- #1

i_emanuel

- 14

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If we have vect (u) which denotes an infinite-dimensional vector space of all vector fields on u. As infinitesimal elements of the continuous group of Diff(u) they form a Lie Algebra. We then can define the bracket of two vector fields in v and w. If in coordinates:

v = [tex]\sum_{i}[/tex]V

w = [tex]\sum_{j}[/tex]W

the components of

[v,w]:=[tex](\sum_{i,j}([/tex] [tex]V^{i}[/tex] [tex]\frac{d}{dx}[/tex][tex]X^{i}[/tex] [tex]W^{j}[/tex] - [tex]W^{i}[/tex] [tex]\frac{d}{dx}[/tex][tex]X^{i}[/tex] [tex]V^{j} )[/tex] [tex]\frac{d}{dx}[/tex]

if the definition is independent of the choice of coordinates is it bilinear by nature? if so it must be antisymmetric [v,w] = -[w,v];

therefore the jacobi identity would yield [v,[u,w]] = [[v,u],w] + [u, [v,w]]

how can i go about verifying this for a lie bracket?

v = [tex]\sum_{i}[/tex]V

^{i}[tex]\partial[/tex][tex]/[/tex][tex]\partial[/tex]X[tex]^{i}[/tex]w = [tex]\sum_{j}[/tex]W

^{j}[tex]\partial[/tex][tex]/[/tex] [tex]\partial[/tex]X[tex]^{j}[/tex]the components of

**[v,w]**[v,w]:=[tex](\sum_{i,j}([/tex] [tex]V^{i}[/tex] [tex]\frac{d}{dx}[/tex][tex]X^{i}[/tex] [tex]W^{j}[/tex] - [tex]W^{i}[/tex] [tex]\frac{d}{dx}[/tex][tex]X^{i}[/tex] [tex]V^{j} )[/tex] [tex]\frac{d}{dx}[/tex]

if the definition is independent of the choice of coordinates is it bilinear by nature? if so it must be antisymmetric [v,w] = -[w,v];

therefore the jacobi identity would yield [v,[u,w]] = [[v,u],w] + [u, [v,w]]

how can i go about verifying this for a lie bracket?

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