Embedding of the rank 1 DAHA into Mat(2,Tq) and its automorphisms

Abstract

In this review paper we show how the Cherednik algebra of type $\check{C_1}{C}_1$ 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 $\check{C_1}{C}_1$ into $Mat(2,{\mathbb{T}}_q)$, i.e. $2\times 2$ matrices with entries in the quantum torus. For $q=1$ this result is equivalent to say that the Cherednik algebra of type $\check{C_1}{C}_1$ is Azumaya of degree $2$ \cite{O}. 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 $\check{C_1}{C}_1$ 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 ${\mathcal{H}}_V,{\mathcal{H}}_{IV},{\mathcal{H}}_{III},{\mathcal{H}}_{II}$ and ${\mathcal{H}}_I,$ defined in arXiv:1307.6140.

Dedicated to Masatoshi Noumi for his 60th birthday