On Leonard Pairs and $q$-Tetrahedron Algebra $\boxtimes_q$
Let $\Fa$ denote an algebraically closed field of characteristic zero, fix a nonzero scalar $q\in \Fa$ that is not a root of unity. Consider the $q$-tetrahedron algebra $\boxtimes_q$ over $\Fa$ with standard generators $\{X_{ij}: i, j \in Z_4, j-i=1 \; or \; j-i=2\}$. Let $V$ denote finite dimensional evaluation module for $\boxtimes_q$.
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Askey-Wilson relations and Leonard pairs
It is known that if $ (A, A^*) $ is a Leonard pair, then the linear transformations $ A $, $ A^* $ satisfy the Askey-Wilson relations $ A^2A^* − \betaAA^*A + A^*A^2 − gamma(AA^* + A^*A) − sigmaA^* = gamma^*A^2 + omegaA + etaI, A^*2A − \betaA^*AA^* + AA^*2 − gamma^*(A^*A + AA^*) − sigma^*A = gammaA^*2 + omegaA^* + eta^*I $, for some scalars $ \beta $, $
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