On the Hall algebra of coherent sheaves on \(\mathbb P^1\) over \(\mathbb F^1\) (Q418921)

In the paper the category \(Coh_n(\mathbb{P}^1)\) of normal coherent sheaves on the monoid scheme \(\mathbb{P}^1\) is defined and studied. This category resembles in most ways a finitary abelian category (but not additive) and hence the Ringel-Hall algebra on it may be defined. The algebra is proved to be 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 may 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 \(\mathbb{U}_v(\hat{sl}_2),\) where \(v^2=q.\)