The Hall algebra of the category of coherent sheaves on the projective line

To an abelian category A of homological dimension one satisfying certain fi niteness conditions, one can associate an algebra, called the Hall algebra. Kapranov studied this algebra when A is the category of coherent sheaves o ver a smooth projective curve defined over a finite field, and observed ana logies with quantum affine algebras. We recover here in an elementary way h is results in the case when the curve is the projective line.