Abstract
The Casimir element of a fusion ring (R,B) gives rise to the so called Casimir matrix C of (R,B). This enables us to construct a generalized Cartan matrix D − C in the sense of Kac for a suitable diagonal matrix D. In this paper, we study some elementary properties of the Casimir matrix C and use them to realize certain fusion rings from the generalized Cartan matrix D−C of finite (resp. affine) type. It turns out that there exists a fusion ring with D−C being of finite (resp. affine) type if and only if D−C has only the form A 2 (resp. A 1 (1) ). We also realize all fusion rings with D−C being a particular generalized Cartan matrix of indefinite type.
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