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I have for a while been sniffing around the edges of Lie groups, -algebras etc, but what tends to put me off is that it always seems to focus on matrix groups and detailed calculations of things, like $exp()$, that I feel ought to have a more general, abstract definition. Thus:

  • Is there a book/books that treat Lie theory in full abstraction - perhaps even from Category theory?
j4nd3r53n
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    For what it's worth, $\exp$ is a generic construction taking a vector from a tangent space on a manifold to a point in a neighborhood. – lisyarus Sep 27 '22 at 07:52
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    @lisyarus: there is no such construction on a manifold. $\exp$ can be defined on a Riemannian manifold, and that definition does not (immediately) specialize to the exponential map on a Lie group. – Qiaochu Yuan Sep 27 '22 at 16:32
  • @QiaochuYuan Indeed, thank you! – lisyarus Sep 27 '22 at 20:31

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Yes, if you want abstract Lie theory there's plenty of abstract Lie theory to go around. You can consult, for example, Serre's Lie Algebras and Lie Groups which IIRC does not even restrict to working over $\mathbb{R}$ or $\mathbb{C}$ in case you want to study $p$-adic Lie groups or similar! Fulton and Harris' Representation Theory is a more pleasant read; they focus a lot on examples which I think is healthy but you will find general definitions and proofs about the exponential map etc. as well. I also hear good things about Knapp's Lie Groups: Beyond an Introduction although as stated in the title it's not an introductory text.

Qiaochu Yuan
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