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Concept
Lie group–Lie algebra correspondence
Summary
In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are isomorphic to each other, but the converse is not necessarily true. One obvious counterexample is and (see real coordinate space and the circle group respectively) which are non-isomorphic to each other as Lie groups but their Lie algebras are isomorphic to each other. However, by restricting our attention to the simply connected Lie groups, the Lie group-Lie algebra correspondence will be one-to-one.
In this article, a Lie group refers to a real Lie group. For the complex and p-adic cases, see complex Lie group and p-adic Lie group. In this article, manifolds (in particular Lie groups) are assumed to be second countable; in particular, they have at most countably many connected components.
There are various ways one can understand the construction of the Lie algebra of a Lie group G. One approach uses left-invariant vector fields. A vector field X on G is said to be invariant under left translations if, for any g, h in G,
where is defined by and is the differential of between tangent spaces.
Let be the set of all left-translation-invariant vector fields on G. It is a real vector space. Moreover, it is closed under Lie bracket; i.e., is left-translation-invariant if X, Y are. Thus, is a Lie subalgebra of the Lie algebra of all vector fields on G and is called the Lie algebra of G. One can understand this more concretely by identifying the space of left-invariant vector fields with the tangent space at the identity, as follows: Given a left-invariant vector field, one can take its value at the identity, and given a tangent vector at the identity, one can extend it to a left-invariant vector field. This correspondence is one-to-one in both directions, so is bijective.
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