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Lie group
About: Lie group is a research topic. Over the lifetime, 18359 publications have been published within this topic receiving 381012 citations.
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TL;DR: In this article, the general structure of phenomenological Lagrangian theories is investigated, and the possible transformation laws of the phenomenological fields under a group are discussed, which is equivalent to finding all (nonlinear) realizations of a (compact, connected, semisimpleasure) Lie group which become linear when restricted to a given subgroup.
Abstract: The general structure of phenomenological Lagrangian theories is investigated, and the possible transformation laws of the phenomenological fields under a group are discussed. The manifold spanned by the phenomenological fields has a special point, called the origin. Allowed changes in the field variables, which do not change the on-shell S matrix, must leave the origin fixed. By a suitable choice of fields, the transformations induced by the group on the manifold of the phenomenological fields can be made to have standard forms, which are described in detail. The mathematical problem is equivalent to that of finding all (nonlinear) realizations of a (compact, connected, semisimple) Lie group which become linear when restricted to a given subgroup. The relation between linear representations and nonlinear realization is discussed. The important special case of the chiral groups SU(2)×SU(2) and SU(3)×SU(3) is considered in detail.
2,096 citations
Book•
01 Jan 1984
TL;DR: In this article, a very concise treatment of riemannian and pseudo-riemannian manifolds and their curvatures is given, along with a discussion of the representation theory of finite groups.
Abstract: This sixth edition illustrates the high degree of interplay between group theory and geometry The reader will benefit from the very concise treatments of riemannian and pseudo-riemannian manifolds and their curvatures, of the representation theory of finite groups, and of indications of recent progress in discrete subgroups of Lie groups
2,036 citations
Book•
01 Jun 1971
TL;DR: Foundations of Differentiable Manifolds and Lie Groups as discussed by the authors provides a clear, detailed, and careful development of the basic facts on manifold theory and Lie groups, including differentiable manifolds, tensors and differentiable forms.
Abstract: Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. It includes differentiable manifolds, tensors and differentiable forms. Lie groups and homogenous spaces, integration on manifolds, and in addition provides a proof of the de Rham theorem via sheaf cohomology theory, and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem. Those interested in any of the diverse areas of mathematics requiring the notion of a differentiable manifold will find this beginning graduate-level text extremely useful.
1,992 citations