4Representations of Lie algebras

III Symmetries, Fields and Particles

4.2 Complexification and correspondence of representa-

tions

So far, we have two things — Lie algebras and Lie groups. Ultimately, the thing

we are interested in is the Lie group, but we hope to simplify the study of a

Lie group by looking at the Lie algebra instead. So we want to understand

how representations of Lie groups correspond to the representations of their Lie

algebras.

If we have a representation

D

:

G → GL

(

V

) of a Lie group

G

, then taking

the derivative at the identity gives us a linear map

ρ

:

T

e

G → T

I

GL

(

V

), i.e. a

map

ρ

:

g → gl

(

V

). To show this is a representation, we need to show that it

preserves the Lie bracket.

Lemma.

Given a representation

D

:

G → GL

(

V

), the induced representation

ρ : g → gl(V ) is a Lie algebra representation.

Proof.

We will again only prove this in the case of a matrix Lie group, so that

we can use the construction we had for the Lie bracket.

We have to check that the bracket is preserved. We take curves

γ

1

, γ

2

:

R → G

passing through I at 0 such that ˙γ

i

(0) = X

i

for i = 1, 2. We write

γ(t) = γ

−1

1

(t)γ

−1

2

(t)γ

1

(t)γ

2

(t) ∈ G.

We can again Taylor expand this to obtain

γ(t) = I + t

2

[X

1

, X

2

] + O(t

3

).

Essentially by the definition of the derivative, applying D to this gives

D(γ(t)) = I + t

2

ρ([X

1

, X

2

]) + O(t

3

).

On the other hand, we can apply D to (∗) before Taylor expanding. We get

D(γ) = D(γ

−1

1

)D(γ

−2

2

)D(γ

1

)D(γ

2

).

So as before, since

D(γ

i

) = I + tρ(X

i

) + O(t

2

),

it follows that

D(γ)(t) = I + t

2

[ρ(X

1

), ρ(X

2

)] + O(t

3

).

So we must have

ρ([X

1

, X

2

]) = [ρ(X

1

), ρ(X

2

)].

How about the other way round? We know that if

ρ

:

g → gl

(

V

) is induced

by D : G → GL(V ), then have

D(exp(X)) = I + tρ(X) + O(t

2

),

while we also have

exp(ρ(X)) = I + rρ(X) + O(t

2

).

So we might expect that we indeed have

D(exp(X)) = exp(ρ(X))

for all X ∈ g.

So we can try to use this formula to construct a representation of

G

. Given

an element

g ∈ G

, we try to write it as

g

=

exp

(

X

) for some

X ∈ g

. We then

define

D(g) = exp(ρ(X)).

For this to be well-defined, we need two things to happen:

(i) Every element in G can be written as exp(X) for some X

(ii) The value of exp(ρ(X)) does not depend on which X we choose.

We first show that if this is well-defined, then it indeed gives us a representation

of G.

To see this, we use the Baker-Campbell-Hausdorff formula to say

D(exp(X) exp(Y )) = exp(ρ(log(exp(X) exp(Y ))))

= exp

ρ

log

exp

X + Y +

1

2

[X, Y ] + ···

= exp

ρ

X + Y +

1

2

[X, Y ] + ···

= exp

ρ(X) + ρ(Y ) +

1

2

[ρ(X), ρ(Y )] + ···

= exp(ρ(X)) exp(ρ(Y ))

= D(exp(X))D(exp(Y )),

where

log

is the inverse to

exp

. By well-defined-ness, it doesn’t matter which

log we pick. Here we need to use the fact that

ρ

preserves the Lie bracket, and

all terms in the Baker-Campbell-Hausdorff formula are made up of Lie brackets.

So when are the two conditions satisfied? For the first condition, we know

that

g

is connected, and the continuous image of a connected space is connected.

So a necessary condition is that

G

must be a connected Lie group. This rules out

groups such as O(

n

). It turns out this is also sufficient. The second condition is

harder, and we will take note of the following result without proof:

Theorem.

Let

G

be a simply connected Lie group with Lie algebra

g

, and let

ρ

:

g → gl

(

V

) be a representation of

g

. Then there is a unique representation

D : G → GL(V ) of G that induces ρ.

So if we only care about simply connected Lie groups, then studying the

representations of its Lie algebra is exactly the same as studying the representa-

tions of the group itself. But even if we care about other groups, we know that

all representations of the group give representations of the algebra. So to find a

representation of the group, we can look at all representations of the algebra,

and see which lift to a representation of the group.

It turns out Lie algebras aren’t simple enough. Real numbers are terrible,

and complex numbers are nice. So what we want to do is to look at the

complexification of the Lie algebra, and study the complex representations of

the complexified Lie algebra.

We will start with the definition of a complexification of a real vector space.

We will provide three different definitions of the definition, from concrete to

abstract, to suit different people’s tastes.

Definition

(Complexification I)

.

Let

V

be a real vector space. We pick a basis

{T

a

}

of

V

. We define the complexification of

V

, written

V

C

as the complex

linear span of {T

a

}, i.e.

V

C

=

n

X

λ

a

T

a

: λ

a

∈ C

o

.

There is a canonical inclusion

V → V

C

given by sending

P

λ

a

T

a

to

P

λ

a

T

a

for

λ

a

∈ R.

This is a rather concrete definition, but the pure mathematicians will not

be happy with such a definition. We can try another definition that does not

involve picking a basis.

Definition

(Complexification II)

.

Let

V

be a real vector space. The complex-

ification of

V

has underlying vector space

V

C

=

V ⊕ V

. Then the action of a

complex number λ = a + bi on (u

1

, u

2

) is given by

λ(u

1

, u

2

) = (au

1

− bu

2

, au

2

+ bu

1

).

This gives

V

C

the structure of a complex vector space. We have an inclusion

V → V

C

by inclusion into the first factor.

Finally, we have a definition that uses some notions from commutative algebra,

which you may know about if you are taking the (non-existent) Commutative

Algebra course. Otherwise, do not bother reading.

Definition

(Complexification III)

.

Let

V

be a real vector space. The com-

plexification of

V

is the tensor product

V ⊗

R

C

, where

C

is viewed as an

(R, C)-bimodule.

To define the Lie algebra structure on the complexification, we simply declare

that

[X + iY, X

0

+ iY

0

] = [X, X

0

] + i([X, Y

0

] + [Y, X

0

]) − [Y, Y

0

]

for X, Y ∈ V ⊆ V

C

.

Whichever definition we decide to choose, we have now a definition, and we

want to look at the representations of the complexification.

Theorem.

Let

g

be a real Lie algebra. Then the complex representations of

g

are exactly the (complex) representations of g

C

.

Explicitly, if

ρ

:

g → gl

(

V

) is a complex representation, then we can extend

it to g

C

by declaring

ρ(X + iY ) = ρ(X) + iρ(Y ).

Conversely, if

ρ

C

:

g

C

→ gl

(

V

), restricting it to

g ⊆ g

C

gives a representation of

g.

Proof. Just stare at it and see that the formula works.

So if we only care about complex representations, which is the case most of

the time, we can study the representations of the complexification instead. This

is much easier. In fact, in the next chapter, we are going to classify all simple

complex Lie algebras and their representations.

Before we end, we note the following definition:

Definition

(Real form)

.

Let

g

be a complex Lie algebra. A real form of

g

is a

real Lie algebra h such that h

C

= g.

Note that a complex Lie algebra can have multiple non-isomorphic real forms

in general.