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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 characteristic different from 2.)
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 ...
It is a consequence of the general result relating Lie group and Lie algebra homomorphisms via the exponential map. [ 5 ] If G is an immersely linear Lie group, then the above computation simplifies: indeed, as noted early, Ad g ( Y ) = g Y g − 1 {\displaystyle \operatorname {Ad} _{g}(Y)=gYg^{-1}} and thus with g = e t X {\displaystyle g ...
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 ...
Let be a Lie algebra and let be a vector space. We let () denote the space of endomorphisms of , that is, the space of all linear maps of to itself. Here, the associative algebra () is turned into a Lie algebra with bracket given by the commutator: [,] = for all s,t in ().
The Pauli matrices abide by the physicists' convention for Lie algebras. In that convention, Lie algebra elements are multiplied by i, the exponential map (below) is defined with an extra factor of i in the exponent and the structure constants remain the same, but the definition of them acquires a factor of i.
Note that a "complex Lie group" is defined as a complex analytic manifold that is also a group whose multiplication and inversion are each given by a holomorphic map. The dimensions in the table below are dimensions over C. Note that every complex Lie group/algebra can also be viewed as a real Lie group/algebra of twice the dimension.
In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism :, satisfying a Leibniz rule.A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.