## Stream: maths

### Topic: lie theory

#### Johan Commelin (Jan 21 2020 at 08:55):

@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 (Jan 21 2020 at 20:14):

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 (Jan 21 2020 at 20:15):

Even the statement would be cool!

I hope so :)

#### Oliver Nash (Jan 21 2020 at 20:16):

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 (Jan 21 2020 at 20:16):

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

#### Oliver Nash (Jan 21 2020 at 20:17):

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 (Jan 21 2020 at 20:18):

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 (Jan 21 2020 at 20:19):

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 (Jan 21 2020 at 20:20):

I really appreciate what you've done so far.

#### Oliver Nash (Jan 21 2020 at 20:20):

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

#### Oliver Nash (Jan 21 2020 at 20:21):

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

#### Kevin Buzzard (Jan 21 2020 at 21:59):

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

Last updated: May 06 2021 at 18:20 UTC