Categorical and K-Theoretic Hall Algebras for Quivers With Potential

Abstract

Given a quiver with potential $(Q,W)$ , Kontsevich–Soibelman constructed a cohomological Hall algebra (CoHA) 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 $(\widetilde{Q}, \widetilde{W})$ whose CoHA is conjecturally the positive half of the Maulik–Okounkov Yangian $Y_{\text {MO}}(\mathfrak {g}_{Q})$ . 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 $(\widetilde{Q}, \widetilde{W})$ to recover the positive part of quantum affine algebra $U_{q}(\widehat {\mathfrak {g}_{Q}})$ defined by Okounkov–Smirnov.

The trace of the affine Hecke category

“…This subsection is motivational, and will not be directly used in the present paper (in more detail, the contents herein are derived category generalizations of the principles laid out in [44]; the generalization itself is straightforward using tools of derived algebraic geometry, but it is beyond the scope on the present paper). See also [37,38] for related constructions and computations. As we will recall in Section 2, the commuting stack Comm 𝑛 of ( 26) parameterizes pairs of commuting 𝑛 × 𝑛 matrices modulo the general linear group 𝐺𝐿 𝑛 .…”

“…Equality ( 39) is a straightforward computation involving pushforward and pullback morphisms; we will explain the main idea of the proof, and leave the details as an exercise to the interested reader (see [32,Proposition 2.7] for a closely related computation). Consider the following diagram (40) where 𝔸 𝑛 2 −𝑛+1 and 𝔸 2𝑛 2 are the affine spaces of coordinates of the matrices 𝑋 and 𝑌 in (37) and (27), respectively, and the maps 𝑗 ′ and 𝑗 denote the respective closed embeddings of 𝑈 and 𝑉 into these affine spaces. Using (30), we have…”

“…Note that this quotient of R precisely matches the support condition for the sheaves (15) on the commuting stack. Indeed, the sheaf  𝐝 arises from (37), with all the eigenvalues of the 𝑋 matrix being equal. Similarly,  𝐝 1 ⋆ ⋯ ⋆  𝐝 𝑟 arises from the analogue of (37) where the eigenvalues of the 𝑋 matrix are equal in blocks of sizes 𝑛 1 , 𝑛 2 , … , 𝑛 𝑟 .…”

Quantum toroidal and shuffle algebras

“…The isomorphism A ∼ = U q ( g Q ) is one of those mathematical statements that most specialists in the field believe, but no proof has yet been written down. For example, conjectural versions of this statement have appeared in [12,Conjecture 1.2] and in [16]. There has been more work on the subject in cohomology, see [2] and the sequence of papers [14,15].…”

On cohomological and K-theoretical Hall algebras of symmetric quivers

“…Similarly, the Z I ≥0 -graded algebra R = ⊕ γ K 0 Gγ (M γ ) with multiplication defined in an analogous way is called the K-theoretical Hall algebra (KHA), it was introduced by Pȃdurariu [7,8].…”