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

By Markus Reineke. 2002

260 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/223d5af5f8811e2a87d060c11daff1917207e792

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Citations in the corpus

Donaldson-Thomas theory and cluster algebras (2010, methodology): “ the following powerful method in Donaldson-Thomas theory for 3-Calabi-Yau categories, which originates with Reineke’s computation of the Betti numbers of the spaces of stable quiver representations ([Rei03]): Starting from a simple categorical statement, provide an identity in the motivic Hall alge”
Wall crossing from Boltzmann black hole halos (2010, methodology): “In the next subsection, we relinquish the gauge theoretical crutch provided by Reineke’s formula and directly quantize the internal degrees of freedom of the bound state.”
On the Hall algebra of an elliptic curve, I (2005, unknown): “ by the formula χ [F] α = q−αβ X Yβ aF , where (α,β) = rank(F),deg(F) ∈ Z2. Proof. Let Γ ⊂ K′ 0 Coh(X) be the semi-group generated by the images of classes of coherent sheaves on X. Following Reineke [Re], consider the associative algebra K Γ,h ,i = nX γ∈Γ aγt γ|a γ ∈ K o where the multiplication is”
Noncommutative projective curves and quantum loop algebras (2002, methodology): “The use of the Harder-Narasimhan filtration in the context of Hall algebras is also noted in [ R ].”
The integrality conjecture and the cohomology of preprojective stacks (2016, methodology): “hen, from Lemma 4.3 and (29) that BPSω -nilp Q,˜ W˜ is pure, of Tate type. A tensor product of pure mixed Hodge modules is pure, and so BPSω-nilp Q,˜ W˜ ⊗L is also pure, of Tate type. It follows from (27) and (28) that H Dim! φ Tr(W)IC M(Q˜)(Q) and L d∈NQ0 Hc(µ −1 Q,d(0)/GL d,Q) are pure, of Tate ”
Smooth models of quiver moduli (2007, methodology): “We also recall some results from [12] which will be used in the following (subsection 2.”
Quantum quivers and Hall / hole halos (2002, background): “[N, t], we have [17] P (t) = (t(2) − 1) ∑”
Moduli of representations of quivers (2008, background): “M st d (Q)) non-empty? For the semistable representations, the Harder-Narasimhan filtration yields a recursive criterion, see [52].”
Four dimensional black hole microstates: from D-branes to spacetime foam (2006, background): “Unlike the case without closed loops [49], a general formula for the number of states in the Higgs branch is not available and hence the relation to the Coulomb branch in this case remains to be studied.”
Split states, entropy enigmas, holes and halos (2007, background): “The latter often are under good mathematical control [93], and their relation to multicentered configurations is well understood in a number of cases [65].”
LECTURES ON HALL ALGEBRAS (2006, background): “···1βs where the sum now ranges over all tuples (β1,...,βs) satisfying µ( Ps l=kβl) >µ(α) for all k= 1,...,s. This formula is known as Reineke’s inversion formula and a proof of it may be found in [R3]. 78 OLIVIER SCHIFFMANN Remarks 4.36. The notions of semistability, Harder-Narasimhan ﬁltration,”
Localization in quiver moduli spaces (2009, methodology): “One easily verifies that this is the same result one obtains by the algorithm from [23], i.”
Affine Mirković-Vilonen polytopes (2011, methodology): “1) whose subquotients are nonzero and semistable, with moreover μ(Tk/Tk−1) decreasing with k (see for instance [53], close to the present context).”
Stability conditions and Stokes factors (2008, background): “From an algebraic perspective, these wall-crossing phenomena give rise to, and may be studied as change of bases within the Ringel–Hall algebra H(A) of A, an idea which has its origins in the work of Reineke [21], and was greatly developed by Joyce [12, 14].”
Counting rational points of quiver moduli (2005, background): “This is contained in [8], namely Corollary 6.”