Hall Algebra Realizations of Quantum Groups

In particular, Q has finitely many indecomposables if and only if the underlying graph of Q is a disjoint union of copies of type ADE graphs. In this case, the above theorem is true for any field. Hence we can identify the isomorphism class of a representation over a Dynkin quiver with a function from the positive roots of ∆ to the nonnegative integers. The path algebra of Q is hereditary, so Ext(M,N) = 0 for all modules M,N and i > 1. Given two representations M,N , define the Euler form