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
(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 ?