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?
--
You received this message because you are subscribed to the Google Groups
"sage-devel" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To view this discussion visit
https://groups.google.com/d/msgid/sage-devel/f9954362-0419-4f54-b784-0c6a0e615ae0n%40googlegroups.com.
