Geometry of Multiplicative Preprojective Algebra

Abstract

Crawley-Boevey and Shaw recently introduced a certain multiplicative analogue of the deformed preprojective algebra, which they called the multiplicative preprojective algebra. In this paper we study the moduli space of (semi)stable representations of such an algebra (the multiplicative quiver variety), which in fact has many similarities to the quiver variety. We show that there exists a complex analytic isomorphism between the nilpotent subvariety of the quiver variety and that of the multiplicative quiver variety (which can be extended to a symplectomorphism between these tubular neighborhoods). We also show that when the quiver is star-shaped, the multiplicative quiver variety parametrizes Simpson's (poly)stable filtered local systems on a punctured Riemann sphere with prescribed filtration type, weight and associated graded local system around each puncture.

51pages; corrected typos and references; changed font; v4 is the same as v3 except margin

References in corpus (7)

• Symplectic Manifolds and Isomonodromic Deformations
• Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevé equation of type VI, Part I
• Quasi-Hamiltonian Geometry of Meromorphic Connections
• Stokes Matrices and Poisson Lie Groups
• Absolutely indecomposable representations and Kac-Moody Lie algebras (with an appendix by Hiraku Nakajima)
• A remark on quiver varieties andweyl groups
• Mixed Hodge polynomials of character varieties

Cited by in corpus (29)

• Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, II
• Multiplicative quiver varieties and generalised Ruijsenaars-Schneider models
• Twisted wild character varieties
• Global Weyl groups and a new theory of multiplicative quiver varieties
• On the combinatorics of exact Lagrangian surfaces
• Calabi-Yau structures on topological Fukaya categories
• Linear differential equations on the Riemann sphere and representations of quivers
• Spin versions of the complex trigonometric Ruijsenaars-Schneider model from cyclic quivers
• Geometry and braiding of Stokes data; Fission and wild character varieties
• Geometry of moduli spaces of meromorphic connections on curves, Stokes data, wild nonabelian Hodge theory, hyperkahler manifolds, isomonodromic deformations, Painleve equations, and relations to Lie theory
• Multiplicative preprojective algebras are 2-Calabi-Yau
• Relative Calabi-Yau completions
• Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systems
• Calabi-Yau structures for multiplicative preprojective algebras
• Quantum Multiplicative Hypertoric Varieties and Localization
• The moduli stack of parabolic bundles over the projective line, quiver representations, and the Deligne-Simpson problem
• Multiplicative preprojective algebras of Dynkin quivers
• Integrable systems on multiplicative quiver varieties from cyclic quivers
• Spectral Description of the Spin Ruijsenaars-Schneider System
• Moduli spaces of meromorphic connections and quiver varieties
• On the noncommutative Poisson geometry of certain wild character varieties
• Lecture Notes on Quiver Representations and Moduli Problems in Algebraic Geometry
• Euler continuants in noncommutative quasi-Poisson geometry
• Moduli spaces of framed logarithmic and parabolic connections on a Riemann surface
• Symmetric cubic surfaces and G_2 character varieties
• Quiver varieties and the character ring of general linear groups over finite fields
• Unfolding of the unramified irregular singular generalized isomonodromic deformation
• Cohomology of non-generic character stacks
• Moduli of regular singular parabolic connections of spectral type on smooth projective curves