Embeddings of root systems. I: Root systems over commutative rings.
type: Item
instance of: scholarly article
title: Embeddings of root systems. I: Root systems over commutative rings.
zbMATH Open document ID: 1181.20036
author: Matthew Dyer
published in: Journal of Algebra
publication date: 2009-08-27T00:00:00Z
review text: The paper under review considers a class of reflection representations of Coxeter groups over commutative unital rings \(R\) and their corresponding root systems. For \(R=\mathbb{R}\), these include the standard (real, reduced) root systems. In general, they also include, for any (unital) subring \(Z\) of \(\mathbb{R}\), certain root systems over coefficient rings \(R\) which are generalized \(Z\)-fusion algebras arising as quotients of coproducts over \(Z\) of Verlinde algebras (over \(Z\)) for \(\text{su}(2)\) WZW conformal field theories at levels determined by entries of the Coxeter matrix, call these \(Z\)-fusion root systems and \(Z\)-fusion reflection representations. Any \(Z\)-fusion root system \(\Phi\) of a Coxeter system \((W,S)\) embeds \(Z\)-linearly in a standard real root system \(\Psi\) (defined over \(Z\)) of a corresponding Coxeter system \((W',S')\). Every Coxeter system \((W,S)\) has \(Z\)-fusion root systems for every subring \(Z\) of \(\mathbb{R}\). In particular, \((W,S)\) has a canonical \(\mathbb{Z}\)-fusion root system \(\Phi\) (its ``universal symmetric root system over commutative rings''); for this \(\Phi\), \((W',S')\) is a simply-laced Coxeter system. In addition, any Coxeter system for which all the entries of its Coxeter matrix are either 1, 2, 3, 4, 6, \(\infty\) or odd primes has a \(\mathbb{Z}\)-fusion root system \(\Phi\) whose coefficient ring \(R\) is a subring of \(\mathbb{R}\), and for which \(\Phi\) is just a standard root system. As an example, the paper describes explicitly an embedding \(\Phi(H_4)\to\Phi(E_8)\) of standard root systems.
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 17B22
zbMATH DE Number: 5599086
zbMATH Keywords: Coxeter groups
zbMATH Keywords: root systems
zbMATH Keywords: fusion rings
zbMATH Keywords: Verlinde algebras
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.10.008
OpenAlex ID: W2032790067
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DOI: 10.1016/J.JALGEBRA.2008.10.008
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description: scientific article; zbMATH DE number 5599086