Copyright © 2010 Rudolf Schmid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give a review of infinite-dimensional Lie groups and algebras and show some applications and examples in mathematical physics. This includes diffeomorphism groups and their natural subgroups like volume-preserving and symplectic transformations, as well as gauge groups and loop groups. Applications include fluid dynamics, Maxwell's equations, and plasma physics. We discuss applications in quantum field theory and relativity (gravity) including BRST and supersymmetries.