lie theory

Stream: maths

[Johan Commelin]

@Oliver Nash I've been following your Lie PRs with curiosity, and I'm a big fan. Could you share in a couple of lines what kind of things you would like to aim for. E.g., may we dream of working towards the classification of semisimple Lie algebras?

[Oliver Nash]

Thanks @Johan Commelin I'm very grateful for the numerous very helpful improvements that you and others have provided. I do have the semi-simple classification in mind but I've been telling myself just to aim for the statement for now, rather than the proof.

[Johan Commelin]

Even the statement would be cool!

[Oliver Nash]

I hope so :)

[Oliver Nash]

I'm a differential geometer at heart though and I'd really like to define the Lie algebra structure on vector fields of a smooth manifold since IMHO until we have that, the tangent bundle is just another vector bundle but once we have that, we're really doing differential geometry.

[Oliver Nash]

(Obviously this doesn't really need any Lie theory.)

[Oliver Nash]

I actually tried this a week or two ago but I learned that there was a moderate amount of work to do even to prove that the continuous/smooth sections of the tangent bundle are an additive group, and that most of this should be done for vector bundles (not yet defined) so I backed off for the time being.

[Oliver Nash]

So for now I'll probably just keep plugging away trying to get to the statement of the classification. I looked for the trace because I was thinking I should be able to state Cartan's criterion for semisimplicity etc.

[Oliver Nash]

Unfortunately (for now) my day job takes up far more of my time than I'd like so my progress has been somewhat glacial but nevertheless, non-zero!

[Johan Commelin]

I really appreciate what you've done so far.

[Oliver Nash]

Not at all, most fun I've had in years!

[Oliver Nash]

Well, perhaps I exaggerate, but I do enjoy it!

[Kevin Buzzard]

I know the feeling. It's an open source project and people contribute when they have time. Thanks a lot!