〈 Berger | Dillon 〉

okay (1) you’ve demonstrated you don’t actually know what a gauge theory is and (2) I’m well aware of @skdh’s arguments, which I find valuable and necessary for progress. So don’t troll my comments in your arm chair, acting like you actually know particle physics

And then there’s me who only knows that a gauge transformation is an element of the Lie group of sections of the automorphism bundle associated to a principal bundle. I’ll never get physics.

You don’t “feel” Lie groups as PDE symmetry groups? That’s the primary intuition I have for them at least. Well, I guess if anybody would (or even could!) have a deep understanding of Lie algebras/groups and not have the PDE intuition, it’d be you lol

I’m open to that interpretation but it’s not the first one I go to. Another I heard is “Lie groups: differential equations :: Galois groups: algebraic equations.” I can see one way in which this holds true but in general I don’t think I’ve thought enough about these perspectives.

Just out of curiosity, how do you think of Lie algebras?

iirc, over R or C, there is an adjunction between FD lie algebras and Lie groups that restricts to an adjoint equivalence between simply connected Lie groups and FD Lie algebras. Now if you’re working over a different base field lol good luck

But to the Lie group thing, do people not just think about them as manifolds enjoying the natural symmetries? I’ve never heard of this analogy you described with Galois groups and I’m keen on hearing how it plays out