Domestic canonical algebras and simple Lie algebras

For each simply-laced Dynkin graph Δ we realize the simple complex Lie algebra of type Δ as a quotient algebra of the complex degenerate composition Lie algebra of a domestic canonical algebra A of type Δ by some ideal I of that is defined via the Hall algebra of A, and give an explicit form of I. Moreover, we show that each root space of has a basis given by the coset of an indecomposable A-module M with root easily computed by the dimension vector of M. Dedicated to Professor Claus Michael Ringel on the occasion of his 60th birthday.