Primitively generated Hall algebras

TL;DR

This paper demonstrates that Hall algebras of finitary exact categories are generated by indecomposable objects and primitive elements, revealing structural similarities to quantum groups and proposing a Lie correspondence conjecture.

Contribution

It shows Hall algebras are generated by primitive elements in a broad class of categories and introduces primitively generated subalgebras with a conjectured Lie correspondence.

Findings

Hall algebras are generated by indecomposable objects

Hall algebras are generated by primitive elements in many categories

Introduction of primitively generated subalgebras and a related conjecture

Abstract

In the present paper we show that Hall algebras of finitary exact categories behave like quantum groups in the sense that they are generated by indecomposable objects. Moreover, for a large class of such categories, Hall algebras are generated by their primitive elements, with respect to the natural comultiplication, even for non-hereditary categories. Finally, we introduce certain primitively generated subalgebras of Hall algebras and conjecture an analogue of "Lie correspondence" for those finitary categories.