On the Cohomological Hall Algebra of Dynkin quivers

Abstract

Consider the Cohomological Hall Algebra as defined by Kontsevich and Soibelman, associated with a Dynkin quiver. We reinterpret the geometry behind the multiplication map in the COHA, and give an iterated residue formula for it. We show natural subalgebras whose product is the whole COHA (except in the $E_8$ case). The dimension count version of this statement is an identity for quantum dilogarithm series, first proved by Reineke. We also show that natural structure constants of the COHA are universal polynomials representing degeneracy loci, a.k.a. quiver polynomials.

References in corpus (6)

• Quantum Dilogarithm
• Abelian Current Algebra and the Virasoro Algebra on the Lattice
• Chern class formulas for quiver varieties
• Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
• Kempf collapsing and quiver loci
• Integration over homogenous spaces for classical Lie groups using iterated residues at infinity

Cited by in corpus (10)

• Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras
• K-theoretic Hall algebras for quivers with potential
• An iterated residue perspective on stable Grothendieck polynomials
• Three combinatorial formulas for type A quiver polynomials and K-polynomials
• Quiver Schur algebras and cohomological Hall algebras
• Partition Identities and Quiver Representations
• Interpolating factorizations for acyclic Donaldson--Thomas invariants
• Quantum dilogarithm identities for n-cycle quivers
• A family of structure isomorphisms for the Cohomological Hall Algebra of an acyclic quiver
• K-polynomials of type A quiver orbit closures and lacing diagrams