John Carlos Baez on Nostr: There's a beautiful way to combine symmetry (that is, group theory) with calculus ...

There's a beautiful way to combine symmetry (that is, group theory) with calculus (that is, differential geometry) in a single theory, which not surprisingly is fundamental to a lot of modern physics. It's called the theory of Lie groups. I've always loved it.

But when you try to do calculations with Lie groups, you're instantly led to a trick which is a grand generalization of a slide rule. A slide rule converts multiplication to addition. You'd like to do this with a Lie group, too. It's harder, because multiplication doesn't need to commute: in a Lie group you can have ab≠ba. But you can still manage. Not surprisingly you're led to think about ab-ba all the time - and this leads to an operation called the Lie bracket,

[a,b] = ab-ba

and gadgets called Lie algebras. Every Lie group has a partner called a Lie algebra, and while the Lie group is more beautiful and conceptual, when you need to calculate you whip out your slide rule: its Lie algebra.

(Experts will roll their eyes at this oversimplification, but tough!)

I dutifully studied Lie algebras in grad school, but I didn't really love them. You need technical things called roots and weights, which seemed rather dry.

Year later I became enamored by these things thanks to conversations with James Dolan, who has a way of making things fun. But I still avoided doing serious calculations using this technology.

Now all of a sudden I need to do this - to write a paper relating the Lie algebra 𝔢₇ to particle physics! 𝔢₇ is a so-called 'exceptional' Lie algebra, meaning that it's connected to the octonions. I like calculating with octonions. But for now I need to calculate using roots and weights. 😩

I'll learn to like it.