Lie-Cartan coordinates of the first kind vs Lie-Cartan coordinates of the second kind
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Lie-Cartan coordinates of the first kind:
$$
R_1 = exp(alpha_1w_1 + alpha_2w_2+alpha_3w_3)
$$
Lie-Cartan coordinates of the second kind:
$$
R_2 = exp(beta_1w_1) exp(beta_2w_2) exp(beta_3w_3)
$$
My problem is are these two different?
For $exp$ we have
$$
b^a+b = b^ab^b,
$$
does this not apply to matrices?
matrices rotations
edited Aug 26 at 19:44
Jendrik Stelzner
7,63121037
asked Aug 26 at 19:41
user2660324
1
add a comment |Â
up vote
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Lie-Cartan coordinates of the first kind:
$$
R_1 = exp(alpha_1w_1 + alpha_2w_2+alpha_3w_3)
$$
Lie-Cartan coordinates of the second kind:
$$
R_2 = exp(beta_1w_1) exp(beta_2w_2) exp(beta_3w_3)
$$
My problem is are these two different?
For $exp$ we have
$$
b^a+b = b^ab^b,
$$
does this not apply to matrices?
matrices rotations
edited Aug 26 at 19:44
Jendrik Stelzner
7,63121037
asked Aug 26 at 19:41
user2660324
1
In general, they differ unless $[w_i, w_j] =0$. We have the equality for matrices if they commute.
– Berci
Aug 26 at 20:02
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Lie-Cartan coordinates of the first kind:
$$
R_1 = exp(alpha_1w_1 + alpha_2w_2+alpha_3w_3)
$$
Lie-Cartan coordinates of the second kind:
$$
R_2 = exp(beta_1w_1) exp(beta_2w_2) exp(beta_3w_3)
$$
My problem is are these two different?
For $exp$ we have
$$
b^a+b = b^ab^b,
$$
does this not apply to matrices?
matrices rotations
edited Aug 26 at 19:44
Jendrik Stelzner
7,63121037
asked Aug 26 at 19:41
user2660324
1
Lie-Cartan coordinates of the first kind:
$$
R_1 = exp(alpha_1w_1 + alpha_2w_2+alpha_3w_3)
$$
Lie-Cartan coordinates of the second kind:
$$
R_2 = exp(beta_1w_1) exp(beta_2w_2) exp(beta_3w_3)
$$
My problem is are these two different?
For $exp$ we have
$$
b^a+b = b^ab^b,
$$
does this not apply to matrices?
matrices rotations
edited Aug 26 at 19:44
Jendrik Stelzner
7,63121037
asked Aug 26 at 19:41
user2660324
1
edited Aug 26 at 19:44
Jendrik Stelzner
7,63121037
edited Aug 26 at 19:44
Jendrik Stelzner
7,63121037
edited Aug 26 at 19:44
Jendrik Stelzner
7,63121037
7,63121037
asked Aug 26 at 19:41
user2660324
1
asked Aug 26 at 19:41
user2660324
1
asked Aug 26 at 19:41
user2660324
1
1
In general, they differ unless $[w_i, w_j] =0$. We have the equality for matrices if they commute.
– Berci
Aug 26 at 20:02
add a comment |Â
In general, they differ unless $[w_i, w_j] =0$. We have the equality for matrices if they commute.
– Berci
Aug 26 at 20:02
In general, they differ unless $[w_i, w_j] =0$. We have the equality for matrices if they commute.
– Berci
Aug 26 at 20:02
In general, they differ unless $[w_i, w_j] =0$. We have the equality for matrices if they commute.
– Berci
Aug 26 at 20:02
add a comment |Â
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In general, they differ unless $[w_i, w_j] =0$. We have the equality for matrices if they commute.
– Berci
Aug 26 at 20:02