Tame quivers and affine enveloping algebras

Let $\mathfrak{g}$ be an affine Kac-Moody algebra with symmetric Cartan datum, ${\mathfrak{n}}^{+}$ be the maximal nilpotent subalgebra of $\mathfrak{g}$. By the Hall algebra approach, we construct integral bases of the $\mathbb{Z}$-form of the enveloping algebra $U({\mathfrak{n}}^{+})$. In particular, the representation theory of tame quivers is essentially used in this paper.