Multiloop Lie algebras and the construction of extended affine Lie algebras

TL;DR

This paper explores the construction of extended affine Lie algebras (EALAs) from multiloop Lie algebras, establishing conditions for their isomorphism and support-isomorphism, and providing criteria for the equivalence of multiloop Lie algebras.

Contribution

It introduces the concept of support-isomorphism for multiloop Lie algebras and characterizes when their associated EALAs are isomorphic, extending previous understanding.

Findings

Two EALAs constructed from multiloop Lie algebras are isomorphic iff the algebras are support-isomorphic.

Provides a necessary and sufficient condition for support-isomorphism of multiloop Lie algebras.

Establishes a framework for constructing EALAs from multiloop Lie algebras beyond traditional conditions.

Abstract

It is known that a multiloop Lie algebra, which constructed using multiloop realization, can be a Lie torus if the given multiloop Lie algebra satisfies several conditions, and it is also known that a family of extended affine Lie algebras (EALAs) are obtained from a Lie torus. In many cases, however, even if a given multiloop Lie algebra does not satisfy these conditions, we can also construct a family of EALAs from it. In this paper, we study this construction, and prove that two families of EALAs constructed from two multiloop Lie algebras coincide up to isomorphisms as EALAs if and only if two multiloop Lie algebras are "support-isomorphic". Also, we give a necessary and sufficient condition for two multiloop Lie algebras to be support-isomorphic.