \(\Psi \circ \beta_* = \alpha \circ \Phi\) in the middle equality and the
bracket-preservation of \(\alpha\) at the last. Combining the two paths,
\[
\alpha\bigl( \Phi[X_1, X_2] \bigr) = \alpha\bigl( [\Phi X_1, \Phi X_2] \bigr) ,
\]
and since \(\alpha\) is injective, \(\Phi[X_1, X_2] = [\Phi X_1, \Phi X_2]\). The top row
composition \(\Phi = \varphi \circ \varepsilon\) is therefore a Lie algebra isomorphism between
\(\mathrm{Lie}(GL(n, \mathbb{C}))\) and \(\mathfrak{gl}(n, \mathbb{C})\).
The proposition closes the analysis of matrix Lie groups internal to this section. The remaining
section of the page handles the case of general Lie subgroups of \(GL(n, \mathbb{R})\), where the
induced Lie algebra homomorphism associated with the inclusion of a subgroup identifies the Lie
algebra of the subgroup with a Lie subalgebra of the ambient matrix Lie algebra.
Lie Subgroups and Classical Examples
A direct application of the induced homomorphism construction is its specialization to the
inclusion of a
Lie subgroup
\(H \subseteq G\). The inclusion \(\iota : H \hookrightarrow G\) is a Lie group homomorphism, so the
induced homomorphism theorem of the previous section produces a Lie algebra homomorphism
\(\iota_* : \mathrm{Lie}(H) \to \mathrm{Lie}(G)\). Sharpening the conclusions of that theorem for
the special case of an inclusion identifies \(\mathrm{Lie}(H)\) with a Lie subalgebra of
\(\mathrm{Lie}(G)\) characterized by a tangent space condition at the identity.
Identifying Lie(H) inside Lie(G)
Theorem (Lie Algebra of a Lie Subgroup)
Suppose \(H \subseteq G\) is a Lie subgroup, and let \(\iota : H \hookrightarrow G\) be the
inclusion. The induced Lie algebra homomorphism
\(\iota_* : \mathrm{Lie}(H) \to \mathrm{Lie}(G)\) is injective, and its image is the Lie
subalgebra
\[
\iota_*\bigl( \mathrm{Lie}(H) \bigr)
= \bigl\{ X \in \mathrm{Lie}(G) : X_e \in T_e H \bigr\} ,
\]
where \(T_e H\) is viewed as a vector subspace of \(T_e G\) under the
tangent space inclusion
\(d\iota_e : T_eH \hookrightarrow T_eG\). Under this identification, \(\mathrm{Lie}(H)\) becomes
the Lie subalgebra of \(\mathrm{Lie}(G)\) consisting of those left-invariant vector fields on
\(G\) whose value at \(e\) is tangent to \(H\).
Proof:
The induced map \(\iota_*\) is a Lie algebra homomorphism by the
induced homomorphism theorem.
We verify the two remaining claims: injectivity, and the explicit characterization of the image.
Injectivity. The inclusion of a Lie subgroup is an immersion at every point; in
particular, the differential \(d\iota_e : T_eH \to T_eG\) is injective. Using the formula
\(\iota_*X = (d\iota_e(X_e))^L\) from the induced homomorphism theorem, suppose
\(\iota_*X = 0\) for some \(X \in \mathrm{Lie}(H)\). Evaluating at \(e\),
\[
0 = (\iota_*X)_e = d\iota_e(X_e) ,
\]
and injectivity of \(d\iota_e\) gives \(X_e = 0\). Since \(X\) is left-invariant on \(H\) and is
determined by its value at the identity via the
evaluation isomorphism,
\(X = 0\) on all of \(H\). Hence \(\iota_*\) is injective.
The image as the tangent-condition subalgebra. One inclusion is immediate: for any
\(X \in \mathrm{Lie}(H)\), the field \(\iota_*X\) has value
\((\iota_*X)_e = d\iota_e(X_e) \in d\iota_e(T_eH)\) at the identity, which lies in \(T_eH\)
under the identification of \(T_eH\) with its image in \(T_eG\). Therefore
\[
\iota_*\bigl( \mathrm{Lie}(H) \bigr)
\subseteq \bigl\{ X \in \mathrm{Lie}(G) : X_e \in T_eH \bigr\} .
