Isotopy for extended affine Lie algebras and Lie tori

TL;DR

This paper explores the relationship between centreless Lie tori and extended affine Lie algebras, establishing a correspondence via isotopy and showing how isotopes of Lie tori relate to isotopes of their coordinate algebras.

Contribution

It introduces the concept of isotopy for centreless Lie tori and demonstrates a one-to-one correspondence with families of EALAs, also relating isotopes of Lie tori to isotopes of their coordinate algebras.

Findings

Establishes a 1-1 correspondence between centreless Lie tori up to isotopy and EALAs up to isomorphism.

Shows that isotopes of Lie tori are coordinatized by isotopes of their coordinate algebras.

Provides foundational insights connecting Lie tori, EALAs, and their isotopic structures.

Abstract

Centreless Lie tori have been used by E. Neher to construct all extended affine Lie algebras (EALAs). In this article, we study notions of isotopes and isotopy for centreless Lie tori, and we use these notions, along with Neher's construction, to show that there is a 1-1 correspondence between centreless Lie tori up to isotopy and families of EALAs up to isomorphism. Also, centreless Lie tori can be coordinatized by unital algebras that are in general nonassociative, and, for many types of centreless Lie tori, there are classical notions of isotopes and isotopy for the coordinate algebras. We show for those types that an isotope of the Lie torus is coordinatized by an isotope of its coordinate algebra. In writing the article, we have not assumed prior knowledge of the theories of EALAs, Lie tori or isotopy. In fact, we hope that this article will help to introduce the reader to these theories and their interconnections.