Distance-regular graphs related to the quantum enveloping algebra of \(sl(2)\). (English) Zbl 0967.05067

A graph \(\Gamma\) is \(2\)-homogeneous whenever for all integers \(i\) there exists a scalar \(\gamma_i\) such that for all vertices \(x,y\) and \(z\) with \(y\) and \(z\) at distance \(i\) from \(x\) and at distance \(2\) from each other there are precisely \(\gamma_i\) neighbours of \(y\) and \(z\) which are at distance \(i-1\) from \(x\).
The first author proved that a distance-regular graph \(\Gamma\) which is not isomorphic to the \(d\)-cube is bipartite and \(2\)-homogeneous iff there exists a complex number \(q\) which is not \(0,1\) or \(-1\) such that the parameters \(c_i\) and \(b_i\) of \(\Gamma\) (for their definition see [A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-regular graphs (Springer-Verlag, Berlin, etc.) (1989; Zbl 0747.05073)]) can be written as algebraic expressions in \(q\); see B. Curtin [Discrete Math. 187, No. 1-3, 39-70 (1998; Zbl 0958.05143)].
In the paper under review the authors consider a distance-regular graph \(\Gamma\) which is not a \(d\)-cube and whose diameter and valency are at least \(3\). For a vertex \(x\) of \(\Gamma\) let \(T = T(x)\) be the Terwilliger algebra. They prove that \(T\) is a homomorphic image of the quantum universal enveloping algebra \(U_q(sl(2))\) iff \(\Gamma\) is bipartite and \(2\)-homogeneous. The scalar \(q\) is as in the criterion of Curtin.


05E30 Association schemes, strongly regular graphs
Full Text: DOI


[1] E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin/Cummings, Menlo Park, 1984.
[2] A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer, New York, 1989. · Zbl 0747.05073
[3] Curtin, B., 2-homogeneous bipartite distance-regular graphs, Discrete Math., 187, 39-70, (1998) · Zbl 0958.05143
[4] J. Go, “The Terwilliger algebra of the hypercube,” preprint. · Zbl 0997.05097
[5] Jimbo, M., Topics from representations of Uq (g)-an introductory guide to physicists, 1-61, (1992), River Edge, NJ
[6] C. Kassel, Quantum Groups, Springer-Verlag, New York, 1995.
[7] Nomura, K., Homogeneous graphs and regular near polygons, J. Combin. Theory Ser. B, 60, 63-71, (1994) · Zbl 0793.05130
[8] Nomura, K., Spin models on bipartite distance-regular graphs, J. Combin. Theory Ser. B, 64, 300-313, (1995) · Zbl 0827.05060
[9] Proctor, R. A., Representations of sl\( (2;\)C) on posets and the Sperner property, SIAM J. Algebraic Discrete Methods, 3, 275-280, (1982) · Zbl 0496.06004
[10] Terwilliger, P., The subconstituent algebra of an association scheme, J. Alg. Combin., 1, 363-388, (1992) · Zbl 0785.05089
[11] Yamazaki, N., Bipartite distance-regular graphs with an eigenvalue of multiplicity \(k,\) J. Combin. Theory Ser. B, 66, 34-37, (1995) · Zbl 0835.05087
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.