3Lie algebras
III Symmetries, Fields and Particles
3.1 Lie algebras
We begin with a rather formal and weird definition of a Lie algebra.
Definition
(Lie algebra)
.
A Lie algebra
g
is a vector space (over
R
or
C
) with
a bracket
[ ·, ·] : g × g → g
satisfying
(i) [X, Y ] = −[Y, X] for all X, Y ∈ g (antisymmetry)
(ii)
[
αX
+
βY, Z
] =
α
[
X, Z
] +
β
[
Y, Z
] for all
X, Y, Z ∈ g
and
α, β ∈ F
((bi)linearity)
(iii)
[
X,
[
Y, Z
]] + [
Y,
[
Z, X
]] + [
Z,
[
X, Y
]] = 0 for all
X, Y, Z ∈ g
.(Jacobi identity)
Note that linearity in the second argument follows from linearity in the first
argument and antisymmetry.
Some (annoying) pure mathematicians will complain that we should state
anti-symmetry as [
X, X
] = 0 instead, which is a stronger condition if we are
working over a field of characteristic 2, but I do not care about such fields.
There isn’t much one can say to motivate the Jacobi identity. It is a property
that our naturally-occurring Lie algebras have, and turns out to be useful when
we want to prove things about Lie algebras.
Example.
Suppose we have a vector space
V
with an associative product (e.g.
a space of matrices with matrix multiplication). We can then turn
V
into a Lie
algebra by defining
[X, Y ] = XY − Y X.
We can then prove the axioms by writing out the expressions.
Definition
(Dimension of Lie algebra)
.
The dimension of a Lie algebra is the
dimension of the underlying vector space.
Given a finite-dimensional Lie algebra, we can pick a basis B for g.
B = {T
a
: a = 1, ··· , dim g}.
Then any X ∈ g can be written as
X = X
a
T
a
=
n
X
a=1
X
a
T
a
,
where X
a
∈ F and n = dim g.
By linearity, the bracket of elements X, Y ∈ g can be computed via
[X, Y ] = X
a
Y
b
[T
a
, T
b
].
In other words, the whole structure of the Lie algebra can be given by the bracket
of basis vectors. We know that [
T
a
, T
b
] is again an element of
g
. So we can write
[T
a
, T
b
] = f
ab
c
T
c
,
where f
ab
c
∈ F are the structure constants.
Definition
(Structure constants)
.
Given a Lie algebra
g
with a basis
B
=
{T
a
}
,
the structure constants are f
ab
c
given by
[T
a
, T
b
] = f
ab
c
T
c
,
By the antisymmetry of the bracket, we know
Proposition.
f
ba
c
= −f
ab
c
.
By writing out the Jacobi identity, we obtain
Proposition.
f
ab
c
f
cd
e
+ f
da
c
f
cb
e
+ f
bd
c
f
ca
e
= 0.
As before, we would like to know when two Lie algebras are the same.
Definition
(Homomorphism of Lie algebras)
.
A homomorphism of Lie algebras
g, h is a linear map f : g → h such that
[f(X), f(Y )] = f([X, Y ]).
Definition
(Isomorphism of Lie algebras)
.
An isomorphism of Lie algebras is a
homomorphism with an inverse that is also a homomorphism. Two Lie algebras
are isomorphic if there is an isomorphism between them.
Similar to how we can have a subgroup, we can also have a subalgebra
h
of
g
.
Definition
(Subalgebra)
.
A subalgebra of a Lie algebra
g
is a vector subspace
that is also a Lie algebra under the bracket.
Recall that in group theory, we have a stronger notion of a normal subgroup,
which are subgroups invariant under conjugation. There is an analogous notion
for subalgebras.
Definition
(Ideal)
.
An ideal of a Lie algebra
g
is a subalgebra
h
such that
[X, Y ] ∈ h for all X ∈ g and Y ∈ h.
Example. Every Lie algebra g has two trivial ideals h = {0} and h = g.
Definition (Derived algebra). The derived algebra of a Lie algebra g is
i = [g, g] = span
F
{[X, Y ] : X, Y ∈ g},
where F = R or C depending on the underlying field.
It is clear that this is an ideal. Note that this may or may not be trivial.
Definition (Center of Lie algebra). The center of a Lie algebra g is given by
ξ(g) = {X ∈ g : [X, Y ] = 0 for all Y ∈ g}.
This is an ideal, by the Jacobi identity.
Definition
(Abelian Lie algebra)
.
A Lie algebra
g
is abelian if [
X, Y
] = 0 for
all X, Y ∈ g. Equivalently, if ξ(g) = g.
Definition
(Simple Lie algebra)
.
A simple Lie algebra is a Lie algebra
g
that is
non-abelian and possesses no non-trivial ideals.
If
g
is simple, then since the center is always an ideal, and it is not
g
since
g
is not abelian, we must have
ξ
(
g
) =
{
0
}
. On the other hand, the derived algebra
is also an ideal, and is non-zero since it is not abelian. So we must have
i
(
g
) =
g
.
We will later see that these are the Lie algebras on which we can define a non-
degenerate invariant inner product. In fact, there is a more general class, known
as the semi-simple Lie algebras, that are exactly those for which non-degenerate
invariant inner products can exist.
These are important in physics because, as we will later see, to define the
Lagrangian of a gauge theory, we need to have a non-degenerate invariant inner
product on the Lie algebra. In other words, we need a (semi-)simple Lie algebra.