Combinatorial analogues of ad-nilpotent ideals for untwisted affine Lie algebras

TL;DR

This paper classifies certain ideals in the Borel subalgebra of untwisted affine Lie algebras using root combinatorics, providing explicit formulas and revealing new combinatorial sequences.

Contribution

It introduces a combinatorial classification of ideals in affine Lie algebras and derives explicit counting formulas, especially for type A.

Findings

Explicit formula for the number of ideals in type A

Connection to Dyck path combinatorics

Introduction of a new integral sequence

Abstract

We study certain types of ideals in the standard Borel subalgebra of an untwisted affine Lie algebra. We classify these ideals in terms of the root combinatorics and give an explicit formula for the number of such ideals in type $A$. The formula involves various aspects of combinatorics of Dyck paths and leads to a new interesting integral sequence.