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Hi all,

I was following Nakahara's book and I really got my mind stuck with something. I would appreciate if anybody could help with this.

The Lie derivative of a vector field [itex]Y[/itex] along the flow [itex]\sigma_t[/itex] of another vector field [itex]X[/itex] is defined as

[tex] L_X Y=lim_{\epsilon\to0}\frac{1}{\epsilon}\left[Y|_{x}-(\sigma_{\epsilon})_* Y|_{\sigma_{-\epsilon}(x)}\right][/tex]

wich is equal to the Lie bracket [itex][X,Y][/itex].

Now when he goes on and defines the left invariant vector field [itex]X[/itex] of a Lie group, this is given by demanding the relation

[tex]L_{a*}X|_g=X|_{a\,g}[/tex]

where [itex]L_a[/itex] is the left translation by [itex]a[/itex].

I guess we can assume that this left translation is the result of a flow of some other field.

My question is that since every left-invariant vector field defines a one parameter flow, why the Lie bracket of two left invariant vector fields doesn't vanish identically?

However, we know that the left invariant vector fields define the Lie algebra of the group and this has non-trivial commutation relations.

Thank you very much for your help.

I was following Nakahara's book and I really got my mind stuck with something. I would appreciate if anybody could help with this.

The Lie derivative of a vector field [itex]Y[/itex] along the flow [itex]\sigma_t[/itex] of another vector field [itex]X[/itex] is defined as

[tex] L_X Y=lim_{\epsilon\to0}\frac{1}{\epsilon}\left[Y|_{x}-(\sigma_{\epsilon})_* Y|_{\sigma_{-\epsilon}(x)}\right][/tex]

wich is equal to the Lie bracket [itex][X,Y][/itex].

Now when he goes on and defines the left invariant vector field [itex]X[/itex] of a Lie group, this is given by demanding the relation

[tex]L_{a*}X|_g=X|_{a\,g}[/tex]

where [itex]L_a[/itex] is the left translation by [itex]a[/itex].

I guess we can assume that this left translation is the result of a flow of some other field.

My question is that since every left-invariant vector field defines a one parameter flow, why the Lie bracket of two left invariant vector fields doesn't vanish identically?

However, we know that the left invariant vector fields define the Lie algebra of the group and this has non-trivial commutation relations.

Thank you very much for your help.

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