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.Comment: Dedicated to Masatoshi Noumi for his 60th birthda

Irregular isomonodromic deformations for Garnier systems and Okamoto's canonical transformations

In this paper we describe the Garnier systems as isomonodromic deformation equations of a linear system with a simple pole at zero and a Poincar\'e rank two singularity at infinity. We discuss the extension of Okamoto's birational canonical transformations to the Garnier systems in more than one variable and to the Schlesinger systems.Comment: 17 page

Quantum ordering for quantum geodesic functions of orbifold Riemann surfaces

We determine the explicit quantum ordering for a special class of quantum geodesic functions corresponding to geodesics joining exactly two orbifold points or holes on a non-compact Riemann surface. We discuss some special cases in which these quantum geodesic functions form sub--algebras of some abstract algebras defined by the reflection equation and we extend our results to the quantisation of matrix elements of the Fuchsian group associated to the Riemann surface in Poincar\'e uniformization. In particular we explore an interesting relation between the deformed $U_q(\mathfrak{sl}_2)$ and the Zhedanov algebra AW(3).Comment: 22 pages; 6 figures in LaTeX; contribution to AMS volume dedicated to the 75th birthday of S.P.Noviko

Confluence on the Painlev\'e Monodromy Manifolds, their Poisson Structure and Quantisation

In this paper we obtain a system of flat coordinates on the monodromy manifold of each of the Painlev\'e equations. This allows us to quantise such manifolds. We produce a quantum confluence procedure between cubics in such a way that quantisation and confluence commute. We also investigate the underlying cluster algebra structure and the relation to the versal deformations of singularities of type $D_4,A_3,A_2$, and $A_1$.Comment: Version 1, 16 pages, 3 figure

On the Reductions and Classical Solutions of the Schlesinger equations

The Schlesinger equations $S_{(n,m)}$ describe monodromy preserving deformations of order $m$ Fuchsian systems with $n+1$ poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of $n$ copies of $m\times m$ matrix algebras equipped with the standard linear Poisson bracket. In this paper we address the problem of reduction of particular solutions of ``more complicated'' Schlesinger equations $S_{(n,m)}$ to ``simpler'' $S_{(n',m')}$ having $n'< n$ or $m' < m$.Comment: 32 pages. To the memory of our friend Andrei Bolibruc

Canonical structure and symmetries of the Schlesinger equations

The Schlesinger equations $S_{(n,m)}$ describe monodromy preserving deformations of order $m$ Fuchsian systems with $n+1$ poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of $n$ copies of $m\times m$ matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation of the general Schlesinger equations $S_{(n,m)}$ for all $n$, $m$ and we compute the action of the symmetries of the Schlesinger equations in these coordinates.Comment: 92 pages, no figures. Theorem 1.2 corrected, other misprints removed. To appear on Comm. Math. Phy