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BPS Lie algebras, perverse filtrations and shuffle algebras

Shivang Jindal, Andrei Neguţ

Abstract

We give an explicit description of the BPS Lie algebra of any quiver with zero potential, by relating the perverse filtration on the cohomological Hall algebra with certain limit conditions on polynomials. Our results also give a partial description of the perverse filtration for arbitrary potential, which we conjecture is complete in the case of tripled quivers with canonical cubic potential.

BPS Lie algebras, perverse filtrations and shuffle algebras

Abstract

We give an explicit description of the BPS Lie algebra of any quiver with zero potential, by relating the perverse filtration on the cohomological Hall algebra with certain limit conditions on polynomials. Our results also give a partial description of the perverse filtration for arbitrary potential, which we conjecture is complete in the case of tripled quivers with canonical cubic potential.

Paper Structure

This paper contains 32 sections, 20 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Cohomological Hall algebras
  3. Perverse filtration
  4. Tripled quivers
  5. Acknowledgements
  6. Shuffle algebras
  7. Notations
  8. Shuffle algebras
  9. Limits
  10. The key property
  11. Perverse Filtrations
  12. Stacks of quiver representations
  13. Potential
  14. Vanishing cycles and constructible sheaves
  15. Cohomological Hall algebras
  16. ...and 17 more sections

Key Result

Theorem 1.1

The filtration defined above matches with the perverse filtration, i.e. for all $d \geq 1$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2: Proposition \ref{['prop:perverse']}
  • Corollary 1.3
  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 34 more