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 + (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 + (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) + (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.