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Critical CoHAs, vertex coalgebras and Deformed Drinfeld coproducts

Shivang Jindal, Sarunas Kaubrys, Alexei Latyntsev

Abstract

We construct a vertex coproduct on the Kontsevich--Soibelman cohomological Hall algebra (CoHA) of a quiver with potential, following Joyce (2018). We show it forms a vertex bialgebra. By applying a vertex algebraic analogue of Majid--Radford bosonisation, we form an extension of the CoHA of quivers with potential which incorporates a Cartan part. In the case of ADE quivers our vertex coproduct recovers Drinfeld's deformed coproduct on the Yangian. We compare the vertex coproduct with a localised coproduct defined by Davison and with the construction of Dotsenko--Mozgovoy when the potential is trivial. Our construction gives a new proof of the cohomological integrality theorem for symmetric quivers with trivial potential.

Critical CoHAs, vertex coalgebras and Deformed Drinfeld coproducts

Abstract

We construct a vertex coproduct on the Kontsevich--Soibelman cohomological Hall algebra (CoHA) of a quiver with potential, following Joyce (2018). We show it forms a vertex bialgebra. By applying a vertex algebraic analogue of Majid--Radford bosonisation, we form an extension of the CoHA of quivers with potential which incorporates a Cartan part. In the case of ADE quivers our vertex coproduct recovers Drinfeld's deformed coproduct on the Yangian. We compare the vertex coproduct with a localised coproduct defined by Davison and with the construction of Dotsenko--Mozgovoy when the potential is trivial. Our construction gives a new proof of the cohomological integrality theorem for symmetric quivers with trivial potential.
Paper Structure (98 sections, 53 theorems, 290 equations)

This paper contains 98 sections, 53 theorems, 290 equations.

Table of Contents

  1. Introduction
  2. Cohomological Hall product and Joyce--Liu vertex coproduct
  3. Localised versus vertex coproducts
  4. Compatibility and bosonisation
  5. Bosonisation
  6. Obtaining Drinfeld's coproduct on ADE Yangians
  7. Outlook: Maulik-Okounkov Yangians
  8. Why should such structure exist?
  9. Quantum groups
  10. Hall algebras
  11. Physics
  12. Relation to the work of Dotsenko--Mozgovoy
  13. Acknowledgements
  14. Preliminaries
  15. Quivers with potential
  16. ...and 83 more sections

Key Result

Theorem 1

Let $Q$ be any quiver with potential $W$ and a torus action that leaves the potential invariant, and assume it satisfies the $T$-equivariant Künneth property eqn:KunnethAssumption.This is known to hold if $T =1$ or in canonical tripled cases, see section sssec:kunneth_assumpt for more discussion.. T defines a coassociative (alias noncolocal or weakly coassociative) vertex coalgebra. Furthermore, $

Theorems & Definitions (93)

  • Theorem 1: Theorem \ref{['quiver_potential_jl']}
  • Theorem 2: Theorem \ref{['thm:DavisonIsJoyce']}
  • Theorem 3: Theorem \ref{['thm:CoHABialgebra']} symmetric case, Theorem \ref{['thm:CoHABialgebra_non_symm']} general case
  • Theorem 4: Theorem \ref{['thm:VertexBosonisation']}
  • Corollary 5: Theorem \ref{['thm:CoHABosonisation']}
  • Theorem 6: Theorem \ref{['thm:DrinfeldJoyce']}
  • Theorem 7: Theorem \ref{['chiral_envelope']}
  • Theorem 8: Theorem \ref{['DM_comp']}
  • Proposition 1.4.3
  • proof
  • ...and 83 more