Chevalley involutions for Lie tori and extended affine Lie algebras

TL;DR

This paper explores the existence and construction of Chevalley involutions in Lie tori and extended affine Lie algebras, extending the modular theory known for finite-dimensional and affine cases.

Contribution

It establishes the existence of Chevalley involutions for all centerless Lie tori of reduced type, advancing the understanding of symmetries in extended affine Lie algebras.

Findings

Chevalley involutions can be lifted from the centerless core to the entire algebra.

Every centerless Lie torus of reduced type admits a Chevalley involution.

The results support the development of modular theory for extended affine Lie algebras.

Abstract

In finite-dimensional simple Lie algebras and affine Kac-Moody Lie algebras, Chevalley involutions are crucial ingredients of the modular theory. Towards establishing the modular theory for extended affine Lie algebras, we investigate the existence of ``Chevalley involutions" for Lie tori and extended affine Lie algebras. We first discuss how to lift a Chevalley involution from the centerless core which is characterized to be a centerless Lie torus to the core and then to the entire extended affine Lie algebra. We then prove that each centerless Lie torus of reduced type admits a Chevalley involution.