Divided Power Integral forms of Affine Algebras

TL;DR

This paper establishes an integral form for affine Kac-Moody algebras generated by divided powers, compares it with existing forms, and explores the structure of subalgebras generated by imaginary vectors, revealing new algebraic properties.

Contribution

It proves the integral form generated by divided powers is well-defined and compares it with Chevalley-based forms, also analyzing the structure of subalgebras in higher ranks.

Findings

Integral form generated by divided powers is established.

Comparison shows forms coincide outside twisted A type.

Subalgebra generated by imaginary vectors is not polynomial in higher rank.

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.