Distance-regular graphs and the \(q\)-tetrahedron algebra
type: Item
instance of: scholarly article
title: Distance-regular graphs and the \(q\)-tetrahedron algebra
zbMATH Open document ID: 1193.17008
author: Tatsuro Ito
author: Paul Terwilliger
published in: European Journal of Combinatorics
publication date: 2009-04-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0608694
review text: The authors define the \(q\)-tetrahedron algebra generated by \(\{x_{ij} \mid i,j\in \mathbb Z_4, j-i=1 \text{ or } j-i=2\}\) and relations such as \(x_{ij}x_{ji}=1\) if \(j-i=2\) and certain Weyl-type and cubic \(q\)-Serre type of relations, as in semisimple Lie algebra theory. They then exhibit four \(\mathbb C\)-algebra homomorphisms from the quantum algebra \(U_q(\widehat{\mathfrak{sl}}_2)\) to the \(q\)-tetrahedron algebra. They then consider \(\Gamma\), a distance-regular graph, e.g., the Platonic solids, and assume that \(\Gamma\) is formally self-dual, i.e., the intersection numbers coincide with the Krein parameters. They impose further restrictions on the intersection numbers of \(\Gamma\). They proceed to define eight matrices indexed by the set of vertices in the graph and with complex coefficients, called \(A, A^*, B, B^*, K, K^*, \Phi, \Psi\). The main result is that there exists a \(q\)-tetrahedron module structure on the standard module \(V\) of the graph \(\Gamma\) where the action is essentially given by (products of) the above eight matrices. Using the homomorphisms, they also provide \(V\) with a \(U_q(\widehat{\mathfrak{sl}}_2)\)-module structure.
reviewed by: Stefano Capparelli
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 5541742
zbMATH Keywords: Distance-regular graphs
zbMATH Keywords: quantum affine algebra
MaRDI profile type: Publication
OpenAlex ID: W2010388467
arXiv ID: math/0608694
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DOI: 10.1016/J.EJC.2008.07.011
description: scientific article