Generalized Cartan Matrices

IsGeneralizedCartanMatrix(C) : AlgMatElt -> BoolElt
Whether the square matrix C is a generalized Cartan matrix.
KacMoodyClass(C) : AlgMatElt -> MonStgElt, ModMatRngElt
The class of the indecomposable generalized Cartan matrix C. The first return value is a string, "a", "b" or "c", corresponding to the three cases described in the introduction Introduction. The second is a positive integral column vector v such that Cv is positive, 0 or negative, respectively (so this return value corresponds to the vectors θ, δ and α in the introduction).
KacMoodyClasses(C) : AlgMatElt -> SeqEnum, SeqEnum, SeqEnum
The class of the possibly decomposable generalized Cartan matrix C. Three sequences are returned: the first is a sequence of strings "a", "b" or "c", describing the class of each component; the second is a positive integral vector v such that Cv is positive, 0 or negative, respectively (see KacMoodyClass).

The third sequence Q contains integral sequences Qi such that the i-th component is formed by taking the rows and columns with index j, for j ∈Qi.

Example AlgLieKM_generalized-cartan (H108E1)

First, we consider an indecomposable Cartan matrix. 
> C := Matrix(Integers(), 3, 3, [2,-1,0, -5,2,-1, 0,-1,2]);
> s, v := KacMoodyClass(C);
> s;
c
> v;
[2]
[5]
[1]
> C*v;
[-1]
[-1]
[-3]
As a second example, we consider a decomposable Cartan matrix. 
> C := CartanMatrix("B2 A~3");
> S, V, Q := KacMoodyClasses(C);
> S;
[ a, b ]
> Q;
[
    [ 1, 2 ],
    [ 3, 4, 5, 6 ]
]
> C1 := Submatrix(C, Q[1], Q[1]);
> KacMoodyClass(C1);
a
> C2 := Submatrix(C, Q[2], Q[2]);
> KacMoodyClass(C2);
b
V2.28, 13 July 2023