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.