Primitively generated Hall algebras

Arkady Berenstein and Jacob Greenstein

Vol. 281 (2016), No. 2, 287–331

DOI: 10.2140/pjm.2016.281.287

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 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

Mathematical Subject Classification 2010

Primary: 16G20, 17B37, 16T15, 16T20

Milestones

Received: 23 March 2015

Accepted: 14 September 2015

Published: 16 February 2016