\]
For the reverse inclusion, suppose \(X \in \mathrm{Lie}(G)\) satisfies \(X_e \in T_eH\). Let
\(Y \in \mathrm{Lie}(H)\) be the left-invariant vector field on \(H\) whose value at the
identity \(e \in H\) equals \(X_e\); such a \(Y\) exists and is unique by the evaluation
isomorphism for \(H\), which uses precisely that \(X_e \in T_eH\) to define the candidate. The
induced field \(\iota_*Y \in \mathrm{Lie}(G)\) has value
\[
(\iota_*Y)_e = d\iota_e(Y_e) = d\iota_e(X_e) = X_e ,
\]
identifying \(T_eH\) with its image in \(T_eG\). Both \(\iota_*Y\) and \(X\) are left-invariant
on \(G\) and agree at the identity, so by the evaluation isomorphism for \(G\) they are equal
as left-invariant vector fields on \(G\). Hence \(X = \iota_*Y\) lies in the image of
\(\iota_*\), and
\[
\bigl\{ X \in \mathrm{Lie}(G) : X_e \in T_eH \bigr\}
\subseteq \iota_*\bigl( \mathrm{Lie}(H) \bigr) .
\]
Combining the two inclusions establishes the claimed equality.
The image \(\iota_*(\mathrm{Lie}(H))\) is closed under the bracket of \(\mathrm{Lie}(G)\), since
\(\iota_*\) is a Lie algebra homomorphism; therefore it is a Lie subalgebra of
\(\mathrm{Lie}(G)\), and the identification \(\mathrm{Lie}(H) \cong \iota_*(\mathrm{Lie}(H))\)
is one of Lie algebras.
The theorem permits a Lie subgroup's Lie algebra to be computed as a tangent space at the identity
inside the ambient Lie algebra, without separately constructing left-invariant vector fields on the
subgroup. Each classical matrix subgroup of \(GL(n, \mathbb{R})\) thereby acquires an explicit
description of its Lie algebra as a subspace of \(\mathfrak{gl}(n, \mathbb{R})\) defined by a
linear condition arising from the defining equations of the subgroup. The next part of this section
applies this principle to the orthogonal group, completing the identification of its Lie algebra
with the skew-symmetric matrix algebra introduced on the linear-algebra side of the series.
The Lie algebra of the orthogonal group
The
orthogonal group
\(O(n) \subseteq GL(n, \mathbb{R})\) is the subgroup of \(n \times n\) real matrices preserving the
Euclidean inner product, equivalently the set of matrices \(A\) satisfying \(A^T A = I_n\). As an
embedded submanifold of \(GL(n, \mathbb{R})\) cut out by a smooth defining equation, \(O(n)\) is a
Lie subgroup of \(GL(n, \mathbb{R})\), so the theorem above applies and identifies
\(\mathrm{Lie}(O(n))\) with a Lie subalgebra of \(\mathrm{Lie}(GL(n, \mathbb{R})))
= \mathfrak{gl}(n, \mathbb{R})\) characterized by a tangent-condition at the identity.
Example (Lie Algebra of \(O(n)\)):
Let \(\mathrm{Sym}(n, \mathbb{R}) \subseteq M(n, \mathbb{R})\) denote the
\(n(n+1)/2\)-dimensional vector subspace of symmetric \(n \times n\) real matrices, and write
\(\Phi : GL(n, \mathbb{R}) \to \mathrm{Sym}(n, \mathbb{R})\) for the smooth map
\(\Phi(A) = A^T A\), whose image lies in \(\mathrm{Sym}(n, \mathbb{R})\) because \(A^T A\) is
symmetric for every \(A\). Then \(O(n) = \Phi^{-1}(I_n)\). To apply the theorem above we
identify \(T_{I_n} O(n)\) inside \(T_{I_n} GL(n, \mathbb{R}) = \mathfrak{gl}(n, \mathbb{R})\) by
differentiating the defining equation. The differential of \(\Phi\) at \(I_n\) is computed by
the product rule for matrix-valued functions: for \(B \in T_{I_n} GL(n, \mathbb{R})\),
\[
d\Phi_{I_n}(B)
= B^T \cdot I_n + I_n^T \cdot B
= B^T + B ,
\]
viewing \(\Phi\) as a function of two matrix arguments \((A_1, A_2) \mapsto A_1^T A_2\) along
the curve \(A_1 = A_2 = I_n + tB\) and computing the derivative at \(t = 0\). The map
\(B \mapsto B^T + B\) sends \(M(n, \mathbb{R})\) onto \(\mathrm{Sym}(n, \mathbb{R})\) —
every symmetric matrix \(S\) is the image \(d\Phi_{I_n}(S/2) = S\) — so \(d\Phi_{I_n}\) is
surjective. The same calculation shows \(d\Phi_A\) is surjective at every
\(A \in GL(n, \mathbb{R})\), making \(\Phi\) a
smooth submersion.
Since \(O(n)\) is a
level set
of \(\Phi\) at a regular value, its tangent space at the identity is the kernel of
\(d\Phi_{I_n}\),
\[
T_{I_n} O(n)
= \bigl\{ B \in \mathfrak{gl}(n, \mathbb{R}) : B^T + B = 0 \bigr\}
= \bigl\{ \text{skew-symmetric } n \times n \text{ real matrices} \bigr\} .
