On the integral form of rank 1 Kac-Moody algebras

Abstract: In this paper we shall prove that the subalgebra generated over the integers by the divided powers of the Drinfeld generators $x_r^{\pm}$ of the Kac-Moody algebra of type $A_2^{(2)}$ is an integral form (strictly smaller than Mitzman's (see [Mi]) of the enveloping algebra, we shall exhibit a basis generalizing the one provided by Garland in [G] for the untwisted affine Kac-Moody algebras, and we shall determine explicitly the commutation relations. Moreover we prove that both in the untwisted and in the twisted case the positive (respectively negative) imaginary part of the integral form is an algebra of polynomials over the integers.