Semi-derived and derived Hall algebras for stable categories

TL;DR

This paper introduces semi-derived Hall algebras for Frobenius categories, exploring their properties, relations to derived Hall algebras, and applications in categorifying quantum groups, especially in cases where derived Hall algebras are not defined.

Contribution

It defines semi-derived Hall algebras, establishes their structure and functoriality, and relates them to twisted tensor products and derived Hall algebras, extending their applicability.

Findings

Semi-derived Hall algebra is a free module over a twisted group algebra.

Isomorphism with twisted tensor product of Grothendieck group and derived Hall algebra.

Application to 2-periodic derived categories and categorification of quantum groups.

Abstract

Given a Frobenius category $\mathcal{F}$ satisfying certain finiteness conditions, we consider the localization of its Hall algebra $\mathcal{H(F)}$ at the classes of all projective-injective objects. We call it the {\it "semi-derived Hall algebra"} $\mathcal{SDH(F, P(F))}.$ We discuss its functoriality properties and show that it is a free module over a twisted group algebra of the Grothendieck group $K_0(\mathcal{P(F)})$ of the full subcategory of projective-injective objects, with a basis parametrized by the isomorphism classes of objects in the stable category $\underline{\mathcal{F}}$. We prove that it is isomorphic to an appropriately twisted tensor product of $\mathbb{Q}K_0(\mathcal{P(F)})$ with the derived Hall algebra (in the sense of To\"{e}n and Xiao-Xu) of $\underline{\mathcal{F}},$ when both of them are well-defined. We discuss some situations where the semi-derived Hall algebra is defined while the derived Hall algebra is not. The main example is the case of $2-$periodic derived category of an abelian category with enough projectives, where the semi-derived Hall algebra was first considered by Bridgeland who used it to categorify quantum groups.