  Let $Q$ be a finite type quiver i.e. ADE Dynkin quiver. Denote by $\Lambda$
its preprojective algebra. It is known that there are finitely many
indecomposable $\Lambda$-modules if and only if $Q$ is of type
$A_1,A_2,A_3,A_4$. In this paper, extending Lusztig's construction of
$U\frak{n}_+$, we study an algebra generated by these indecomposable
submodules. It turns out that it forms the universal enveloping algebra of some
nilpotent Lie algebra inside the function algebra on Lusztig's nilpotent
scheme. The defining relations of the corresponding nilpotent Lie algebra for
type $A_1, A_2,A_3,A_4$ are given here.
