## Lie Groups

Now we come to one of the most broadly useful and fascinating structures on all of mathematics: Lie groups. These are objects which are both smooth manifolds and groups in a compatible way. The fancy way to say it is, of course, that a Lie group is a group object in the category of smooth manifolds.

To be a little more explicit, a Lie group is a smooth -dimensional manifold equipped with a multiplication and an inversion which satisfy all the usual group axioms (wow, it’s been a while since I wrote that stuff down) and are also smooth maps between manifolds. Of course, when we write we mean the product manifold.

We can use these to construct some other useful maps. For instance, if is any particular element we know that we have a smooth inclusion defined by . Composing this with the multiplication map we get a smooth map defined by , which we call “left-translation by “. Similarly we get a smooth right-translation .

[...] a Lie group is a smooth manifold we know that the collection of vector fields form a Lie algebra. But this is [...]

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[...] Lie groups are groups, they have representations — homomorphisms to the general linear group of some [...]

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