The cohomological Hall algebra of a preprojective algebra

Abstract

We introduce for each quiver $Q$ and each algebraic oriented cohomology theory $A$, the cohomological Hall algebra (CoHA) of $Q$, as the $A$-homology of the moduli of representations of the preprojective algebra of $Q$. This generalizes the $K$-theoretic Hall algebra of commuting varieties defined by Schiffmann-Vasserot. When $A$ is the Morava $K$-theory, we show evidence that this algebra is a candidate for Lusztig's reformulated conjecture on modular representations of algebraic groups. We construct an action of the preprojective CoHA on the $A$-homology of Nakajima quiver varieties. We compare this with the action of the Borel subalgebra of Yangian when $A$ is the intersection theory. We also give a shuffle algebra description of this CoHA in terms of the underlying formal group law of $A$. As applications, we obtain a shuffle description of the Yangian.

V6: 44 pages; section 6 added; errors in section 8 corrected; minor revisions throughout; final version

References in corpus (13)

• Calabi-Yau algebras
• Lectures on Nakajima's Quiver Varieties
• Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
• On the center of quiver-Hecke algebras
• Surface defects and elliptic quantum groups
• On two cohomological Hall algebras
• Current algebras and categorified quantum groups
• Algebraic Cobordism of Classifying Spaces
• Quiver varieties and elliptic quantum groups
• Quantum Algebras and Cyclic Quiver Varieties
• The critical CoHA of a quiver with potential
• Elliptic quantum groups and their finite-dimensional representations
• Representations of the elliptic affine Hecke algebras

Cited by in corpus (24)

• Cohomological Hall algebras, vertex algebras and instantons
• Toroidal and Elliptic Quiver BPS Algebras and Beyond
• On two cohomological Hall algebras
• Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces
• The $R$-matrix presentation for the Yangian of a simple Lie algebra
• On cohomological Hall algebras of quivers : Yangians
• Quantum toroidal algebras and solvable structures in gauge/string theory
• K-theoretic Hall algebras for quivers with potential
• Quiver varieties and elliptic quantum groups
• BPS Lie algebras and the less perverse filtration on the preprojective CoHA
• BPS States Meet Generalized Cohomology
• The formal shift operator on the Yangian double
• Quiver Hall-Littlewood functions and Kostka-Shoji polynomials
• Cohomological Hall algebra of Higgs sheaves on a curve
• Quiver Schur algebras and cohomological Hall algebras
• Cohomological Hall algebras and vertex algebras
• Framed cohomological Hall algebras and cohomological stable envelopes
• Hilbert schemes of nonreduced divisors in Calabi-Yau threefolds and W-algebras
• The PBW theorem for the affine Yangians
• Frobenii on Morava $E$-theoretical quantum groups
• Shuffle algebras for quivers as quantum groups
• Cohomological Hall algebras of quivers and Yangians
• Coherent sheaves on surfaces, COHAs and deformed $W_{1+\infty }$-algebras
• Shuffle algebras for quivers and R-matrices