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Distance-regular graphs of ▫$q$▫-Racah type and the universal Askey-Wilson algebra
Terwilliger, Paul ; Žitnik, Arjana
Z ▫${{\mathbb C}}$▫ označimo obseg kompleksnih števil. Naj bo ▫$q \in \mathbb {C}$▫ takšen, da ▫$q \ne 0$▫ in▫ $q^4 \ne 1$▫. ▫${{\mathbb C}}$▫-algebra ▫$\Delta_q$▫ je definirana z generatorji in ...relacijami na naslednji način. Generatorje imenujemo▫ $A, B, C$▫. Relacije povedo, da je vsak od▫ $$ A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}, \ \ \ B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}}, \ \ \ C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}} $$▫ centralen element algebre▫ $\Delta_q$▫. Algebra ▫$\Delta_q$▫ se imenuje univerzalna Askey-Wilsonova algebra. Naj bo ▫$\Gamma$▫ razdaljno regularen graf tipa ▫$q$▫-Racah. Izberemo vozlišče ▫$x$▫ grafa ▫$\Gamma$▫ in naj bo ▫$T=T(x)$▫ ustrezna Terwilligerjeva algebra. V članku opišemo povezavo med ▫$\Delta_q$▫ in ▫$T$▫. V primeru, da je vsak nerazcepen ▫$T$▫-modul tanek, najdemo surjektiven homomorfizem ▫${{\mathbb C}}$▫-algeber ▫$\Delta_q \to T$▫. To nam da tudi delovanje algebre ▫$\Delta_q$▫ na standardnem modulu ▫$T$▫.
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