On conjugacy of Cartan subalgebras in extended affine Lie algebras

TL;DR

This paper proves that all Cartan subalgebras in extended affine Lie algebras are conjugate, extending the known conjugacy results from finite-dimensional and affine Kac-Moody Lie algebras to this broader class.

Contribution

It establishes the conjugacy of Cartan subalgebras in extended affine Lie algebras, a key step in understanding their structure and invariants.

Findings

All Cartan subalgebras are conjugate in extended affine Lie algebras.

Root systems and Cartan matrices are invariants for these algebras.

Generalizes conjugacy results from finite-dimensional and affine cases.

Abstract

That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter-Dynkin diagrams and indecomposable irreducible root systems, is arguably one of the most elegant results in mathematics. The definition of the root system is done by fixing a Cartan subalgebra of the given Lie algebra. The remarkable fact is that (up to isomorphism) this construction is independent of the choice of the Cartan subalgebra. The modern way of establishing this fact is by showing that all Cartan subalgebras are conjugate. For symmetrizable Kac-Moody Lie algebras, with the appropriate definition of Cartan subalgebra, conjugacy has been established by Peterson and Kac. An immediate consequence of this result is that the root systems and generalized Cartan matrices are invariants of the Kac-Moody Lie algebras. The purpose of this paper is to establish conjugacy of Cartan subalgebras for extended affine Lie algebras; a natural class of Lie algebras that generalizes the finite-dimensional simple Lie algebra and affine Kac-Moody Lie algebras.