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The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups. The ordinary exponential function of mathematical analysis is a special case of the exponential map when G {\displaystyle G} is the multiplicative group of positive real numbers (whose Lie algebra is the additive group ...
This glossary of geography terms is a list of definitions of terms and concepts used in geography and related fields, including Earth science, oceanography, cartography, and human geography, as well as those describing spatial dimension, topographical features, natural resources, and the collection, analysis, and visualization of geographic ...
exponential map (Lie theory) from a Lie algebra to a Lie group, More generally, in a manifold with an affine connection, (), where is a geodesic with initial velocity X, is sometimes also called the exponential map. The above two are special cases of this with respect to appropriate affine connections.
Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4; Kac, Victor (1990). Infinite dimensional Lie algebras (3rd ed.). Cambridge University Press. ISBN 0-521-46693-8. Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN ...
The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the ...
Since most of the interesting examples of Lie groups can be realized as matrix Lie groups, some textbooks restrict attention to this class, including those of Hall, [11] Rossmann, [12] and Stillwell. [13] Restricting attention to matrix Lie groups simplifies the definition of the Lie algebra and the exponential map.
One setting in which the Lie algebra representation is well understood is that of semisimple (or reductive) Lie groups, where the associated Lie algebra representation forms a (g,K)-module. Examples of unitary representations arise in quantum mechanics and quantum field theory, but also in Fourier analysis as shown in the following example.
Here the zeros on the ends represent the zero Lie algebra (containing only the zero vector 0) and the maps are the obvious ones; maps 0 to 0 and maps all elements of to 0. With this definition, it follows automatically that i is a monomorphism and s is an epimorphism.