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
nonhereditary categories. Finally, we introduce certain primitively generated
subalgebras of Hall algebras and conjecture an analogue of “Lie correspondence” for
those finitary categories.
Keywords
Hall algebra, exact category, PBW property, Nichols algebra