The extended affine Lie algebra associated with a connected non-negative unit form

TL;DR

This paper constructs extended affine Lie algebras from connected non-negative unit forms, generalizing affine Kac-Moody algebras, and establishes a correspondence between these algebras and the forms.

Contribution

It introduces a method to build extended affine Lie algebras from non-negative unit forms and relates them via Chevalley bases and generalized Serre relations.

Findings

Constructed extended affine Lie algebras from non-negative unit forms.

Established a one-to-one correspondence between these algebras and equivalence classes of forms.

Generalized the construction to arbitrary nullity and simply-laced Dynkin types.

Abstract

Given a connected non-negative unit form we construct an extended affine Lie algebra by giving a Chevalley basis for it. We also obtain this algebra as a quotient of an algebra defined by means of generalized Serre relations by M. Barot, D. Kussin and H. Lenzing. This is done in an analogous way to the construction of the simply-laced affine Kac-Moody algebras. Thus, we obtain a family of extended affine Lie algebras of simply-laced Dynkin type and arbitrary nullity. Furthermore, there is a one-to-one correspondence between these Lie algebras and the equivalence classes of connected non-negative unit forms.