Lie Algebras: A Walkthrough - The Structures

[fresh_42]

Lie algebra theory is to a large extend the classification of the semisimple Lie algebras which are direct sums of the simple algebras listed in the previous paragraph, i.e. to show that those are all simple Lie algebras there are. Their counterpart are solvable Lie algebras, e.g. the Heisenberg algebra $\mathfrak{H}=\langle X,Y,Z\,:\, [X,Y]=Z\rangle\,.$ They have less structure each and are less structured as a whole as well. In physics, they don't play such a prominent role as simple Lie algebras do, although the reader might have recognized, that e.g. the Poincaré algebra - the tangent space of the Poincaré group at its identity matrix - wasn't among the simple ones. It isn't among the solvable Lie algebras either like $\mathfrak{H}$ is, so what is it then? It is the tangent space of the Lorentz group plus translations: something orthogonal plus something Abelian (solvable).

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[Wrichik Basu]

That is quite fast! Thanks for the next part. I couldn't even finish the first when you posted the second.

By the way, how many parts will be there in total?

 

[fresh_42]

That is quite fast! Thanks for the next part. I couldn't even finish the first when you posted the second.

By the way, how many parts will be there in total?

Three. The next (and as of yet last part) will be "Representations", but I only have the rough concept and two pages yet, so it will take a bit longer. The difficulty is to get through without slipping into too many technical details.

Theoretically one could add even more parts, e.g. cohomologies, but for these I'd have to (re-)learn them first and I'm not sure, whether these are interesting enough. Lie groups would be another possibility, but they are a subject on their own. So I will stick with the three parts - as titled "A Walkthrough".

 

[A. Neumaier]

''The Killing-form defines angels.''

I guess you meant angles...

 

[fresh_42]

''The Killing-form defines angels.''

I guess you meant angles...

Thank you. Seems I cannot completely eliminate this one, it happens to me from time to time.