\]
Applying the theorem, \(\mathrm{Lie}(O(n))\) is canonically isomorphic to the Lie subalgebra of
\(\mathfrak{gl}(n, \mathbb{R})\) consisting of all skew-symmetric matrices, with the matrix
commutator as bracket.
Closing the Linear-Algebra–Smooth-Manifold Loop
The Lie subalgebra of \(\mathfrak{gl}(n, \mathbb{R})\) consisting of skew-symmetric matrices is
precisely the
algebra \(\mathfrak{so}(n)\)
introduced on the linear-algebra side of the series, where it was defined directly as a
commutator-closed subspace of the matrix algebra. The example above proves that this
purely algebraic object coincides, under the identifications of this page, with the Lie algebra
of the orthogonal group as a Lie subgroup of \(GL(n, \mathbb{R})\): the smooth-manifold
construction \(O(n) \mapsto \mathrm{Lie}(O(n))\) and the linear-algebra construction
\(\mathfrak{so}(n) = \{\text{skew-symmetric matrices}\}\) produce the same Lie algebra.
A point worth noting is that bracket-closure of the skew-symmetric matrices — the property
that the commutator of two skew-symmetric matrices is again skew-symmetric — was verified
on the linear-algebra side as a direct matrix computation, but in the present setting it
requires no separate verification. The Lie subgroup theorem produces the subalgebra structure
automatically from the immersed-submanifold structure of \(O(n)\) and the bracket-preservation
of the inclusion-induced homomorphism. The same automatic closure holds for every classical
matrix subgroup of \(GL(n, \mathbb{R})\), each of which inherits its Lie algebra structure from
the ambient \(\mathfrak{gl}(n, \mathbb{R})\) by way of a tangent-condition at the identity.
The structure theorem for finite-dimensional Lie algebras
The matrix Lie groups examined above — \(GL(n, \mathbb{R})\), \(GL(n, \mathbb{C})\), \(O(n)\)
— give rise to Lie algebras that are themselves matrix algebras or subalgebras of matrix
algebras. The natural question is whether every Lie group's Lie algebra arises this way, that is,
whether every finite-dimensional real Lie algebra can be realized as a Lie subalgebra of some
\(\mathfrak{gl}(n, \mathbb{R})\). The answer is affirmative, and is the content of a structural
theorem about Lie algebras that does not depend on differential-geometric considerations.
Theorem (Ado, and Matrix Realization of Lie Algebras)
Ado's Theorem: Every finite-dimensional real Lie algebra admits a faithful
finite-dimensional representation, i.e., an injective Lie algebra homomorphism into
\(\mathfrak{gl}(V)\) for some finite-dimensional real vector space \(V\).
Corollary: Every finite-dimensional real Lie algebra is isomorphic to a Lie
subalgebra of \(\mathfrak{gl}(n, \mathbb{R})\) for some \(n\), with the commutator bracket.
The corollary follows from Ado's Theorem by choosing a basis for the representing vector space
\(V\), which identifies \(\mathfrak{gl}(V)\) with \(\mathfrak{gl}(n, \mathbb{R})\) for
\(n = \dim V\); composing the faithful representation with this identification produces the desired
injective homomorphism. We state Ado's Theorem without proof: the argument requires algebraic
methods beyond the scope of this page, principally the structure theory of finite-dimensional Lie
algebras over a field.
An Asymmetry Between Lie Groups and Lie Algebras
The corollary to Ado's Theorem states that every finite-dimensional real Lie algebra is, up to
isomorphism, a matrix Lie algebra. The analogous statement for Lie groups is false: there exist
Lie groups that are not isomorphic to any Lie subgroup of \(GL(n, \mathbb{R})\) for any \(n\).
The simply-connected covers of certain Lie groups, and certain solvable Lie groups, fail to
admit faithful finite-dimensional representations, even though their Lie algebras do.
The asymmetry is structurally informative. Lie algebras are linear objects governed by finite
amounts of algebraic data — a vector space, a bracket, the Jacobi identity — and
the realization theorem makes them tractable through matrix computation. Lie groups carry
additional global topological and analytic structure (covering spaces, fundamental groups, the
possible failure of global integrability of a left-invariant connection) that is not seen by
the Lie algebra alone. The correspondence \(G \mapsto \mathrm{Lie}(G)\) developed across this
portion of the smooth-manifold track is therefore a local construction that loses global
information; the recovery of \(G\) from \(\mathrm{Lie}(G)\), to the extent that it is possible,
is the content of the integrability theorems for Lie algebras (sometimes called the third
fundamental theorem of Lie theory), which are taken up in subsequent developments. The site
treats the local correspondence as the content of the present series and notes the global
correspondence as a destination rather than a completed result.