ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sat, 04 Sep 2010 08:39:19 +0200Structure constants for unitary groupshttps://ask.sagemath.org/question/7661/structure-constants-for-unitary-groups/I want to define a generalized cross product in sage such the a_i=f^{ijk} b_j c_k, where f^{ijk} are the structure constants of the SU(3) group. Are the structure constants for unitary group predefined is sage. If not what is the best way to define such a generalized cross product?
Thanks in advance.
Edit:
Sorry for not being clear about the question. As Mitesh rightly pointed out I am trying to do a High energy calculation. I have two eight dimensional vectors (say b and c). I want to define a generalized product of these two vectors as described in the original post. Here f are structure constants of SU(3) Lie algebra.
When I try the series of commands suggested by niles
gap> e6 := SimpleLieAlgebra("E",6,Rationals);
gap> StructureConstantsTable(Basis(e6));
I get a error in sage but it works if I open gap in a terminal. I think I can manage by copy pasting the results in sage.
Thanks a lot again for the help
Wed, 01 Sep 2010 20:09:04 +0200https://ask.sagemath.org/question/7661/structure-constants-for-unitary-groups/Comment by niles for <p>I want to define a generalized cross product in sage such the a_i=f^{ijk} b_j c_k, where f^{ijk} are the structure constants of the SU(3) group. Are the structure constants for unitary group predefined is sage. If not what is the best way to define such a generalized cross product?
Thanks in advance.</p>
<p>Edit:
Sorry for not being clear about the question. As Mitesh rightly pointed out I am trying to do a High energy calculation. I have two eight dimensional vectors (say b and c). I want to define a generalized product of these two vectors as described in the original post. Here f are structure constants of SU(3) Lie algebra.</p>
<p>When I try the series of commands suggested by niles</p>
<p>gap> e6 := SimpleLieAlgebra("E",6,Rationals);
gap> StructureConstantsTable(Basis(e6));</p>
<p>I get a error in sage but it works if I open gap in a terminal. I think I can manage by copy pasting the results in sage. </p>
<p>Thanks a lot again for the help</p>
https://ask.sagemath.org/question/7661/structure-constants-for-unitary-groups/?comment=22710#post-id-22710ok; sorry for the confusion. I've reworked my answer to use the GAP interface, so it should now work from within sage.Sat, 04 Sep 2010 08:39:19 +0200https://ask.sagemath.org/question/7661/structure-constants-for-unitary-groups/?comment=22710#post-id-22710Comment by Mitesh Patel for <p>I want to define a generalized cross product in sage such the a_i=f^{ijk} b_j c_k, where f^{ijk} are the structure constants of the SU(3) group. Are the structure constants for unitary group predefined is sage. If not what is the best way to define such a generalized cross product?
Thanks in advance.</p>
<p>Edit:
Sorry for not being clear about the question. As Mitesh rightly pointed out I am trying to do a High energy calculation. I have two eight dimensional vectors (say b and c). I want to define a generalized product of these two vectors as described in the original post. Here f are structure constants of SU(3) Lie algebra.</p>
<p>When I try the series of commands suggested by niles</p>
<p>gap> e6 := SimpleLieAlgebra("E",6,Rationals);
gap> StructureConstantsTable(Basis(e6));</p>
<p>I get a error in sage but it works if I open gap in a terminal. I think I can manage by copy pasting the results in sage. </p>
<p>Thanks a lot again for the help</p>
https://ask.sagemath.org/question/7661/structure-constants-for-unitary-groups/?comment=22716#post-id-22716Partly out of curiosity: Are you performing calculations in high-energy physics (e.g., with quarks and gluons)?Fri, 03 Sep 2010 17:16:28 +0200https://ask.sagemath.org/question/7661/structure-constants-for-unitary-groups/?comment=22716#post-id-22716Answer by BWW for <p>I want to define a generalized cross product in sage such the a_i=f^{ijk} b_j c_k, where f^{ijk} are the structure constants of the SU(3) group. Are the structure constants for unitary group predefined is sage. If not what is the best way to define such a generalized cross product?
