$0$-affine quantum groups as K-theoretic Hall algebras

TL;DR

This paper establishes a connection between $0$-affine quantum groups and K-theoretic Hall algebras for type A quivers, constructing categorical actions that lead to semiorthogonal decompositions, with applications to partial flag varieties.

Contribution

It provides a new realization of the positive part of $0$-affine quantum groups as K-theoretic Hall algebras and constructs categorical actions inducing semiorthogonal decompositions.

Findings

Realization of $0$-affine quantum groups as K-theoretic Hall algebras

Construction of categorical actions on weight categories

Application to derived categories of partial flag varieties

Abstract

In this note, we show that the positive part of Arkhipov-Mazin's $0$-affine quantum group can be realized as the K-theoretic Hall algebra of the type $A$ Dynkin quiver. We then construct a categorical action of this positive part and demonstrate that such an action induces semiorthogonal decompositions on the corresponding weight categories. As a main example, we study the bounded derived category of coherent sheaves on $n$-step partial flag varieties.