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

*(English)*Zbl 0967.05067A 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.

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.

Reviewer: B.Baumeister (Halle)

##### MSC:

05E30 | Association schemes, strongly regular graphs |

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\textit{B. Curtin} and \textit{K. Nomura}, J. Algebr. Comb. 12, No. 1, 25--36 (2000; Zbl 0967.05067)

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##### References:

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