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So good morning and welcome back. Today we'll turn to a topic of the utmost importance in theoretical physics,

but also for our further development of differential geometry, and that's Lie theory.

So we arrive to chapter four, Lie theory.

Now, from the point of view of differential geometry, what we start with is Lie groups.

You may have met or not met Lie algebras in physics, but we'll start with Lie groups,

and once you start with a Lie group, to a Lie group there is an associated Lie algebra.

So the question is, what is a Lie group? Well, that's a very simple one,

and you certainly met several Lie groups already, maybe without knowing it.

The definition is the following. A Lie group consisting of a set G and a group operation blob,

a Lie group is, first of all, a group with the group operation blob.

And that means that the group operation blob satisfies the properties A and I,

so blob is associative, blob, there's a neutral element in G,

such that that neutral element blobbed with any other element, yields that other element again,

and there's an inverse element, for each element in G there is an inverse element,

such that taking the element and blobbing it with the inverse, you get back the neutral element, that's a group,

and only sometimes is there also a commutative law, but that's a special case,

that's then a so-called commutative or abelian, so that's just the same, it's just a synonym, or abelian group.

So nothing spectacular about this. A Lie group is a group, but it's more than a group,

because in addition, this underlying set G is much more than a set G, is actually a smooth manifold,

that means there is a topology given on G, and also a smooth structure, so a maximal smooth atlas.

G is a smooth manifold, and that's not all of it, and the operation, well, the blob operation,

we can encode this in a map, and the maps, actually two maps we're considering,

and the map's the first one, we call it mu, like multiplication, mu, that takes a pair of elements in G,

and sends it to G, and does so, G1, G2, by sending them to the blob product of the two, that's a map,

and that's just a group operation, but now if G is a smooth manifold, then you can actually make G cross G,

is then clearly also a topological manifold, but you can also inherit a smooth atlas,

so inherits a smooth atlas from G, because G has a smooth atlas, so you can equip the Cartesian product

of the underlying topological manifolds, again with a smooth atlas, with the result that the little cloud here

is again a smooth manifold. Now, if this is a smooth manifold, and this is a smooth manifold,

we have a map mu between smooth manifolds, and this map mu can be, we can stipulate, or require, to be smooth,

and the map's mu, and, and I give you another map, I want to be smooth, and the map, let's call it I, like inverse,

that takes an element of G to its inverse, so takes an element G and maps it to G inverse,

whose existence is guaranteed by this axiom, and the maps mu and I are both smooth maps.

Aha, so, in a sense, this is to be expected, if you have a group, and then you have extra structure

on the underlying set, you want a kind of compatibility condition with that extra structure,

in this case a multiplication, or a, well, it's just a generic name for this blob operation,

and you want that to be smooth as well, and so here this inverse, of course, is also an operation we'll use a lot,

so also require that this be a smooth map. And any such structure with this combined properties is called a Lie group.

Okay? Now, examples, well, a very simple example is, as G, you simply take R to the N,

but because we want to construct a Lie group, we take Rn considered as a smooth manifold,

and we know there is a topology understood, and there's a smooth atlas understood,

we suppress that in the rotation, consider it as a smooth manifold, and as the blob operation,

we simply take the addition on Rn, so you compose two elements in here by simply adding them component-wise.

Now, this makes this, G with the blob, makes this a commutative Lie group,

because the addition and the inverse, which is, of course, taking the opposite vector, are smooth,

and sometimes one calls this the translation group, called the N-dimensional translation group,

and when you study quantum mechanics, you talked a lot about the momentum operator

generating translations, right, of the wave function, or even in classical mechanics,

you take the Poisson bracket, you plug in into the first slot, you plug in a momentum, roughly speaking,