标题： 椭圆扩展仿射李代数类型$A_1$的 Chevalley 基础 显示英文标题

标题： Chevalley bases for elliptic extended affine Lie algebras of type $A_1$

摘要： 我们研究类型为$A_1$的扩展仿射李代数的 Chevalley 基。近年来，对于秩大于一的扩展仿射李代数的整数结构概念已被成功探索。然而，对于秩为一的情况，情况变得更加微妙。在本工作中，我们考虑类型为$A_1$的扩展仿射李代数，其类型为{空集}$2$，称为椭圆扩展仿射李代数。这些李代数通过应用一些特定的 Jordan 代数（Tits-Kantor-Koecher (TKK) 构造）来构建：量子环面的加法代数、带有对合的 Laurent 多项式环的厄米特 Jordan 代数，以及与半格相关联的 Jordan 代数。通过对这些元素的检查，我们确定了相应椭圆扩展仿射李代数的零空间的适当基，从而建立了这些李代数的 Chevalley 基。 显示英文摘要

摘要： We investigate Chevalley bases for extended affine Lie algebras of type $A_1$.The concept of integral structures for extended affine Lie algebras of rank greater than one has been successfully explored in recent years. However, for the rank one it has turned out that the situation becomes more delicate. In this work, we consider $A_1$-type extended affine Lie algebras of {nullity} $2$, known as elliptic extended affine Lie algebras. These Lie algebras are build using the Tits-Kantor-Koecher (TKK) construction by applying some specific Jordan algebras: the plus algebra of a quantum torus, the Hermitian Jordan algebra of the ring of Laurent polynomials equipped with an involution, and the Jordan algebra associated with a semilattice. By examining these ingredient we determine appropriate bases for null spaces of the corresponding elliptic extended affine Lie algebra leading to the establishment of Chevalley bases for these Lie algebras.