Lie group–Lie algebra correspondence

In mathematics, Lie group–Lie algebra correspondence allows one to study Lie groups, which are geometric objects, in terms of Lie algebras, which are linear objects. 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 ${\displaystyle l_{g}:G\to G,x\mapsto gx}$ and ${\displaystyle (dl_{g})_{h}:T_{h}G\to T_{gh}G}$ is the differential of ${\displaystyle l_{g}}$ between tangent spaces. (In other words, it is ${\displaystyle l_{g}}$-related to itself for any g in G.)

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