5.5 Complexification

对照

Conventional physics discussion of Lie algebra representations proceed by assuming complex coefficients are allowed in all calculations, since we are interested in complex representations. An important subtlety is that the Lie algebra is a real vector space, often in a confusing way, as a subspace of complex matrices. To properly keep track of what is going on one needs to understand the notion of “complexification” of a vector space or Lie algebra. In some cases this is easily understood as just going from real to complex coefficients, but in other cases a more complicated construction is necessary. The reader is advised that it might be a good idea to just skim this section at first reading, coming back to it later only as needed to make sense of exactly how things work when these subtleties make an appearance in a concrete problem.

The way we have defined a Lie algebra $\mathfrak{g}$, it is a real vector space, not a complex vector space. Even if $G$ is a group of complex matrices, its tangent space at the identity will not necessarily be a complex vector space. Consider for example the cases $G=U(1)$ and $G=SU(2)$, where $\mathfrak{u}(1)=\mathbf{R}$ and $\mathfrak{su}(2)={\mathbf{R}}^3$. While the tangent space to the group $GL(n,\mathbf{C})$ of all invertible complex matrices is a complex vector space ($M(n,\mathbf{C})$, all $n$ by $n$ matrices), imposing some condition such as unitarity picks out a subspace of $M(n,\mathbf{C})$ which generally is just a real vector space, not a complex one. So the adjoint representation $(Ad,\mathfrak{g})$ is in general not a complex representation, but a real representation, with

$$Ad(g)\in GL(\mathfrak{g})=GL(dim\mathfrak{g},\mathbf{R})$$

The derivative of this is the Lie algebra representation

$$ad:X\in \mathfrak{g}\to ad(X)\in \mathfrak{gl}(dim\mathfrak{g},\mathbf{R})$$

and once we pick a basis of $\mathfrak{g}$, we can identify $\mathfrak{gl}(\dim \mathfrak{g},\mathbf{R})=M(\dim \mathfrak{g},\mathbf{R})$. So, for each $X\in \mathfrak{g}$ we get a real linear operator on a real vector space.

We most often would like to work with not real representations, but complex representations, since it is for these that Schur’s lemma applies (the proof of 2.1 also applies to the Lie algebra case), and representation operators can be diagonalized. To get from a real Lie algebra representation to a complex one, we can “complexify”, extending the action of real scalars to complex scalars. If we are working with real matrices, complexification is nothing but allowing complex entries and using the same rules for multiplying matrices as before.

More generally, for any real vector space we can define:

Definition · Complexification

Definition. The complexification $V_{\mathbf{C}}$ of a real vector space $V$ is the space of pairs $(v_1,v_2)$ of elements of $V$ with multiplication by $a+bi\in \mathbf{C}$ given by

$$(a+ib)(v_1,v_2)=\left(a{v}_1-b{v}_2,a{v}_2+b{v}_1\right)$$

One should think of the complexification of $V$ as

$$V_{\mathbf{C}}=V+iV$$

with $v_1$ in the first copy of $V,v_2$ in the second copy. Then the rule for multiplication by a complex number comes from the standard rules for complex multiplication.

Given a real Lie algebra $\mathfrak{g}$, the complexification ${\mathfrak{g}}_{\mathbf{C}}$ is pairs of elements $(X,Y)$ of $\mathfrak{g}$, with the above rule for multiplication by complex scalars, which can be thought of as

$${\mathfrak{g}}_{\mathbf{C}}=\mathfrak{g}+i\mathfrak{g}$$

The Lie bracket on $\mathfrak{g}$ extends to a Lie bracket on ${\mathfrak{g}}_{\mathbf{C}}$ by the rule

$$[(X_1,Y_1),(X_2,Y_2)]=([X_1,X_2]-[Y_1,Y_2],[X_1,Y_2]+[Y_1,X_2])$$

which can be understood by the calculation

$$[X_1+i{Y}_1,X_2+i{Y}_2]=[X_1,X_2]-[Y_1,Y_2]+i([X_1,Y_2]+[Y_1,X_2])$$

With this Lie bracket ${\mathfrak{g}}_{\mathbf{C}}$ is a Lie algebra over the complex numbers.

