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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.

##### MSC:
 5e+30 Association schemes, strongly regular graphs
##### Keywords:
distance-regular graph; Terwilliger algebra; quantum group
Full Text:
##### References:
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