Structure constants for unitary groups

edit

You 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:

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'>

Now you can do the following with either tg or ts, but note the difference in indexing (GAP lists start indexing at 1!):

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)

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. 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. 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_.

I 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.