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 \(\theta\), \(\delta\) and \(\alpha\) 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 \(Q_i\) such that the \(i\)-th component is formed by taking the rows and columns with index \(j\), for \(j \in Q_i\).

Example: Generalized Cartan (ex-89fcf7)

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]

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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

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