The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli

Abstract

Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the Harder-Narasimhan recursion is constructed inside the quantized enveloping algebra of a Kac-Moody algebra. This leads to a canonical orthogonal system, the HN system, in this algebra. Using a resolution of the recursion, an explicit formula for the HN system is given. As an application, explicit formulas for Betti numbers of the cohomology of quiver moduli are derived, generalizing several results on the cohomology of quotients in 'linear algebra type' situations.

22 pages

Cited by in corpus (46)

• Split States, Entropy Enigmas, Holes and Halos
• Witten Index and Wall Crossing
• Wall-Crossing from Boltzmann Black Hole Halos
• From Black Holes to Quivers
• Stability conditions and quantum dilogarithm identities for Dynkin quivers
• Stability conditions and Stokes factors
• Wild Wall Crossing and BPS Giants
• On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants
• Donaldson-Thomas theory and cluster algebras
• Line Defects, Tropicalization, and Multi-Centered Quiver Quantum Mechanics
• Vafa-Witten invariants from exceptional collections
• Exponential Networks and Representations of Quivers
• Four ways across the wall
• The Coulomb Branch Formula for Quiver Moduli Spaces
• Exact Results in Quiver Quantum Mechanics and BPS Bound State Counting
• Semi-Stable Chow-Hall Algebras of Quivers and Quantized Donaldson-Thomas Invariants
• A-branes, foliations and localization
• Abelianization of BPS Quivers and the Refined Higgs Index
• BPS Hall Algebra of Scattering Hall States
• Counting Higgs bundles and type A quiver bundles
• Generalized quiver mutations and single-centered indices
• Cambrian combinatorics on quiver representations (type A)
• Torsion pairs and quasi-abelian categories
• Linearity of stability conditions
• MPS degeneration formula for quiver moduli and refined GW/Kronecker correspondence
• Witten Index for Noncompact Dynamics
• Fano quiver moduli
• Donaldson-Thomas invariants from tropical disks
• Remarks on Hall algebras of triangulated categories
• Quiver indices and Abelianization from Jeffrey-Kirwan residues
• Multi-centered black holes, scaling solutions and pure-Higgs indices from localization
• An algebraic approach to Harder-Narasimhan filtrations
• Motivic Donaldson-Thomas invariants and Kac conjecture
• Chow Rings of Fine Quiver Moduli are Tautologically Presented
• Stability and Indecomposability of the Representations of Quivers of $A_n$-type
• On higher torsion classes
• Attractor invariants, brane tilings and crystals
• Partial semiorthogonal decompositions for quiver moduli
• Vector fields and admissible embeddings for quiver moduli
• Harder-Narasimhan Filtrations of Persistence Modules
• On Some Computations of Higher Rank Refined Donaldson-Thomas Invariants
• Motivic classes of Nakajima quiver varieties
• Moduli spaces of parabolic bundles over ${\mathbb{P}}^1$ with five marked points
• Quiver Structure of Heterotic Moduli
• Stratifying systems and Jordan-Hölder extriangulated categories
• $e$-polynomials of character varieties