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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 ...
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 ...
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.
The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows.
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 ...
Now, for the case of a representation of a Lie algebra, we simply drop all the gradings and the (−1) to the some power factors. A Lie (super)algebra is an algebra and it has an adjoint representation of itself. This is a representation on an algebra: the (anti)derivation property is the super Jacobi identity.
The definition of a Lie algebra can be reformulated more abstractly in the language of category theory. Namely, one can define a Lie algebra in terms of linear maps—that is, morphisms in the category of vector spaces—without considering individual elements. (In this section, the field over which the algebra is defined is assumed to be of ...
The Chevalley group construction of Lie groups in terms of their Dynkin diagram does not yield some of the classical groups, namely the unitary groups and the non-split orthogonal groups. The Steinberg groups construct the unitary groups 2 A n , while the other orthogonal groups are constructed as 2 D n , where in both cases this refers to ...