By B. Kolman
Introduces the techniques and strategies of the Lie conception in a sort available to the nonspecialist by means of retaining mathematical must haves to a minimal. even supposing the authors have focused on providing effects whereas omitting many of the proofs, they've got compensated for those omissions through together with many references to the unique literature. Their remedy is directed towards the reader looking a extensive view of the topic instead of tricky information regarding technical info. Illustrations of varied issues of the Lie conception itself are discovered during the ebook in fabric on functions.
In this reprint variation, the authors have resisted the temptation of together with extra themes. with the exception of correcting a number of minor misprints, the nature of the ebook, in particular its specialise in classical illustration conception and its computational elements, has now not been replaced.
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Additional info for A Survey of Lie Groups and Lie Algebra with Applications and Computational Methods
From the theory of topological groups it is known that every open subgroup is closed, and hence open subgroups of Lie groups are also Lie subgroups . Similarly, a homomorphism between Lie groups in general need not be an analytic mapping. A homomorphism between Lie groups is called an analytic homomorphism if the coordinates of the image of a point are analytic functions of the coordinates of the point. Analytic homomorphisms of Lie groups induce homomorphisms of the corresponding Lie algebras.
Weyl, a large variety of quantization procedures have been studied over the years . One of the most useful of these quantization methods, due to Dirac, makes use of Lie algebraic ideas. The Dirac correspondence principle relates the Poisson brackets of classical mechanics to quantum mechanical commutators , , . In this theory the classical Lie algebra of dynamical variables is related to a Lie algebra of operators in Hilbert space in quantum mechanics. By using this correspondence principle, we can often discuss symmetries of analogous systems in a parallel fashion.
For example, in relativistic quantum field theory, a basic problem is that of finding all wave equations invariant under the Lorentz group . The problem simplifies somewhat for the case of zero-mass wave equations because in that case relativistic invariance implies in variance under the bigger group of conformal transformations . The problem of finding differential equations invariant under a given Lie group also comes up in quite different fields. We mention here some work by Hoffman on pattern recognition in visual perception .
A Survey of Lie Groups and Lie Algebra with Applications and Computational Methods by B. Kolman