Comment: My question is more of a conceptual '(adsbygoogle = window.adsbygoogle || []).push({}); whydo we do this' rather than a technical 'howdo we do this.'

1. The problem statement, all variables and given/known data

Given a lie group [tex]G[/tex] parameterized by [tex]x_1, ... x_n[/tex], give a basis of left-invariant vector fields.

2. Relevant equations

We have a basis for the vector fields at the identity, namely the Lie algebra: [tex]v_1, ..., v_n[/tex]. For a general group element [tex]g[/tex], we can write [tex]g^{-1}

\frac{\partial g}{\partial x_i}[/tex].

3. The attempt at a solution

Apparently the procedure is to write [tex]v_i = A^{ij}g^{-1}

\frac{\partial g}{\partial x_j}[/tex], where the [tex]A^{ij}[/tex] are just coefficients. Then we can somewhat magically read off the left invariant vector fields [tex]w_i[/tex] via:

[tex]w_i = A^{ij}\frac{\partial}{\partial x_j}[/tex]

Where I have assumed a sum over repeated indices (though haven't been careful with upper or lower indices).

Why is this the correct procedure? I'm especially concerned about this thing [tex]\frac{\partial g}{\partial x_j}[/tex]... is it an element of the tangent space or is it dual to the tangent space?

Thanks for any thoughts,

Flip

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# Homework Help: Left invariant vector fields of a lie group

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