On realization of fusion rings from generalized Cartan matrices
The matrix for left multiplication of the central element of a ring \( R\) with respect to a basis \(B\) is the Casimir matrix \(C\) of \((R,B)\). The information in \(C\) allows to classify \(R\). This is used here to determine fusion rings from generalized Cartan matrices \(D - C\) for \(R\) where \(D\) is a suitable diagonal matrix. In particular, special generalized Cartan matrices of types \(A_n\), \(D_n\), and \(E_n\) are studied as regards ring realization. Unfortunately, these matrix types are not defined, nor explained in the paper. But they are available in \textit{V. G. Kac}'s book [Infinite dimensional Lie algebras. 3rd ed. Cambridge etc.: Cambridge University Press (1990; Zbl 0716.17022)] in Section 4.8. The studied matrices are tridiagonal of size \(n\) by \(n\) except for a full small leading diagonal block and they contain only small integers.
scientific article; zbMATH DE number 6715540
scientific article