Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations

Abstract

To a finite quiver equipped with a positive integer on each of its vertices, we associate a holomorphic symplectic manifold having some parameters. This coincides with Nakajima's quiver variety with no stability parameter/framing if the integers attached on the vertices are all equal to one. The construction of reflection functors for quiver varieties are generalized to our case, in which these relate to simple reflections in the Weyl group of some symmetrizable, possibly non-symmetric Kac-Moody algebra. The moduli spaces of meromorphic connections on the rank 2 trivial bundle over the Riemann sphere are described as our manifolds. In our picture, the list of Dynkin diagrams for Painlevé equations is slightly different from (but equivalent to) Okamoto's.

References in corpus (7)

• Dual Isomonodromic Deformations and Moment Maps to Loop Algebras
• Irregular connections and Kac-Moody root systems
• Quivers and difference Painleve equations
• Duality for the general isomonodromy problem
• Fourier transform and middle convolution for irregular D-modules
• Moduli spaces for linear differential equations and the Painlevé equations
• Symmetry in the Painlevé systems and their extensions to four-dimensional systems

Cited by in corpus (4)

• Quivers with relations for symmetrizable Cartan matrices I : Foundations
• Linear differential equations on the Riemann sphere and representations of quivers
• Diagrams for nonabelian Hodge spaces on the affine line
• Regularity of quantum tau-functions generated by quantum birational Weyl group actions