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**why**do we do this' rather than a technical '

**how**do we do this.'

**1. Homework Statement**

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

**2. Homework 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