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 ...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 A2 （resp. A1^（1）））. We also realize all fusion rings with D - C being a particular generalized Cartan matrix of indefinite type.展开更多
The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible Z＋-modules over the near-group fusion ring K（Z3, 3） are explicitly classified. It turns...The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible Z＋-modules over the near-group fusion ring K（Z3, 3） are explicitly classified. It turns out that there are only four inequivalent irreducible Z＋-modules of rank 2 and two inequivalent irreducible Z＋-modules of rank 4 over K（Z3, 3）.展开更多
基金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)NSFC(Grant No.11471282)
文摘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 A2 （resp. A1^（1）））. We also realize all fusion rings with D - C being a particular generalized Cartan matrix of indefinite type.
基金This work was supported in part by the National Natural Science Foundation of China （Grant No. 11471282）.
文摘The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible Z＋-modules over the near-group fusion ring K（Z3, 3） are explicitly classified. It turns out that there are only four inequivalent irreducible Z＋-modules of rank 2 and two inequivalent irreducible Z＋-modules of rank 4 over K（Z3, 3）.
