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So good morning and welcome back. After the last lecture some of you asked me about representations

of Lie Algebras and Lie Groups and I decided to have a brief section on this because indeed

mostly you meet Lie Algebras and Lie Groups in physics in terms of their representations

and very often they're even defined in terms of their representations.

This is something we didn't do. We defined a Lie Group as a smooth manifold with a group law

that is compatible with the smooth structure and we defined the Lie Algebra as the left invariant vector fields

and of course we could pull all of this back to the tangent space at the identity with a suitably induced bracket

but what on earth are these representations? So we should comment on that because it's very important.

So we have a section 4.6, linear representations of Lie Algebras and Lie Groups.

And we start with the Algebras and then we briefly comment on the groups.

Now again you can give a at least half semester long lecture on this topic.

So I'll present mainly no proofs but I'll introduce the important concepts.

So the key concept is the following definition. Let L with our abstract Lie bracket be a Lie Algebra.

Then a representation of this Lie Algebra is the following.

It is a linear map which we may call rows. So let's say a representation row of this algebra.

This Lie Algebra is a linear map row that goes from L, why can I say linear?

Well because L in particular is a vector space, goes into the endomorphisms of some vector space V.

So this is some vector space V over the real or the complex numbers such that, does that make sense?

Well if V is a vector space, let's say some finite dimensional vector space and then we have a finite dimensional representation.

Does that make sense? Well L is a vector space and V, well that's a vector space because that's all the linear maps from V to V

which by themselves constitute a vector space again.

Such that we need somehow compatibility with the Lie Algebra bracket such that I can first take the bracket of two Lie Algebra elements,

let's call them A and B and then the result will be a Lie Algebra element and I map it and I require that this be the same as first mapping one of them

and mapping the other of them and then the result being put into brackets.

These are now of course not the differential geometric brackets because these are not vector fields, they're just endomorphisms on V

and where this bracket is defined, so this is defined as whenever we write this, now we mean this is an endomorphism, this is an endomorphism

so what I can certainly do, I can execute one after the other, I can compose them and then I can again compose them

but in the opposite order and that's of course what we know as a commutator, you take this and apply it to this minus this apply it to this,

this is of course again an endomorphism so this bracket here is a bracket, is a bilinear map from end V, end V into end V

that is the bracket here goes from end V cross end V into end V and of course it's bilinear, it's easily checked

and in fact by construction it's anti-symmetric and you can also check that it satisfies the Jacobi identity.

Aha, that means why do I map into the endomorphisms? Well because the endomorphisms by virtue of having a natural composition on them

can be immediately made into a Lie algebra, okay, so and what I formulated here is that I say a representation of an abstract Lie algebra

is a linear mapping into end V which itself carries a Lie bracket, namely this commutator bracket and I require compatibility

that I can first take the bracket and then map, I can map the elements and then take the bracket and should be the same outcome.

Okay and so the, this row is then called the representation and the vector space, the particular vector space that we used here,

the particular vector space V that comes with a representation is called the representation space, the representation space.

Okay, so that is maybe confusing because one might have thought well you map the Lie algebra to end V so you might say end V is the representation space

and you represent the L, well that is what it is but the terminology says this V is the representation space, okay.

And the idea behind it is that once you have a row of some Lie algebra elements, some row of A, you can act on the representation space V

because row of A, of a little a is an endomorphism in V so the endomorphism can act on a vector and produce a new vector, that's the idea.

Okay, so that's the abstract definition of a linear representation so we should call this then a linear representation because we do this by endomorphisms on a vector space.

Okay, so example or examples.

Well, so we discussed a great length the SL2C Lie algebra and we found that it has three generators, well three basis vectors, this is three dimensional algebra and we wrote down these commutation relations for it.

This was 2x2, this is x1, x3, this minus 2x3 and this is x2, x3 is x1 and we derive this from the left invariant vector fields and the structure was inherited after all from the product structure on the Lie group capital SL2C.

But this is a Lie algebra and now these three vector fields, these three vectors we had only the interpretation, these are the representatives of the left invariant vector fields at the identity, we didn't know more.

Now a representation, a representation because there in general there are many many representations, a representation of SL2C is provided by, so let's call it rho and it takes the first Lie algebra basis vector,

because it's a linear map it suffices to prescribe it on a basis, right, so I prescribe it on the basis, I say this one is 1 minus 1, 0, 0, on the second one it's 0, 1, 0, 0 and rho on the third one is 0, 0, 1, 0.

What I'm obviously doing here, I'm providing a representation SL2C which chooses as its representation space let's say C2 and obviously I can give elements in C2 with respect to a canonical basis because this is C cross C I'm acting on,