Introduction
The Lie algebras we have constructed so far have been real vector spaces. Each arose as
the
tangent space at the identity
of a matrix Lie group — the set of initial velocities \(\gamma'(0)\) of smooth curves
\(\gamma\) through the identity — and a velocity vector can be scaled by any
real number, since \(t \mapsto \gamma(rt)\) reparametrizes the curve and
multiplies its initial velocity by \(r \in \mathbb{R}\). The elements of these algebras
may well be matrices with complex entries; what makes the algebra real is that
only real scalar multiples of its elements are guaranteed to lie in it again.
This restriction is not a formality. Consider
\(\mathfrak{su}(n)\),
the traceless skew-Hermitian matrices, where \(A^* = -A\). If \(A \in \mathfrak{su}(n)\)
and we multiply by the imaginary unit, then \((iA)^* = -i A^* = -i(-A) = iA\): the matrix
\(iA\) is Hermitian, not skew-Hermitian, so \(iA \notin \mathfrak{su}(n)\) unless
\(A = 0\). Multiplication by \(i\) carries the algebra entirely outside itself. The same
obstruction afflicts
\(\mathfrak{u}(n)\)
and the other compact algebras: they are closed under real scalar multiplication but not
under multiplication by all complex numbers, and so they are not vector spaces over
\(\mathbb{C}\).
There is, however, a natural way to manufacture a complex Lie algebra from a real one:
adjoin the missing imaginary multiples. The resulting object — the
complexification \(\mathfrak{g}_{\mathbb{C}}\) — is a genuine Lie algebra
over \(\mathbb{C}\), and it turns out to absorb several of our real algebras into a single
complex one. We will find that the traceless complex matrices \(\mathfrak{sl}(n;
\mathbb{C})\) are exactly the complexification of \(\mathfrak{su}(n)\), and that the full
matrix algebra \(\mathfrak{gl}(n; \mathbb{C})\) is the complexification of
\(\mathfrak{u}(n)\). Two real algebras of differing flavor thus share a single complex
home.
The motivation for the construction is not merely tidiness. When one classifies the ways
a group can act linearly — the subject of the representation theory that follows — the
cleanest results are obtained over \(\mathbb{C}\), where every operator has a full
complement of eigenvalues. A real Lie algebra and its complexification have, in a precise
sense to be developed later, the same complex representations; replacing a real
algebra such as \(\mathfrak{su}(2)\) by its complexification \(\mathfrak{sl}(2;
\mathbb{C})\) loses nothing and gains the analytic comfort of an algebraically closed
field. The present page builds the complexification and identifies it for the classical
algebras; the payoff — that the irreducible representations are most naturally classified
on the complexified algebra — is taken up in the pages on representation theory.
The Complexification
We saw that a real matrix Lie algebra is in general not closed under multiplication by
\(i\). The remedy is to enlarge it by formally adjoining all imaginary multiples of its
elements. Concretely, for a real matrix Lie algebra \(\mathfrak{g}\) — a real subspace
of \(M_n(\mathbb{C})\), closed under the
commutator bracket —
we consider the set of all matrices of the form \(A + iB\) with \(A, B \in \mathfrak{g}\).
Definition: Complexification of a Real Lie Algebra
Let \(\mathfrak{g} \subseteq M_n(\mathbb{C})\) be a real matrix Lie algebra. Its
complexification is the set
\[
\mathfrak{g}_{\mathbb{C}} = \mathfrak{g} + i\,\mathfrak{g}
= \{ A + iB : A, B \in \mathfrak{g} \},
\]
regarded as a complex vector space under the usual addition and scalar
multiplication of matrices, and equipped with the commutator bracket
\([X, Y] = XY - YX\).
The notation \(\mathfrak{g} + i\,\mathfrak{g}\) carries a claim that must be justified:
that this set is a complex vector space and that the commutator makes it a Lie algebra.
Neither is automatic — \(\mathfrak{g}\) itself is closed under the bracket but is not a
complex vector space, so we must check that adjoining the imaginary multiples repairs the
one defect without spoiling the other.
Proposition (\(\mathfrak{g}_{\mathbb{C}}\) is a Complex Lie Algebra)
Let \(\mathfrak{g} \subseteq M_n(\mathbb{C})\) be a real matrix Lie algebra. Then its
complexification \(\mathfrak{g}_{\mathbb{C}} = \mathfrak{g} + i\,\mathfrak{g}\) is a
Lie algebra
over \(\mathbb{C}\) under the commutator bracket.
