Lie theory MOC

Lie group

A Lie group 𝐺 is a topological group 𝐺 on an analytic manifold 𝐺 such that the group operation ( β‹…) :𝐺 ×𝐺 →𝐺 and the inverse map ( βˆ’)βˆ’1 :𝐺 →𝐺 are analytic. #m/def/lie Equivalently, the map π‘”β„Ž β†¦π‘”β„Žβˆ’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 coΓΆrdinate chart (π‘ˆ,πœ‘). For if (π‘ˆ,πœ‘) is a chart with πœ‘(𝑒) =βƒ—πŸŽ, then (π‘”π‘ˆ,πœ‘π‘”) may be defined as

πœ‘π‘”:π‘”π‘ˆβ†’πœ‘π‘”(π‘”π‘ˆ)β„Žβ†¦πœ‘π‘”(π‘”βˆ’1β„Ž)

so that πœ‘π‘”(𝑔) =βƒ—πŸŽ.

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

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