Distance-Regular Graphs Related to the Quantum Enveloping Algebra of sl(2)
Brian Curtin
and Kazumasa Nomura
DOI: 10.1023/A:1008707417118
Abstract
We investigate a connection between distance-regular graphs and U q( sl(2)), the quantum universal enveloping algebra of the Lie algebra sl(2). Let T = T( x) \mathcal{T} = \mathcal{T}(x) ( x) denote the Terwilliger algebra of T \mathcal{T} is generated by certain matrices satisfying the defining relations of U q( sl(2)) if and only if 
is bipartite and 2-homogeneous.
is bipartite and 2-homogeneous.Pages: 25–36
Keywords: distance-regular graph; Terwilliger algebra; quantum group
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References
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2. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer, New York, 1989.
3. B. Curtin, “2-homogeneous bipartite distance-regular graphs,” Discrete Math. 187 (1998), 39-70.
4. J. Go, “The Terwilliger algebra of the hypercube,” preprint.
5. M. Jimbo, “Topics from representations of Uq (g)-an introductory guide to physicists,” Quantum Group and Quantum Integrable Systems, Nankai Lectures Math. Phys., World Sci. Publishing, River Edge, NJ, 1992, pp. 1-61.
6. C. Kassel, Quantum Groups, Springer-Verlag, New York, 1995.
7. K. Nomura, “Homogeneous graphs and regular near polygons,” J. Combin. Theory Ser. B 60 (1994), 63-71.
8. K. Nomura, “Spin models on bipartite distance-regular graphs,” J. Combin. Theory Ser. B 64 (1995), 300-313.
9. R.A. Proctor, “Representations of sl(2, C) on posets and the Sperner property,” SIAM J. Algebraic Discrete Methods 3(2) (1982), 275-280.
10. P. Terwilliger, “The subconstituent algebra of an association scheme,” J. Alg. Combin. 1 (1992), 363-388; 2 (1993), 73-103; 2 (1993), 177-210.
11. N. Yamazaki, “Bipartite distance-regular graphs with an eigenvalue of multiplicity k,” J. Combin. Theory Ser. B 66 (1995), 34-37.