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Derived Hall algebras of root categories

Jiayi Chen, Ming Lu, Shiquan Ruan

Abstract

For a finitary hereditary abelian category $\mathcal{A}$, we define a derived Hall algebra of its root category by counting the triangles and using the octahedral axiom, which is proved to be isomorphic to the Drinfeld double of Hall algebra of $\mathcal{A}$. When applied to finite-dimensional nilpotent representations of the Jordan quiver or coherent sheaves over elliptic curves, these algebras provide categorical realizations of the ring of Laurent symmetric functions and also double affine Hecke algebras.

Derived Hall algebras of root categories

Abstract

For a finitary hereditary abelian category , we define a derived Hall algebra of its root category by counting the triangles and using the octahedral axiom, which is proved to be isomorphic to the Drinfeld double of Hall algebra of . When applied to finite-dimensional nilpotent representations of the Jordan quiver or coherent sheaves over elliptic curves, these algebras provide categorical realizations of the ring of Laurent symmetric functions and also double affine Hecke algebras.
Paper Structure (20 sections, 22 theorems, 137 equations)

This paper contains 20 sections, 22 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Main results
  4. Comparison with previous works
  5. Future works
  6. Organization
  7. Acknowledgments
  8. Preliminaries
  9. Root categories
  10. A useful proposition
  11. Derived Hall algebras of root categories
  12. Derived Hall numbers
  13. Derived Hall algebras
  14. Twisted derived Hall algebras
  15. Drinfeld double of Hall algebras
  16. ...and 5 more sections

Key Result

Proposition 2.1

Given an object $X^\bullet$ and a triangle in $\mathcal{R}(\mathcal{A})$, we have where $\delta_0=\widehat{\operatorname{Ker}\nolimits H^0(l)}$.

Theorems & Definitions (41)

  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2: XX08
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • ...and 31 more