Thanks in advance.</p>
<p>Edit:
Sorry for not being clear about the question. As Mitesh rightly pointed out I am trying to do a High energy calculation. I have two eight dimensional vectors (say b and c). I want to define a generalized product of these two vectors as described in the original post. Here f are structure constants of SU(3) Lie algebra.</p>
<p>When I try the series of commands suggested by niles</p>
<p>gap> e6 := SimpleLieAlgebra("E",6,Rationals);
gap> StructureConstantsTable(Basis(e6));</p>
<p>I get a error in sage but it works if I open gap in a terminal. I think I can manage by copy pasting the results in sage. </p>
<p>Thanks a lot again for the help</p>
https://ask.sagemath.org/question/7661/structure-constants-for-unitary-groups/?answer=11611#post-id-11611I suspect that you have not had a response because your question is not clear. First of all from the expression you give it looks as though you want the structure constants of a Lie algebra. Secondly the cross product of vectors in 3D gives the Lie algebra of SO(3) or SU(2). The Lie algebra of SU(3) has dimension 8. Your question asks for the definition of the structure constants which is a mathematical question. If you are asking a mathematical question you would be better off on another site, probably http://math.stackexchange.com/. If you are asking about an implementation in sage you should be clear about the mathematics. Fri, 03 Sep 2010 12:13:32 +0200https://ask.sagemath.org/question/7661/structure-constants-for-unitary-groups/?answer=11611#post-id-11611Answer by niles for <p>I want to define a generalized cross product in sage such the a_i=f^{ijk} b_j c_k, where f^{ijk} are the structure constants of the SU(3) group. Are the structure constants for unitary group predefined is sage. If not what is the best way to define such a generalized cross product?
Thanks in advance.</p>
<p>Edit:
Sorry for not being clear about the question. As Mitesh rightly pointed out I am trying to do a High energy calculation. I have two eight dimensional vectors (say b and c). I want to define a generalized product of these two vectors as described in the original post. Here f are structure constants of SU(3) Lie algebra.</p>
<p>When I try the series of commands suggested by niles</p>
<p>gap> e6 := SimpleLieAlgebra("E",6,Rationals);
gap> StructureConstantsTable(Basis(e6));</p>
<p>I get a error in sage but it works if I open gap in a terminal. I think I can manage by copy pasting the results in sage. </p>
<p>Thanks a lot again for the help</p>
https://ask.sagemath.org/question/7661/structure-constants-for-unitary-groups/?answer=11614#post-id-11614You might also be asking whether the functionality of GAP for Lie groups, Lie algebras, and structure constants is accessible from sage; for that, the answer is "yes!". If you can construct the corresponding Lie algebra in GAP, then you can get its structure constants by using `StruccctureConstantsTable`. For example:
<pre>
sage: a3 = gap.SimpleLieAlgebra('"A"',3,'Rationals') # note extra quotes around the string "A"
sage: tg = gap.StructureConstantsTable(gap.Basis(a3))
sage: tg.parent()
Gap
sage: type(tg)
<class 'sage.interfaces.gap.GapElement'>
sage: ts = tg.sage()
sage: type(ts)
<type 'list'>
</pre>
Now you can do the following with either `tg` or `ts`, but note the difference in indexing (GAP lists start indexing at 1!):
<pre>
sage: ts[3][7]
[[1], [-1]]
sage: tg[3][7]
[ [ ], [ ] ]
sage: tg[4][8]
[ [ 1 ], [ -1 ] ]
sage: ts[0]
[[[], []], [[4], [-1]], [[], []], [[], []], [[6], [-1]], [[], []], [[13], [1]], [[], []], [[], []], [[8], [1]], [[], []], [[11], [1]], [[1], [-2]], [[1], [1]], [[], []]]
sage: tg[0]
Traceback (most recent call last)
...
TypeError: Gap produced error output
List Element: <position> must be positive (not a 0)
</pre>
Once you know what relations you're looking for, you could create this algebra by first defining a polynomial ring on the `a_i, b_j, c_k` and an ideal determined by your relations, and then constructing the quotient. Documentation is [here](http://www.sagemath.org/doc/reference/sage/rings/quotient_ring.html). Or, if speed is an issue, maybe you could do the work directly in GAP. You can see the following link to read more about the [GAP interface](http://www.sagemath.org/doc/reference/sage/interfaces/gap.html). Or maybe someone else has a better solution.
p.s. I do agree, it might help to clarify your question; and watch out for _underscores_.Fri, 03 Sep 2010 14:48:10 +0200https://ask.sagemath.org/question/7661/structure-constants-for-unitary-groups/?answer=11614#post-id-11614