A class of simple Lie algebras attached to unit forms

Abstract

Let n≥3.The complex Lie algebra, which is attached to a unit form q(x_{\mathrm{1}},x_{\mathrm{2}},\dots ,x_n)={\displaystyle {\sum }_{i=\mathrm{1}}^n{x}_i^{\mathrm{2}}}+{\displaystyle {\sum }_{\mathrm{1}\le i\le j\le n}{(-\mathrm{1})}^{j-i}}{x}_i{x}_j and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type {\mathbb{A}}_n,and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.

Key words： Nakayama algebras; finite-dimensional simple Lie algebras; Ringel-Hall Lie algebras

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