I am just about to finish an introductory book 'Lie groups, Lie algberas & Representations' by Brian Hall and am curious to know what are the current directions of research in this area.

I learn that the finite dimensional complex semisimple Lie algebras have been classified (upto isomorphism), what if we relax the following conditions (separately in each case)

(i) finite dimensionality

(ii) semisimplicity

(iii) complexity (should I call it complex-ness ?)

What about classification of all finite dimensional Lie groups ?

What are the main open questions which motivate the current work in this area ?

And finally : What is the relation of Lie theory to other branches of Mathematics ? Are there areas of Mathematics where Lie theory plays a central/important role ?

  • $\begingroup$ A comment, because this answer is very incomplete. I believe that finite dimensional semisimple real Lie algebras are also classified, but I don't know about arbitrary fields. Lie theory turns up all the time in research on representation theory, and on differential or algebraic geometry. There is also a very strong (and not immediately obvious from the definitions) connection between Lie theory and cluster algebras. Lots of work is currently being done on this connection. $\endgroup$ – mdp Nov 13 '13 at 14:14

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