I studied Lie Groups and Lie Algebras as part of my Masters back in the 1970s. Although very elegant and beautiful, it seemed to its own little world, I never saw the connection with other branches of mathematics (or at least from my course).

Can anyone explain to me how Lie Groups fit in—in a nutshell—or provide me with some useful references I can chew over?

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    $\begingroup$ Did you read why Sophus Lie studied continuous transformation groups and their linearizations? $\endgroup$ – Artem Jun 12 '15 at 3:30
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    $\begingroup$ One of my students who went on to graduate school to study analytic number theory, or, maybe algebraic number theory. Or, well, I don't know, but, Lie algebra representation is important to him. Kind of surprising to me. On the other hand, another of my students is working towards some graduate work in differential geometry. He also needs Lie theory. Essentially, differentiation puts you in contact with derivations and derivations go hand in hand with Lie algebras. Besides that, and maybe in tune with that, Lie algebras approximate Lie groups and groups, well, they be everywhere. $\endgroup$ – James S. Cook Jun 12 '15 at 4:31
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    $\begingroup$ Oh, but, more to your reference request, and a bit off topic, but you might enjoy the paper by Baez on Octonions. It perhaps will scratch your itch. I want all my students to read math.ucr.edu/home/baez/octonions/oct.pdf $\endgroup$ – James S. Cook Jun 12 '15 at 4:33
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    $\begingroup$ Olver's book Applications of Lie Groups to Differential Equations, perhaps? $\endgroup$ – Hans Lundmark Jun 12 '15 at 6:10

Lie theory connects to almost every other branch of mathematics! It's almost absurdly well connected! Just off the top of my head:

And this is just mathematics. Lie theory is also hugely relevant to physics as well (which is related to some of the stuff above) since it's an important ingredient in the study of gauge theory; for example, the Standard Model is a gauge theory.

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    $\begingroup$ I wouldnt have much of a job without lie theory here in particle phenomonology. It is wonderfully beautiful stuff $\endgroup$ – Triatticus Jun 13 '15 at 8:39

There are many connections to other branches of mathematics. Many interesting answers have been given to the MO-question "Why study Lie algebras". It is also worth to have a look at the prerequisites for the Langlands program, where Lie groups and Lie algebras are on top of the list.


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