Lie 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.

Let G be a Lie group. 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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