On realization of fusion rings from generalized Cartan matrices
type: Item
instance of: scholarly article
title: On realization of fusion rings from generalized Cartan matrices
zbMATH Open document ID: 1377.15020
publication date: 2017-05-15T00:00:00Z
review text: 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.
reviewed by: Frank Uhlig
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 6715540
zbMATH Keywords: fusion ring
zbMATH Keywords: Cartan matrix
zbMATH Keywords: Casimir matrix
zbMATH Keywords: ring realizations
author: Zhihua Wang
author: Libin Li
MaRDI profile type: Publication
full work available at URL: s10114-016-6088-9
OpenAlex ID: W2470167333
cites work: Q4523528
cites work: Conformal field theory
cites work: Nilpotent fusion categories
cites work: Q4040877
cites work: Pseudo-unitary non-self-dual fusion categories of rank 4
cites work: 1-supertransitive subfactors with index at most \(6\frac{1}{5}\)
cites work: Some applications of Frobenius algebras to Hopf algebras
cites work: Classical and quantum conformal field theory
cites work: Fusion categories of rank 2
cites work: Pivotal fusion categories of rank 3
cites work: On formal codegrees of fusion categories
cites work: Near-group categories
DOI: 10.1007/S10114-016-6088-9
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6715540