  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.
