Abstract

For any free oriented Borel–Moore homology theory A, we construct an associative product on the A-theory of the stack of Higgs torsion sheaves over a projective curve C. We show that the resulting algebra Amathbf{Ha}_C^0 admits a natural shuffle presentation, and prove it is faithful when A is replaced with usual Borel–Moore homology groups. We also introduce moduli spaces of stable triples, heavily inspired by Nakajima quiver varieties, whose A-theory admits an Amathbf{Ha}_C^0-action. These triples can be interpreted as certain sheaves on mathbb {P}_C(omega _Coplus mathcal {O}_C). In particular, we obtain an action of Amathbf{Ha}_C^0 on the cohomology of Hilbert schemes of points on T^*C.

Highlights

  • Let C be a hereditary abelian category over finite field Fq, such that all Hom- and Extspaces have finite dimension

  • In the case C = Rep Q, where Q is a quiver of Dynkin type, a famous theorem by Ringel [44] describes the Hall algebra H(Rep Q) as the positive half of the quantum group Uν(gQ), specialized at ν = q1/2

  • Page 3 of 67 30 and its restriction to v H (L(v, w)) is the irreducible highest module of weight w. This action can be extended to a much bigger algebra, so-called Yangian. This can be achieved by realizing it inside the cohomological Hall algebra [48,52], isomorphic to v H (T ∗ Repv Q) as a vector space

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Summary

Introduction

Let C be a hereditary abelian category over finite field Fq , such that all Hom- and Extspaces have finite dimension. This action can be extended to a much bigger algebra, so-called Yangian This can be achieved by realizing it inside the cohomological Hall algebra [48,52], isomorphic to v H (T ∗ Repv Q) as a vector space (see [31] for another perspective on Yangians). (1) construct a (bi-)algebra structure AHaC on A(Higgs C); (2) define a suitable stability condition on T ∗Coh←FC, where Coh←FC is the stack of pairs (E, α) with E ∈ Coh C, α ∈ Hom(F, E); (3) construct an action of the Drinfeld double D(AHaC ) on the A-theory A(M) of the moduli of stable objects.

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Coherent sheaves and Quot-schemes
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Global shuffle algebra
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Injectivity of shuffle presentation
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Moduli of stable Higgs triples
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Quiver sheaves
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