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Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics

DOI: 10.1155/2010/280362

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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. 1. Introduction Lie groups play an important role in physical systems both as phase spaces and as symmetry groups. Infinite-dimensional Lie groups occur in the study of dynamical systems with an infinite number of degrees of freedom such as PDEs and in field theories. For such infinite-dimensional dynamical systems, diffeomorphism groups and various extensions and variations thereof, such as gauge groups, loop groups, and groups of Fourier integral operators, occur as symmetry groups and phase spaces. Symmetries are fundamental for Hamiltonian systems. They provide conservation laws (Noether currents) and reduce the number of degrees of freedom, that is, the dimension of the phase space. The topics selected for review aim to illustrate some of the ways infinite-dimensional geometry and global analysis can be used in mathematical problems of physical interest. The topics selected are the following.(1)Infinite-Dimensional Lie Groups.(2)Lie Groups as Symmetry Groups of Hamiltonian Systems.(3)Applications.(4)Gauge Theories, the Standard Model, and Gravity.(5)SUSY (supersymmetry). 2. Infinite-Dimensional Lie Groups 2.1. Basic Definitions A general theory of infinite-dimensional Lie groups is hardly developed. Even Bourbaki [1] only develops a theory of infinite-dimensional manifolds, but all of the important theorems about Lie groups are stated for finite-dimensional ones. An infinite-dimensional Lie group is a group and an infinite-dimensional manifold with smooth group operations Such a Lie group is locally diffeomorphic to an infinite-dimensional vector space. This can be a Banach space whose topology is given by a norm , a Hilbert space whose topology is given by an inner product , or a Frechet space whose topology is given by a metric but not by a norm. Depending on the choice of the topology on , we talk about Banach, Hilbert, or Frechet Lie groups, respectively. The Lie algebra?? of a Lie group is defined as left invariant vector fields on (tangent space at the identity ). The isomorphism is given (as in finite dimensions) by and the Lie bracket on is induced by the Lie bracket of

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