Lattice of dominant weights of affine Kac-Moody algebras

TL;DR

This paper investigates the structure of the poset of dominant weights in affine Kac-Moody algebras, providing explicit descriptions of relations and analyzing basic cell structures for type A algebras.

Contribution

It offers a detailed description of the covering relations in the dominant weight poset and examines the structure of basic cells for untwisted affine Kac-Moody algebras of type A.

Findings

Explicit description of covering relations in the dominant weight poset

Structural analysis of basic cells in type A affine Kac-Moody algebras

Enhanced understanding of the partial ordering in affine Kac-Moody algebra weights

Abstract

The dual space of the Cartan subalgebra in a Kac-Moody algebra has a partial ordering defined by the rule that two elements are related if and only if their difference is a non-negative or non-positive integer linear combination of simple roots. In this paper, we study the subposet formed by dominant weights in affine Kac-Moody algebras. We give a more explicit description of the covering relations in this poset. We also study the structure of basic cells in this poset of dominant weights for untwisted affine Kac-Moody algebras of type A.