Green's theorem on Hall algebras
Claus Michael Ringel
Abstract. Let k be a finite field and \Lambda a hereditary finitary k-algebra.
Let P be the set of isomorphism classes of finite \Lambda-modules. We define
a multiplication on the Q-space with basis P by counting the number of
submodules U of a given module V with prescribed isomorphism classes both of
V/U and U. In this way we obtain the so called Hall algebra H = H(\Lambda,Q)
with coefficients in Q. Besides H, we are also interested in the subalgebra C
generated by the subset I of all isomorphism classes of simple \Lambda-modules;
this subalgebra is called the corresponding composition algebra, since it
encodes the number of composition series of all \Lambda-modules.
...