X,Y and Z operators of some algebra

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SUMMARY

The discussion focuses on the algebraic properties of the X, Y, and Z operators, specifically their commutation relations defined by the equation [X_i, Y_j] = ε_ijk X_k. The change of coordinates is defined using polar coordinates where X = r cos(u), Y = r sin(u), and Z = Z. The main inquiry revolves around determining the commutation relations for the new operators r, u, and Z, specifically [r, u], [r, z], and [u, z]. The discussion highlights the need for clarity on the implications of these transformations within the algebraic framework.

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let be X,Y and Z operators of some algebra so

[tex][X_i , Y_j]= \epsilon _ijkX_k[/tex] where i,j and k range over X, Y and Z

then i define the change of coordinates

[tex]X= rcos(u)[/tex]

[tex]Y= rsin(u)[/tex]

[tex]Z=Z[/tex]

again r, u and Z are new operators, the problem is , how can i find for example what is the value of [tex][r, u][/tex] or [tex][r,z][/tex] or [tex][u,z][/tex] ??
 
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If X, Y, and Z are in "some algebra", then what do "r cos(u)" and "r sin(u)" mean?
 

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