/sci/ - Science & Math

[Anonymous No.5935841]

/sci/ I want to learn more about Lie groups but I don't know where to start.

I have a decent algebra background (Galois theory, some module theory, even less algebraic geometry) but not so much analysis. I know the basics of point-set topology and real/complex analysis. Where do I go from here? Do I need algebraic topology? Should I study manifolds alone first? Do I need experience with PDEs or any fancy differential equation theorems?

Book recommendations are welcome. Thanks.

[Anonymous No.5935866]

>>5935841
You should study some differential topology first. Try Tu or Lee. Both are pretty good, and both contain material on Lie groups. Tu is the most accessible of the two.
It's also advisable to search mathoverflow and math.stackexchange for threads asking for book suggestions.

[Anonymous No.5935876]

>>5935841
I just gave you a couple of book suggestions, but I figured I'd answer your questions as well.


>Where do I go from here?
Smooth manifolds.
>Do I need algebraic topology?
Nope.
>Should I study manifolds alone first?
Yes.
>Do I need experience with PDEs or any fancy differential equation theorems?
In principle, some results on smooth manifolds (for example the local existence of flows) depend on theorems on differential equations, but you can understand these theorems without studying their proofs, and most texts on smooth manifolds state the necessary results in full. So you can learn about smooth manifolds without having studied PDEs beforehand.

[Anonymous No.5935899]

>>5935876
Thanks. Are the books on differential topology mentioned above about smooth manifolds? Is there a distinction between studying differential topology and smooth manifolds at all?

[Anonymous No.5935915]

>>5935876
>>Do I need algebraic topology?
>Nope.
>>Should I study manifolds alone first?
>Yes.

That doesn't really make sense to me.
Manifolds are the domain of differential topology (I know, not exactly) which is a subset of algebraic topology. I would suggest studying some basic algebraic topology first - just an undergrad course dealing with the basics of homotopy and (co)homology (simplicial, singular, de Rham). This will make you familiar with manifolds. You then want some basic representation theory; not even a full course - just finite groups. Then study Lie groups and their representations together.
I'm still in this final stage, so no expert, but I have been assured that the way to a Lie group's heart is through its representations (or those of its Lie algebra).

[Anonymous No.5936086]

>>5935841
You don't need algebraic topology. You should not study manifolds alone. You do not need PDEs. You need Fulton and Harris- Representation Theory: A First Course

[Anonymous No.5936105]

Brian C Hall lie groups, lie algebras, representations: an elementary introduction

[Anonymous No.5936110]

>>5936086
No offense man but i fucking hate this book, or any book by joe harris for that matter. Read the book i posted above op. everyone is right that you should read of lie algebras too and learn of the exp map and the lie derivative, itd be like alg geometry only talking about coordinate rings. Anyway, fuck harris/fulton, omg.

[Anonymous No.5936119]

One other point, Mumford's classic tome on abelian varieties is a great source if you're interested in the (way cooler imo) algebraic set-up. Most of the lie/exp stuff is formal so you don't need such fine topologies as those in the classic set-up but you'll still have the conceptualization you'd learn of in the above texts. It has the bonus of giving you access to the finite characteristic setting, as well as the appropriate sophistication on algebraic groups (alg version of lie) that grants you access to some of the cooler string theory material current presently.

[Anonymous No.5936232]

>>5936110
None taken. But everyone and his brother seems to treat Fulton and Harris like a bible.

Texts I found useful were Warner's "Foundations of Differentiable Manifolds and Lie Groups"
and Olver's "Applications of Lie Groups to Differential Equations"

[Anonymous No.5936312]

Anyone have some good online references/videos?