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# Lie group

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A Lie group G is an analytic manifold that is also a group such that inversion is analytic, and the product is analytic on the product manifold.

A vector field X is said to be left invariant if it commutes with left translation: Define L_g[f](x)= f(gx), then a vector field is left invariant if X L_g = L_g X for all g in G.

The set of all vector fields on an analytic manifold is a Lie algebra. The subalgebra of all left invariant vector fields is called the Lie algebra associated to G, and is usually denoted by a gothic g.

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Last edited July 27, 2001 1:33 pm (diff)
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