Proof:
Closed under addition. For \(A_1 + iB_1\) and \(A_2 + iB_2\) in
\(\mathfrak{g}_{\mathbb{C}}\), the sum is \((A_1 + A_2) + i(B_1 + B_2)\); since
\(\mathfrak{g}\) is a real vector space, \(A_1 + A_2\) and \(B_1 + B_2\) lie in
\(\mathfrak{g}\), so the sum lies in \(\mathfrak{g}_{\mathbb{C}}\).
Closed under complex scalar multiplication. For a complex scalar
\(a + ib\) with \(a, b \in \mathbb{R}\) and an element \(A + iB \in
\mathfrak{g}_{\mathbb{C}}\),
\[
(a + ib)(A + iB) = (aA - bB) + i(bA + aB).
\]
Because \(\mathfrak{g}\) is closed under real scalar multiplication and addition,
both \(aA - bB\) and \(bA + aB\) lie in \(\mathfrak{g}\), so the product lies in
\(\mathfrak{g}_{\mathbb{C}}\). Thus \(\mathfrak{g}_{\mathbb{C}}\) is a complex vector
space. (Multiplication by \(i\), the case \(a = 0\), \(b = 1\), sends \(A + iB\) to
\(-B + iA\), which is again in \(\mathfrak{g}_{\mathbb{C}}\) — the defect that
\(\mathfrak{g}\) alone suffered is now repaired.)
Closed under the bracket. The commutator is bilinear over
\(\mathbb{C}\), so for \(A_1 + iB_1\) and \(A_2 + iB_2\) we may expand
\[
\begin{align*}
[A_1 + iB_1,\, A_2 + iB_2]
&= [A_1, A_2] - [B_1, B_2] + i\bigl([B_1, A_2] + [A_1, B_2]\bigr).
\end{align*}
\]
Each of \([A_1, A_2]\), \([B_1, B_2]\), \([B_1, A_2]\), and \([A_1, B_2]\) lies in
\(\mathfrak{g}\), because \(\mathfrak{g}\) is closed under the bracket. Hence the
real part \([A_1, A_2] - [B_1, B_2]\) and the imaginary part \([B_1, A_2] + [A_1,
B_2]\) both lie in \(\mathfrak{g}\), and the whole bracket lies in
\(\mathfrak{g}_{\mathbb{C}}\).
Finally, the
bracket axioms —
bilinearity, antisymmetry, and the Jacobi identity — hold for the commutator on all
of \(M_n(\mathbb{C})\), hence in particular on the subspace
\(\mathfrak{g}_{\mathbb{C}}\). Bilinearity now holds over \(\mathbb{C}\) rather than
merely over \(\mathbb{R}\), since \(\mathfrak{g}_{\mathbb{C}}\) is a complex vector
space. Therefore \(\mathfrak{g}_{\mathbb{C}}\) is a Lie algebra over \(\mathbb{C}\).
One point in the construction deserves emphasis. We wrote each element uniquely as
\(A + iB\) with \(A, B \in \mathfrak{g}\); this presumes that \(\mathfrak{g} \cap
i\,\mathfrak{g} = \{0\}\), so that the real and imaginary parts are unambiguous. For the
classical algebras this holds, and in those cases the real dimension of
\(\mathfrak{g}_{\mathbb{C}}\) is twice that of \(\mathfrak{g}\), while its complex
dimension equals the real dimension of \(\mathfrak{g}\). The next section identifies
\(\mathfrak{g}_{\mathbb{C}}\) explicitly for the families that matter to us, where this
doubling will be visible in the matrix descriptions.
Complexifying the Classical Algebras
We now compute the complexification for the two families that will recur in the
representation theory ahead. The results are clean: complexifying a compact real algebra
removes the skew-Hermitian constraint and leaves only the algebraic condition — tracelessness,
or none at all. Throughout, \(A^*\) denotes the conjugate transpose, matching the convention
used for the
compact classical algebras.
Proposition (\(\mathfrak{gl}(n; \mathbb{C})\) as a Complexification)
The full complex matrix algebra is the complexification of \(\mathfrak{u}(n)\):
\[
\mathfrak{gl}(n; \mathbb{C}) = M_n(\mathbb{C}) = \mathfrak{u}(n) + i\,\mathfrak{u}(n).
