2013/07/23 by Marta Mazzocco, Mazzocco, Marta · 2 citations
Mathematics · Physics and Astronomy · #16T99 #33D52 #33D80 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #math-ph #math.MP #math.QA #msc:16T99 #msc:33D52 #msc:33D80 #nlin.SI
paper · pdf · doi:10.48550/arxiv.1307.6140
Much less technical version, lots of formulae made more efficient
arxiv created 2015/10/19 · arxiv updated 2015/10/20
In this paper we produce seven new algebras as confluences of the Cherednik algebra of type \checkC1C1 and we characterise their spherical-sub-algebras. The limit of the spherical sub-algebra of the Cherednik algebra of type \checkC1C1 is the monodromy manifold of the Painlevé VI equation. Here we prove that by considering the limits of the spherical sub-algebras of our new confluent algebras, one obtains the monodromy manifolds of all other Painlevé differential equations. Moreover, we introduce confluent versions of the Zhedanov algebra and prove that each of them (quotiented by their Casimir) is isomorphic to the corresponding spherical sub-algebra of our new confluent Cherednik algebras. We show that in the basic representation our confluent Zhedanov algebras act as symmetries of certain elements of the q-Askey scheme, thus setting a stepping stone towards the solution of the open problem of finding the corresponding quantum algebra for each element of the q-Askey scheme. These results establish a new link between the theory of the Painlevé equations and the theory of the q-Askey scheme and shed light on the reasons behind the occurrence of special polynomials in the Painlevé theory.