2016/03/11 by Marta Mazzocco, Mazzocco, Marta
Mathematics · Physics and Astronomy · #16T99 #33D52 #33D80 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Complex Variables (math.CV) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Representation Theory (math.RT) #math-ph #math.CV #math.MP #math.RT #msc:16T99 #msc:33D52 #msc:33D80 #nlin.SI
paper · pdf · doi:10.48550/arxiv.1603.03770
Dedicated to Masatoshi Noumi for his 60th birthday
arxiv created 2016/03/11 · openalex publication_date 2016/03/11 · arxiv updated 2016/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this review paper we show how the Cherednik algebra of type \checkC1C1 appears naturally as quantisation of the group algebra of the monodromy group associated to the sixth Painlevé equation. This fact naturally leads to an embedding of the Cherednik algebra of type \checkC1C1 into Mat(2,\mathbb Tq), i.e. 2× 2 matrices with entries in the quantum torus. For q=1 this result is equivalent to say that the Cherednik algebra of type \checkC1C1 is Azumaya of degree 2 \citeO. By quantising the action of the braid group and of the Okamoto transformations on the monodromy group associated to the sixth Painlevé equation we study the automorphisms of the Cherednik algebra of type \checkC1C1 and conjecture the existence of a new automorphism. Inspired by the confluences of the Painlevé equations, we produce similar embeddings for the confluent Cherednik algebras \mathcal HV,\mathcal HIV,\mathcal HIII,\mathcal HII and \mathcal HI, defined in arXiv:1307.6140.