Hi Emil,
Kind of. So what you are looking for is the compact real form I think:
sage: L = LieAlgebra(QQ, cartan_type=['A',1], representation="compact real")
sage: list(L.basis())
[
[ 0 1] [ i 0] [0 i]
[-1 0], [ 0 -i], [i 0]
]
sage: L = LieAlgebra(QQ, cartan_type=['A',2], representation="compact real")
sage: list(L.basis())
[
[ 0 1 0] [ 0 0 1] [ 0 0 0] [ i 0 0] [0 i 0] [0 0 i]
[-1 0 0] [ 0 0 0] [ 0 0 1] [ 0 0 0] [i 0 0] [0 0 0]
[ 0 0 0], [-1 0 0], [ 0 -1 0], [ 0 0 -i], [0 0 0], [i 0 0],
[ 0 0 0] [0 0 0]
[ 0 i 0] [0 0 i]
[ 0 0 -i], [0 i 0]
]
Now the i in the above is just a little bit of extra processing as it is
considered as a basis in QQ, not QQ[i]. You can of couse look at the Lie
algebras over the corresponding field extension to get the elements you
want. Now this doesn't give you the generators for the Lie groups (a
priori), and it is not implemented for E_8 since that is not given as a
matrix Lie algebra (although that could be fixed by using the adjoint
representation). Also, it is not easy to extract the matrix (at least, I
didn't need it for what I wanted to compute):
sage: L = LieAlgebra(QQ, cartan_type=['A',1], representation="compact real")
sage: M = list(L.basis())[-1]
sage: M
[0 i]
[i 0]
sage: (M._real, M._imag)
(
[0 0] [0 1]
[0 0], [1 0]
)
Implementing getting the matrix would make a nice improvement to Sage.
Best,
Travis
On Thursday, May 28, 2020 at 9:52:14 AM UTC+10, Emil Kozuch wrote:
>
> I am sorry if this question has been asked before.
>
> Is there a way to get the fundamental Lie algebra generators in sage
> (those that physicists use, eq. the Paule matrices, Gellmann matrices and
> so forth)?
> The only thing I can find in the documentation is the generators in the
> Chevalley basis, but they do not satisfy the orthogonality condition
> tr(T^aT^b)=\delta_{ab}.
>
> I am looking for the generators for all the semisimple Lie groups
> (A_n,B_n,C_n,D_n,G_2,F_4,E_6,E_7,E_8).
> The reason i need this is because I am trying to verify some calculations
> done with the birdtrack method for Lie algebras.
>
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