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Advances in Mathematical Physics
Volume 2010, Article ID 280362, 35 pages
Review Article

Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics

Department of Mathematics, Emory University, Atlanta, GA 30032, USA

Received 8 March 2010; Accepted 16 May 2010

Academic Editor: G. A. Goldin

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.


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.