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.
Basics
The Lie algebra of a Lie group
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 space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
s.
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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
; i.e.,