\]
Proof:
Every member of \(\mathfrak{u}(n) + i\,\mathfrak{u}(n)\) is an \(n \times n\) complex
matrix, so the inclusion \(\mathfrak{u}(n) + i\,\mathfrak{u}(n) \subseteq
M_n(\mathbb{C})\) is immediate. For the reverse inclusion, we must write an arbitrary
\(X \in M_n(\mathbb{C})\) as \(X_1 + iX_2\) with \(X_1, X_2 \in \mathfrak{u}(n)\),
that is, with \(X_1\) and \(X_2\) both skew-Hermitian. Set
\[
X_1 = \frac{X - X^*}{2}, \qquad X_2 = \frac{X + X^*}{2i}.
\]
A direct check gives \(X_1 + iX_2 = \frac{X - X^*}{2} + \frac{X + X^*}{2} = X\). Each
of \(X_1, X_2\) is skew-Hermitian: using \((X^*)^* = X\),
\[
\begin{align*}
X_1^* &= \frac{X^* - X}{2} = -X_1, \\\\
X_2^* &= \frac{X^* + X}{\overline{2i}} = \frac{X^* + X}{-2i} = -X_2.
\end{align*}
\]
Hence \(X_1, X_2 \in \mathfrak{u}(n)\) and \(X = X_1 + iX_2 \in \mathfrak{u}(n) +
i\,\mathfrak{u}(n)\).
Proposition (\(\mathfrak{sl}(n; \mathbb{C})\) as a Complexification)
The traceless complex matrix algebra is the complexification of \(\mathfrak{su}(n)\):
\[
\mathfrak{sl}(n; \mathbb{C}) = \{ X \in M_n(\mathbb{C}) : \mathrm{tr}(X) = 0 \}
= \mathfrak{su}(n) + i\,\mathfrak{su}(n).
\]
Proof:
Any \(X \in \mathfrak{su}(n)\) is traceless, so any \(X_1 + iX_2\) with \(X_1, X_2 \in
\mathfrak{su}(n)\) is traceless as well; this gives \(\mathfrak{su}(n) +
i\,\mathfrak{su}(n) \subseteq \mathfrak{sl}(n; \mathbb{C})\). Conversely, let \(X \in
M_n(\mathbb{C})\) be traceless, and form \(X_1\) and \(X_2\) by the same formulas as
above. They are skew-Hermitian, as just shown; moreover they are traceless, because
\(\mathrm{tr}(X) = 0\) forces \(\mathrm{tr}(X^*) = \overline{\mathrm{tr}(X)} = 0\),
whence
\[
\mathrm{tr}(X_1) = \frac{\mathrm{tr}(X) - \mathrm{tr}(X^*)}{2} = 0,
\qquad
\mathrm{tr}(X_2) = \frac{\mathrm{tr}(X) + \mathrm{tr}(X^*)}{2i} = 0.
\]
Thus \(X_1, X_2 \in \mathfrak{su}(n)\) and \(X = X_1 + iX_2 \in \mathfrak{su}(n) +
i\,\mathfrak{su}(n)\).
The decomposition \(X = X_1 + iX_2\) is moreover unique. If \(X = X_1 +
iX_2\) with \(X_1, X_2\) skew-Hermitian, then taking the conjugate transpose of both
sides and using \(X_1^* = -X_1\), \(X_2^* = -X_2\) gives \(X^* = -X_1 + iX_2\). Adding and
subtracting the two equations isolates the parts:
\[
\begin{align*}
X - X^* &= 2X_1, \\\\
X + X^* &= 2iX_2,
\end{align*}
\]
so \(X_1\) and \(X_2\) are determined by \(X\) alone. The formulas used in the proofs are
therefore the only possible choice, and the assignment \(X \mapsto (X_1, X_2)\)
is a bijection between \(\mathfrak{sl}(n; \mathbb{C})\) and ordered pairs \((X_1, X_2)\)
of elements of \(\mathfrak{su}(n)\). In particular the real dimension doubles:
\(\dim_{\mathbb{R}} \mathfrak{sl}(n; \mathbb{C}) = 2(n^2 - 1) = 2 \dim_{\mathbb{R}}
\mathfrak{su}(n)\), consistent with the general remark that complexification doubles the
real dimension.
