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On realization of fusion rings from generalized Cartan matrices

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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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Acknowledgements

We thank the referees for their careful reading of this article and for valuable comments.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Taizhou University, Taizhou, 225300, P. R. China

    Zhi Hua Wang

  2. School of Mathematical Science, Yangzhou University, Yangzhou, 225002, P. R. China

    Li Bin Li

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  1. Zhi Hua Wang
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  2. Li Bin Li
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Correspondence to Zhi Hua Wang.

Additional information

Supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 15KJB110013); the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20150537) and NSFC (Grant No. 11471282)

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Wang, Z.H., Li, L.B. On realization of fusion rings from generalized Cartan matrices. Acta. Math. Sin.-English Ser. 33, 362–376 (2017). https://doi.org/10.1007/s10114-016-6088-9

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  • DOI: https://doi.org/10.1007/s10114-016-6088-9

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