Categorical and K-theoretic Hall algebras for quivers with potential

Abstract

Given a quiver with potential (Q, W ), Kontsevich-Soibelman constructed a Hall algebra on the critical cohomology of the stack of representations of (Q, W ). Special cases of this construction are related to work of Nakajima, Varagnolo, Schiffmann-Vasserot, Maulik-Okounkov, Yang-Zhao etc. about geometric constructions of Yangians and their representations; indeed, given a quiver Q, there exists an associated pair Q, W whose CoHA is conjecturally the positive half of the Maulik-Okounkov Yangian YMO(gQ).For a quiver with potential (Q, W ), we follow a suggestion of Kontsevich-Soibelman and study a categorification of the above algebra constructed using categories of singularities. Its Grothendieck group is a K-theoretic Hall algebra (KHA) for quivers with potential. We construct representations using framed quivers and we prove a wall-crossing theorem for KHAs. We expect the KHA for Q, W to recover the positive part of quantum affine algebra Uq ( gQ) defined by Okounkov-Smirnov.

K-theoretic Hall algebras, quantum groups and super quantum groups

“…From Proposition 2.4 and Lemma 4.3, we deduce that O Zps ij q A ì puqA j pvq " O Zps ji q A j pvqA ì puq and, then, that A ì puqA j pvq " A j pvqA ì puq h ij pu{vq (4. 16) In a similar way we can prove that A í puqA j pvq " A j pvqA í puq h ij pu{vq ´1. (4.17)…”

Critical convolution algebras and quantum loop groups

“…1.3]. Part (d) follows from the Koszul equivalence (=dimensional reduction) in [23], see also [39], [54] for a version closer to our setting. Parts (e), (f) follow from (a)-(d).…”

“…K-theoretical Hall algebras of triple quivers with potentials. We first recall the definition of KHA's following [39], [54]. Let H be the moduli stack of representations of r Q.…”

“…We'll also need K-theoretical Hall algebras of a quiver with potential. They were introduced by Padurariu in [39]. It was proved there that Isik's Koszul duality (=dimensional reduction) in [23] implies that the K-theoretical Hall algebras of triple quivers with the potential w " ε rα, α ˚s, see (3.43) below for the notation, coincide with the K-theretical Hall algebras of preprojective algebras.…”

“…Then, we consider the particular cases of triple quivers. For triple quivers the critical convolution algebra depends on the choice of a potential w. This potential gives rise to a nilpotent K-theoretical Hall algebra U pwq `in the sense of [39], [54]. We expect that the K-theoretical critical convolution algebra is equipped with an algebra homomorphism from a double U pwq of U pwq `, i.e., the algebra U pwq should admit a triangular decomposition U pwq " U pwq `b U pwq 0 b U pwq ẃhere U pwq ´is the opposite of U pwq `and U pwq 0 is commutative.…”