Two Real Forms of One Complex Algebra
The two propositions exhibit a phenomenon worth naming. The compact algebra
\(\mathfrak{su}(n)\) and the noncompact algebra \(\mathfrak{sl}(n; \mathbb{R})\) of
traceless real matrices are quite different as real Lie algebras — one consists of
skew-Hermitian matrices, the other of real traceless matrices — yet both sit inside
\(\mathfrak{sl}(n; \mathbb{C})\) and both complexify to it. For \(\mathfrak{sl}(n;
\mathbb{R})\) this is immediate by splitting any traceless complex matrix into its
real and imaginary parts, each of which is real and traceless; for
\(\mathfrak{su}(n)\) it is the proposition just proved. They are two
real forms of the same complex Lie algebra. This is the structural
reason a single complex classification can serve several real groups at once: a
representation analysis carried out over \(\mathbb{C}\), on \(\mathfrak{sl}(n;
\mathbb{C})\), descends to information about every real form, including the compact
\(\mathfrak{su}(n)\).
The Example \(\mathfrak{su}(2)_{\mathbb{C}} \cong \mathfrak{sl}(2; \mathbb{C})\)
The case \(n = 2\) is the one we will lean on most heavily, because it is the smallest
setting in which the classification of irreducible representations is already
nontrivial, and because \(\mathfrak{su}(2)\) is the infinitesimal symmetry behind
three-dimensional rotations. Specializing the proposition of the previous section to
\(n = 2\) gives
\[
\mathfrak{su}(2) + i\,\mathfrak{su}(2) = \mathfrak{sl}(2; \mathbb{C}),
\]
the \(2 \times 2\) traceless complex matrices. We make the identification concrete using
the basis of \(\mathfrak{su}(2)\) already in hand.
On the
previous page
we used the basis
\[
F_1 = \begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix}, \qquad
F_2 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad
F_3 = \begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix}
\]
of \(\mathfrak{su}(2)\), each matrix skew-Hermitian and traceless, with brackets
\[
\begin{align*}
[F_1, F_2] &= 2F_3, \\\\
[F_2, F_3] &= 2F_1, \\\\
[F_3, F_1] &= 2F_2.
\end{align*}
\]
As a real Lie algebra, \(\mathfrak{su}(2)\) is the three-dimensional real span of
\(F_1, F_2, F_3\). Its complexification allows complex coefficients on the same three
matrices: \(\mathfrak{sl}(2; \mathbb{C})\) is their complex span. The real
dimension doubles from \(3\) to \(6\) — equivalently, the complex dimension is \(3\) —
in agreement with the general count \(\dim_{\mathbb{R}} \mathfrak{sl}(2; \mathbb{C}) =
2(2^2 - 1) = 6\).
Passing to complex coefficients makes available a basis unavailable over \(\mathbb{R}\),
the one in which the representation theory of \(\mathfrak{sl}(2; \mathbb{C})\) is
customarily organized. Writing
\[
H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad
E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad
F = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},
\]
one has \(H = iF_3\), \(E = \tfrac{i}{2}F_1 - \tfrac{1}{2}F_2\), and \(F = \tfrac{i}{2}F_1
+ \tfrac{1}{2}F_2\); the imaginary coefficients show explicitly that \(H, E, F\) live in
the complexification and not in \(\mathfrak{su}(2)\) itself. Their brackets are
\[
\begin{align*}
[H, E] &= 2E, \\\\
[H, F] &= -2F, \\\\
[E, F] &= H.
\end{align*}
\]
The element \(H\) is diagonal, so the operator \([H, \,\cdot\,]\) is diagonalizable with
real eigenvalues — exactly the structure that the eigenvalues of \(H\) being real, rather
than imaginary, makes available. This is the concrete payoff of complexification: the
diagonalizable element \(H\), absent from the compact real form, is what organizes the
eigenvalue bookkeeping on which the classification of irreducible representations rests.
The contrast with the
real isomorphism \(\mathfrak{su}(2) \cong \mathfrak{so}(3)\)
of the previous page is worth drawing. There, two real Lie algebras were found
identical through a real change of basis, reflecting that \(SU(2)\) and \(SO(3)\) share
the same infinitesimal rotational structure. Here we instead enlarge
\(\mathfrak{su}(2)\) to a complex algebra, gaining the diagonalizable generator \(H\) at
the cost of leaving the compact real form. The two operations answer different questions:
the real isomorphism asks which real algebras coincide, while complexification asks for
the complex algebra in which the representation theory of all of them is most cleanly
posed.