Domestic canonical algebras and simple Lie algebras

For each simply-laced Dynkin graph $\mathrm{\Delta }$ we realize the simple complex Lie algebra of type $\mathrm{\Delta }$ as a quotient algebra of the complex degenerate composition Lie algebra $L(A{)}_1^{\mathbb{C}}$ of a domestic canonical algebra $A$ of type $\mathrm{\Delta }$ by some ideal $I$ of $L(A{)}_1^{\mathbb{C}}$ that is defined via the Hall algebra of $A$, and give an explicit form of $I$. Moreover, we show that each root space of $L(A{)}_1^{\mathbb{C}}\mathrm{/}I$ has a basis given by the coset of an indecomposable $A$-module $M$ with root easily computed by the dimension vector of $M$.