4Representations of Lie algebras

III Symmetries, Fields and Particles 4.1 Representations of Lie groups and algebras
So far, we have just talked about Lie groups and Lie algebras abstractly. But we
know these groups don’t just sit there doing nothing. They act on things. For
example, the group
GL
(
n, R
) acts on the vector space
R
n
in the obvious way. In
general, the action of a group on a vector space is known as a representation.
Definition
(Representation of group)
.
Let
G
be a group and
V
be a (finite-
dimensional) vector space over a field
F
. A representation of
G
on
V
is given
by specifying invertible linear maps
D
(
g
) :
V V
(i.e.
D
(
g
)
GL
(
V
)) for each
g G such that
D(gh) = D(g)D(h)
for all
g, h G
. In the case where
G
is a Lie group and
F
=
R
or
C
, we require
that the map D : G GL(V ) is smooth.
The space
V
is known as the representation space, and we often write the
representation as the pair (V, D).
Here if we pick a basis
{e
1
, ··· , e
n
}
for
V
, then we can identify
GL
(
V
) with
GL
(
n, F
), and this obtains a canonical smooth structure when
F
=
R
or
C
. This
smooth structure does not depend on the basis chosen.
In general, the map D need not be injective or surjective.
Proposition.
Let
D
be a representation of a group
G
. Then
D
(
e
) =
I
and
D(g
1
) = D(g)
1
.
Proof. We have
D(e) = D(ee) = D(e)D(e).
Since D(e) is invertible, multiplying by the inverse gives
D(e) = I.
Similarly, we have
D(g)D(g
1
) = D(gg
1
) = D(e) = I.
So it follows that D(g)
1
= D(g
1
).
Now why do we care about representations? Mathematically, we can learn
a lot about a group in terms of its possible representations. However, from a
physical point of view, knowing about representations of a group is also very
important. When studying field theory, our fields take value in a fixed vector
space
V
, and when we change coordinates, our field will “transform accordingly”
according to some rules. For example, we say scalar fields “don’t transform”,
but, say, the electromagnetic field tensor transforms “as a 2-tensor”.
We can describe the spacetime symmetries by a group
G
, so that specifying
a “change of coordinates” is equivalent to giving an element of
G
. For example,
in special relativity, changing coordinates corresponds to giving an element of
the Lorentz group O(3, 1).
Now if we want to say how our objects in
V
transform when we change
coordinates, this is exactly the same as specifying a representation of
G
on
V
!
So understanding what representations are available lets us know what kinds of
fields we can have.
The problem, however, is that representations of Lie groups are very hard.
Lie groups are very big geometric structures with a lot of internal complexity.
So instead, we might try to find representations of their Lie algebras instead.
Definition
(Representation of Lie algebra)
.
Let
g
be a Lie algebra. A represen-
tation ρ of g on a vector space V is a collection of linear maps
ρ(X) gl(V ),
for each
X g
, i.e.
ρ
(
X
) :
V V
is a linear map, not necessarily invertible.
These are required to satisfy the conditions
[ρ(X
1
), ρ(X
2
)] = ρ([X
1
, X
2
])
and
ρ(αX
1
+ βX
2
) = αρ(X
1
) + βρ(X
2
).
The vector space
V
is known as the representation space. Similarly, we often
write the representation as (V, ρ).
Note that it is possible to talk about a complex representation of a real Lie
algebra, because any complex Lie algebra (namely
gl
(
V
) for a complex vector
space
V
) can be thought of as a real Lie algebra by “forgetting” that we can
multiply by complex numbers, and indeed this is often what we care about.
Definition
(Dimension of representation)
.
The dimension of a representation
is the dimension of the representation space.
We will later see that a representation of a Lie group gives rise to a represen-
tation of the Lie algebra. The representation is not too hard to obtain if we
have a representation
D
:
G GL
(
V
) of the Lie group, taking the derivative of
this map gives us the confusingly-denoted D
e
D
:
T
e
G T
e
(
GL
(
V
)), which is
a map
g gl
(
V
). To check that this is indeed a representation, we will have
to see that it respects the Lie bracket. We will do this later when we study the
relation between representations of Lie groups and Lie algebras.
Before we do that, we look at some important examples of representations of
Lie algebras.
Definition
(Trivial representation)
.
Let
g
be a Lie algebra of dimension
D
.
The trivial representation is the representation
d
0
:
g F
given by
d
0
(
X
) = 0
for all X g. This has dimension 1.
Definition
(Fundamental representation)
.
Let
g
=
L
(
G
) for
G Mat
n
(
F
). The
fundamental representation is given by d
f
: g Mat
n
(F) given by
d
f
(X) = X
This has dim(d
f
) = n.
Definition
.
All Lie algebras come with an adjoint
representation
d
of dimension
dim
(
g
) =
D
. This is given by mapping
X g
to the linear map
X
: g g
Y 7→ [X, Y ]
By linearity of the bracket, this is indeed a linear map g gl(g).
g
comes from a Lie group
G
. Writing
Aut
(
G
) for all the isomorphisms
G G
, we
know there is a homomorphism
Φ : G Aut(G)
g 7→ Φ
g
given by conjugation:
Φ
g
(x) = gxg
1
.
Now by taking the derivative, we can turn each Φ
g
into a linear isomorphism
g g, i.e. an element of GL(g). So we found ourselves a homomorphism
which is a representation of the Lie group
G
! It is an exercise to show that
the corresponding representation of the Lie algebra
g
representation.
Thus, if we view conjugation as a natural action of a group on itself, then
the adjoint representation is the natural representation of g over itself.
Proposition. The adjoint representation is a representation.
Proof.
Since the bracket is linear in both components, we know the adjoint
representation is a linear map g gl(g). It remains to show that
X
Y
[X,Y ]
.
But the Jacobi identity says
X
Y
](Z) = [X, [Y, Z]] [Y, [X, Z]] = [[X, Y ], Z] = ad
[X,Y ]
(Z).
We will eventually want to find all representations of a Lie algebra. To do
so, we need the notion of when two representations are “the same”.
Definition
(Homomorphism of representations)
.
Let (
V
1
, ρ
1
)
,
(
V
2
, ρ
2
) be rep-
resentations of
g
. A homomorphism
f
: (
V
1
, ρ
1
)
(
V
2
, ρ
2
) is a linear map
f : V
1
V
2
such that for all X g, we have
f(ρ
1
(X)(v)) = ρ
2
(X)(f(v))
for all v V
1
. Alternatively, we can write this as
f ρ
1
= ρ
2
f.
In other words, the following diagram commutes for all X g:
V
1
V
2
V
1
V
2
f
ρ
1
(X) ρ
2
(X)
f
Then we can define
Definition
(Isomorphism of representations)
.
Two
g
-vector spaces
V
1
, V
2
are
isomorphic if there is an invertible homomorphism f : V
1
V
2
.
In particular, isomorphic representations have the same dimension.
If we pick a basis for
V
1
and
V
2
, and write the matrices for the representations
as
R
1
(
X
) and
R
2
(
X
), then they are isomorphic if there exists a non-singular
matrix S such that
R
2
(X) = SR
1
(X)S
1
for all X g.
We are going to look at special representations that are “indecomposable”.
Definition
(Invariant subspace)
.
Let
ρ
be a representation of a Lie algebra
g
with representation space
V
. An invariant subspace is a subspace
U V
such
that
ρ(X)u U
for all X g and u U.
The trivial subspaces are U = {0} and V .
Definition
(Irreducible representation)
.
An irreducible representation is a rep-
resentation with no non-trivial invariant subspaces. They are referred to as
irreps.