[Paper Review] Stability conditions for preprojective algebras and root systems of Kac-Moody Lie algebras

Akishi Ikeda|arXiv (Cornell University)|Feb 6, 2014

Algebraic structures and combinatorial models14 references6 citations

TL;DR

This paper establishes that the distinguished connected component of the space of stability conditions on the bounded derived category of nilpotent modules over a preprojective algebra associated with a quiver without loops is a covering space over an open subset of the complexified Grothendieck group, determined by the root system of the corresponding Kac-Moody Lie algebra. The covering structure is described via the action of the Weyl group and the braid group, generalizing earlier results for ADE and affine ADE quivers to arbitrary quivers without loops.

ABSTRACT

The aim of this paper is to study the space of stability conditions on the bounded derived category of nilpotent modules over the preprojective algebra associated with a quiver without loops. We describe this space as a covering space of some open set determined by the root system of the Kac-Moody Lie algebra associated with the quiver.

Motivation & Objective

• To describe the structure of the space of stability conditions on the bounded derived category of nilpotent modules over a preprojective algebra associated with a quiver without loops.
• To generalize previous results on stability conditions for ADE and affine ADE quivers to arbitrary quivers without loops.
• To identify the distinguished connected component of the stability manifold as a covering space over an open subset of the complexified Grothendieck group, determined by the root system of the associated Kac-Moody Lie algebra.
• To clarify the action of the Weyl group and the braid group on this covering space, particularly through the deck transformations.

Proposed method

• The paper uses Bridgeland’s framework for stability conditions on triangulated categories, focusing on the derived category of finite-dimensional nilpotent modules over the preprojective algebra.
• It identifies the Grothendieck group $ K( cal{D}_Q) $ with the root lattice $ L_Q $ of the Kac-Moody Lie algebra associated with the quiver $ Q $, via the Euler form.
• The space of stability conditions $ \mathrm{Stab}^\circ(\mathcal{D}_Q) $ is shown to map locally isomorphically to an open subset $ X_{\mathrm{reg}} \subset \mathrm{Hom}_{\mathbb{Z}}(K(\mathcal{D}_Q), \mathbb{C}) $, avoiding hyperplanes defined by roots.
• The Weyl group $ W $ acts on $ X_{\mathrm{reg}} $, and the covering structure is established via the action of $ \mathbb{Z}[2] \times \mathrm{Br}(\mathcal{D}_Q) $, with the braid group acting freely and properly discontinuously.
• The proof relies on the support property and the behavior of central charges on semistable objects, using ray convergence arguments and the structure of real and imaginary roots.
• The key technical tools include the normalized chamber $ C^N $, the central charge map $ \pi $, and the identification of $ \mathrm{Stab}(\mathcal{A}_Q)^N $ with $ C^N $, ensuring injectivity of the map to $ X_{\mathrm{reg}}/W $.

Experimental results

Research questions

• RQ1 How is the space of stability conditions on the derived category of nilpotent modules over a preprojective algebra structured for a quiver without loops?
• RQ2 What is the relationship between the stability manifold and the root system of the associated Kac-Moody Lie algebra?
• RQ3 How do the Weyl group and the braid group act on the space of stability conditions in this context?
• RQ4 Can the covering structure of the stability manifold be described explicitly in terms of the root system and the Grothendieck group?
• RQ5 What is the role of the imaginary cone and the set of real roots in determining the domain of the central charge map?

Key findings

• The distinguished connected component $ \mathrm{Stab}^\circ(\mathcal{D}_Q) $ of the space of stability conditions is a covering space over the open subset $ X_{\mathrm{reg}} \subset \mathrm{Hom}_{\mathbb{Z}}(K(\mathcal{D}_Q), \mathbb{C}) $, which is defined by removing hyperplanes orthogonal to non-zero roots.
• The covering map factors through the quotient $ X_{\mathrm{reg}}/W $, where $ W $ is the Weyl group of the Kac-Moody Lie algebra associated with the quiver $ Q $.
• The action of $ \mathbb{Z}[2] \times \mathrm{Br}(\mathcal{D}_Q) $ on $ \mathrm{Stab}^\circ(\mathcal{D}_Q) $ is free and properly discontinuous, and the quotient space is isomorphic to $ X_{\mathrm{reg}}/W $.
• The central charge map $ \pi: \mathrm{Stab}^\circ(\mathcal{D}_Q) \to X_{\mathrm{reg}} $ is a local isomorphism, and the global structure is determined by the deck transformations induced by the Weyl group and the braid group.
• The normalized chamber $ C^N \subset X_{\mathrm{reg}} $ is isomorphic to $ \mathrm{Stab}(\mathcal{A}_Q)^N $, the set of stability conditions with heart $ \mathcal{A}_Q $, and this allows the identification of the global covering structure.
• The proof relies on contradiction arguments involving the support property and the behavior of central charges on rays of roots, showing that no stability condition can have a non-zero central charge on a ray in the imaginary cone or positive real root cone.