On the Hall algebra of coherent sheaves on P^1 over F_1

Abstract

We define and study the category $Coh_n(\Pone)$ of normal coherent sheaves on the monoid scheme $\Pone$ (equivalently, the $\mathfrak{M}_0$-scheme $\Pone / \fun$ in the sense of Connes-Consani-Marcolli \cite{CCM}). 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 $Coh_n(\Pone)$. 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 $L {\mathfrak{gl}}_2$ 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 $\mathbb{P}^1$ over $\mathbb{F}_q$ to a non-standard quantum Borel inside the quantum loop algebra $\mathbb{U}_{\nu} (\slthat)$, where $\nu^2=q$.