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

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 Painlev\'e 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 Painlev\'e 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 Painlev\'e 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.