You’re using the matrix representation, which is not the Chevalley basis
(as linked). So I don’t see the inconsistency right now as the element
constructed in g_{\theta} can be a scalar multiple I think. Or have I
overly simplified something? Feel free to email me directly and/or cc me on
any Lie algebra tickets. I’m always happy to improve the Lie algebras in
Sage. (Any errors are almost certainly my fault anyways.)
Best,
Travis
On Friday, November 28, 2025 at 9:14:22 PM UTC+9 [email protected] wrote:
> I am finding an inconsistency between the conventions for the Chevalley
> basis in
> https://doc.sagemath.org/html/en/reference/algebras/sage/algebras/lie_algebras/classical_lie_algebra.html#sage.algebras.lie_algebras.classical_lie_algebra.LieAlgebraChevalleyBasis
>
> and the (co)highest root. Namely consider the following code
>
> ct = ['C', 2]
> rank = ct[1]
> g = LieAlgebra(QQ, cartan_type=ct, representation='matrix')
>
> # We get the basis elements e_Theta, f_Theta, and calculate their
> commutator
> e_Theta, f_Theta = [g.highest_root_basis_elt(pos=TF) for TF in [True,
> False]]
> h_Theta = e_Theta.bracket(f_Theta)
>
> # We calculate h_Theta alternatively by the definition in terms of
> coroots,
> Theta_check = RootSystem(ct).root_lattice().cohighest_root()
> h_Theta_def = sum(Theta_check[i]*g.h(i) for i in range(1, rank+1))
>
> scale_dict = {'A':1, 'B':4, 'C':4^(rank-1), 'D':1, 'E':1, 'F':64, 'G':36}
> h_Theta == scale_dict[ct[0]]*h_Theta_def
>
> This shows an inconsistent scaling occuring for Lie algebras which aren't
> of type ADE. Have I misunderstood some part of the conventions (the
> classical lie algebra doc page does have some typsetting issues, which
> someone, perhaps me, should address in a small ticket) or is there an issue
> with the scale of the basis elements?
>
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