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fliptomato
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Comment: My question is more of a conceptual 'why do we do this' rather than a technical 'how do we do this.'
Given a lie group [tex]G[/tex] parameterized by [tex]x_1, ... x_n[/tex], give a basis of left-invariant vector fields.
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].
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
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.
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].
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