For many of the cases we will be interested in, this level of abstraction is not really needed, since they have the property that $V$ will be given as a subspace of a complex vector space, with the property that $V\cap iV=0$ , in which case $V_{\mathbf{C}}$ will just be the larger subspace you get by taking complex linear combinations of elements of $V.$ . For example, $\mathfrak{gl}(n,\mathbf{R})$ , the Lie algebra of real $n$ by $n$ matrices, is a subspace of $\mathfrak{gl}(n,\mathbf{C})$ , the complex matrices, and one can see that

$$\mathfrak{gl}(n,\mathbf{R}{)}_{\mathbf{C}}=\mathfrak{gl}(n,\mathbf{C})$$

Recalling our discussion from section 5.2.2 of $\mathfrak{u}(n)$ , a real Lie algebra, with elements certain complex matrices (the skew-Hermitian ones), multiplication by $i$ gives the Hermitian ones, and complexifying will give all complex matrices so

$$\mathfrak{u}(n{)}_{\mathbf{C}}=\mathfrak{gl}(n,\mathbf{C})$$

This example shows that two different real Lie algebras $(\mathfrak{u}(n)$ and $\mathfrak{gl}(n,\mathbf{R}))$ may have the same complexification. For yet another example, $\mathfrak{so}(n)$ is the Lie algebra of all real antisymmetric matrices, $\mathfrak{so}(n{)}_{\mathbf{C}}$ is the Lie algebra of all complex antisymmetric matrices.

For an example where the general definition is needed and the situation becomes easily confusing, consider the case of $\mathfrak{gl}(n,\mathbf{C})$ , thinking of it as a Lie algebra and thus a real vector space. The complexification of this real vector space will have twice the (real) dimension, so

$$\mathfrak{gl}(n,\mathbf{C}{)}_{\mathbf{C}}=\mathfrak{gl}(n,\mathbf{C})+i\mathfrak{gl}(n,\mathbf{C})$$

will not be what you get by just allowing complex coefficients $\left(\mathfrak{gl}(n,\mathbf{C})\right)$ , but something built out of two copies of this.

Given a representation ${\pi }^{\prime }$ of a real Lie algebra $\mathfrak{g}$, it can be extended to a representation of ${\mathfrak{g}}_{\mathbf{C}}$ by complex linearity, defining

$${\pi }^{\prime }(X+iY)={\pi }^{\prime }(X)+i{\pi }^{\prime }(Y)$$

If the original representation was on a complex vector space $V$, the extended one will act on the same space. If the original representation was on a real vector space $V$, the extended one will act on the complexification $V_{\mathbf{C}}$ . Some of the examples of these phenomena that we will encounter are the following:

• The adjoint representation

$$ad:\mathfrak{g}\to \mathfrak{gl}(dim\mathfrak{g},\mathbf{R})=M(dim\mathfrak{g},\mathbf{R})$$

extends to a complex representation

$$ad:{\mathfrak{g}}_{\mathbf{C}}\to \mathfrak{gl}(dim\mathfrak{g},\mathbf{C})=M(dim\mathfrak{g},\mathbf{C})$$

• Complex $n$ dimensional representations

$${\pi }^{\prime }:\mathfrak{su}(2)\to M(n,\mathbf{C})$$

of $\mathfrak{su}(2)$ extend to representations

$${\pi }^{\prime }:\mathfrak{su}(2{)}_{\mathbf{C}}=\mathfrak{sl}(2,\mathbf{C})\to M(n,\mathbf{C})$$

Doing this allows one to classify the finite dimensional irreducible representations of $\mathfrak{su}(2)$ by studying $\mathfrak{sl}(2,\mathbf{C})$ representations (see section 8.1.2).

• We will see that complex representations of a real Lie algebra called the Heisenberg Lie algebra play a central role in quantum theory and in quantum field theory. An important technique for constructing such representations (using so-called “annihilation” and “creation” operators) does so by extending the representation to the complexification of the Heisenberg Lie algebra (see section 22.4).

• Quantum field theories based on complex fields start with a Heisenberg Lie algebra that is already complex (see chapter 37 for the case of nonrelativistic fields, section 44.1.2 for relativistic fields). The use of annihilation and creation operators for such theories thus involves complexifying a Lie algebra that is already complex, requiring the use of the general notion of complexification discussed in this section.

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上一节：5.4 Lie algebra representations · 下一节：5.6 For further reading

附录：约定与记号

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来源版本：2025-10-20

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