What are applications of Lie groups/algebras in mathematics?

Solution 1:

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

• analysis (e.g. harmonic analysis and the Peter-Weyl theorem),
• algebraic topology (e.g. principal bundles and characteristic classes),
• algebraic geometry (e.g. algebraic groups and flag varieties),
• combinatorics (e.g. root systems and Coxeter groups),
• differential geometry (e.g. connections and Chern-Weil theory),
• number theory (e.g. automorphic forms and the Langlands program),
• low-dimensional topology (e.g. quantum groups and Chern-Simons theory),
• Riemannian geometry (e.g. holonomy and symmetric spaces),
• finite group theory (e.g. the finite simple groups of Lie type)...

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.

Solution 2:

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.