Vol. 281, No. 2, 2016

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Primitively generated Hall algebras

Arkady Berenstein and Jacob Greenstein

Vol. 281 (2016), No. 2, 287–331

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.

Hall algebra, exact category, PBW property, Nichols algebra
Mathematical Subject Classification 2010
Primary: 16G20, 17B37, 16T15, 16T20
Received: 23 March 2015
Accepted: 14 September 2015
Published: 16 February 2016
Arkady Berenstein
Department of Mathematics
University of Oregon
Eugene, OR 97403-1222
United States
Jacob Greenstein
Department of Mathematics
University of California
Riverside, CA 92521-0135
United States