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Categorification of quiver diagonalization and Koszul algebras

Vladimir Dotsenko, Evgeny Feigin, Piotr Kucharski, Markus Reineke

Abstract

In earlier work of three of the authors of the present paper, a supercommutative quadratic algebra was associated to each symmetric quiver, and a new proof of positivity of motivic Donaldson-Thomas invariants of symmetric quivers was given using the so called numerical Koszul property of these algebras. It was furthermore conjectured that for each symmetric quiver such an algebra is Koszul. In this work, we lift the linking and unlinking operations on symmetric quivers of Ekholm, Longhi and the third author to the level of quadratic algebras, and use those lifts to prove the Koszulness conjecture.

Categorification of quiver diagonalization and Koszul algebras

Abstract

In earlier work of three of the authors of the present paper, a supercommutative quadratic algebra was associated to each symmetric quiver, and a new proof of positivity of motivic Donaldson-Thomas invariants of symmetric quivers was given using the so called numerical Koszul property of these algebras. It was furthermore conjectured that for each symmetric quiver such an algebra is Koszul. In this work, we lift the linking and unlinking operations on symmetric quivers of Ekholm, Longhi and the third author to the level of quadratic algebras, and use those lifts to prove the Koszulness conjecture.
Paper Structure (13 sections, 9 theorems, 57 equations)

This paper contains 13 sections, 9 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Recollections
  3. Graded vector spaces and algebras
  4. Poincaré series
  5. Quadratic algebras associated to symmetric quivers
  6. Koszul algebras
  7. Linking and filtrations
  8. Linking
  9. The filtration theorem
  10. Unlinking and partial resolutions
  11. Unlinking
  12. The partial resolution theorem
  13. The Koszulness conjecture

Key Result

Proposition 2.1

In the ring $\mathbb{Q}(q^{\frac{1}{2}})[[x_i\colon i\in Q_0]]$, we have

Theorems & Definitions (15)

  • Proposition 2.1: MR4499100
  • Theorem 3.1: MR4089349
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 4.1: MR4089349
  • Proposition 4.2
  • proof
  • Theorem 4.3
  • ...and 5 more