Realization of level one representations of $U_q(\hat{\mathfrak g})$ at a root of unity

TL;DR

This paper explicitly constructs level one representations of quantum affine algebras at roots of unity using vertex operators, demonstrating that their q-dimension matches the Weyl-Kac formula and extending realizations to affine Kac-Moody algebras in finite characteristic.

Contribution

It provides an explicit construction of level one representations at roots of unity and extends the realization to affine Kac-Moody algebras in finite characteristic.

Findings

q-dimension matches Weyl-Kac character formula

Explicit Lusztig lattice construction using vertex operators

Realization of affine Kac-Moody algebras in finite characteristic

Abstract

Using vertex operators, we construct explicitly Lusztig's $\mathbb Z[q, q^{-1}]$-lattice for the level one irreducible representations of quantum affine algebras of ADE type. We then realize the level one irreducible modules at roots of unity and show that the q-dimension is still given by the Weyl-Kac character formula. As a consequence we also answer the corresponding question of realizing the affine Kac-Moody Lie algebras of simply laced type at level one in finite characteristic.