Characters for extended affine Lie algebras; a combinatorial approach

TL;DR

This paper investigates the characters and automorphisms of extended affine Lie algebras, revealing that finite order Cartan automorphisms are diagonal and their associated combinatorial maps are characters, highlighting structural differences from affine Lie algebras.

Contribution

It provides a combinatorial approach to understanding characters and automorphisms in extended affine Lie algebras, establishing that finite order Cartan automorphisms are diagonal and correspond to characters.

Findings

Finite order Cartan automorphisms are diagonal in extended affine Lie algebras.

Associated combinatorial maps of these automorphisms are characters.

Distinct behavior compared to affine Lie algebras.

Abstract

The behavior of objects associated with general extended affine Lie algebras is typically distinct from their counterparts in affine Lie algebras. Our research focuses on studying characters and Cartan automorphisms, which appear in the study of Chevalley involutions and Chevalley bases for extended affine Lie algebras. We show that for almost all extended affine Lie algebras, any finite order Cartan automorphism is diagonal, and its corresponding combinatorial map is a character.