Lie group

A Lie group $𝐺$ is a topological group $𝐺$ on an analytic manifold $𝐺$ such that the group operation $(\cdot) : 𝐺 \times 𝐺 \to 𝐺$ and the inverse map $(-) - 1 : 𝐺 \to 𝐺$ are analytic. lie Equivalently, the map $𝑔 ℎ \mapsto 𝑔 ℎ - 1$ is analytic.

Charts and atlases of a Lie group

Due to properties of a Lie group as a Homogenous space, it is possible to build an atlas for the entire group from a single coördinate chart $(𝑈, 𝜑)$ . For if $(𝑈, 𝜑)$ is a chart with $𝜑 (𝑒) = ⃗ 𝟎$ , then $(𝑔 𝑈, 𝜑 𝑔)$ may be defined as
$𝜑 𝑔 : 𝑔 𝑈 \to 𝜑 𝑔 (𝑔 𝑈) ℎ \mapsto 𝜑 𝑔 (𝑔 - 1 ℎ)$
so that $𝜑 𝑔 (𝑔) = ⃗ 𝟎$ .

Every Lie group has a corresponding Lie algebra given by the Tangent space at identity. clarify

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Lie group

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