On the Hall algebra of coherent sheaves on P1 over F1

We define and study the category Coh(n)(P-1) of normal coherent sheaves on the monoid scheme P-1 (equivalently, the m(0)-scheme P-1/F-1 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 Coh(n)(P-1). 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 Lgl(2) 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 P-1 over F-q to a non-standard quantum Borel inside the quantum loop algebra U-t,((sI(2)) over cap), where v(2) = q. (C) 2011 Elsevier B.V. All rights reserved.