  To an abelian category A of homological dimension 1 satisfying certain
finiteness conditions, one can associate an algebra, called the Hall algebra.
Kapranov studied this algebra when A is the category of coherent sheaves over a
smooth projective curve defined over a finite field, and observed analogies
with quantum affine algebras. We recover here in an elementary way his results
in the case when the curve is the projective line.
