Project Euclid

We give a survey of infinite dimensional Lie groups and show some applications and examples in mathematical physics. This includes diffeomorphism groups and their natural subgroups like volume preserving, symplectic and contact transformations, as well as gauge groups, quantomorphisms and loop groups. Various applications include fluid dynamics, Maxwell’s equations, plasma physics and BRST symmetries in quantum field theory. We discuss the Lie group structures of pseudodifferential and Fourier integral operators, both on compact and non- compact manifolds and give applications to the KdV equation and quantization.