On the Hall algebra of coherent sheaves on P1 over F1

We define and study the category Cohn(P1) of normal coherent sheaves on the monoid scheme P1 (equivalently, the M0-scheme P1/F1 in the sense of Connes–Consani–Marcolli, Connes (2009) [2]). This category resembles in most ways a finitary abelian category, but is not additive. As an application, we define and study the Hall algebra of Cohn(P1). We show that it is isomorphic as a Hopf algebra to the enveloping algebra of the product of a non-standard Borel in the loop algebra Lgl2 and an abelian Lie algebra on infinitely many generators. This should be viewed as a (q=1) version of Kapranov’s result relating (a certain subalgebra of) the Ringel–Hall algebra of P1 over Fq to a non-standard quantum Borel inside the quantum loop algebra , where ν2=q.