# Simply-laced isomonodromy systems

@article{Boalch2011SimplylacedIS, title={Simply-laced isomonodromy systems}, author={Philip P. Boalch}, journal={Publications math{\'e}matiques de l'IH{\'E}S}, year={2011}, volume={116}, pages={1-68} }

A new class of isomonodromy equations will be introduced and shown to admit Kac–Moody Weyl group symmetries. This puts into a general context some results of Okamoto on the 4th, 5th and 6th Painlevé equations, and shows where such Kac–Moody Weyl groups and root systems occur “in nature”. A key point is that one may go beyond the class of affine Kac–Moody root systems. As examples, by considering certain hyperbolic Kac–Moody Dynkin diagrams, we find there is a sequence of higher order Painlev…

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## References

SHOWING 1-10 OF 71 REFERENCES

### G-bundles, isomonodromy, and quantum Weyl groups

- Mathematics
- 2001

It is now twenty years since Jimbo, Miwa, and Ueno [23] generalized Schlesinger’s equations (governing isomonodromic deformations of logarithmic connections on vector bundles over the Riemann sphere)…

### Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras

- Mathematics
- 1994

To Professor Shoshichi Kobayashi on his 60th birthday 1. Introduction. In this paper we shall introduce a new family of varieties, which we call quiver varieties, and study their geometric…

### Dual isomonodromic deformations and moment maps to loop algebras

- Mathematics
- 1994

The Hamiltonian structure of the monodromy preserving deformation equations of Jimboet al [JMMS] is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to…

### Studies on the Painlevé equations

- Mathematics
- 1986

SummaryIn this series of papers, we study birational canonical transformations of the Painlevé system ℋ, that is, the Hamiltonian system associated with the Painlevé differential equations. We…

### From Klein to Painlevé Via Fourier, Laplace and Jimbo

- Mathematics
- 2003

We describe a method for constructing explicit algebraic solutions to the sixth Painlevé equation, generalising that of Dubrovin and Mazzocco. There are basically two steps. First we explain how to…

### Multiplicative preprojective algebras, middle convolution and the Deligne–Simpson problem

- Mathematics
- 2004

### On Kleinian Singularities and Quivers

- Mathematics
- 1998

Starting from McKay’s observation on the description of (an essential part of) the representation theory of binary polyhedral groups Γ in terms of extended Coxeter-Dynkin-Witt diagrams \(\tilde…

### Sheaves on ALE Spaces and Quiver Varieties

- Mathematics
- 2007

We identify a quiver variety of an affine type with a framed moduli space of torsion free sheaves on an ALE space, a fiber of a simultaneous resolution of the semi-universal deformation of C/Γ. This…