Divided Power Integral forms of Affine Algebras

Abstract: In this paper, we shall prove that the integral subalgebra generated by the divided powers of the Drinfeld generators of an affine Kac-Moody algebra is an integral form. We compare this integral form with the analogous one derived from the Chevalley generators studied by Mitzman and Garland. We shall prove that the integral forms coincide outside the twisted A type, and that it is strictly smaller in the latter case. Moreover, if the rank of the algebra is greater than one, a completely new fact emerges: the subalgebra generated by the imaginary vectors is, in fact, not a polynomial algebra, and we describe its structure. To address this problem, we introduce two other integral forms in the low-rank case in order to obtain the desired polynomial property.