The structure of a Lie algebra attached to a unit form
Received:November 04, 2017  Revised:August 12, 2018
Key Word: Nakayama algebras, finite dimensional simple Lie algebras, Ringel-Hall Lie algebras
Fund ProjectL:The National Natural Science Foundation of China (General Program, Key Program, Major Research Plan),The National Basic Research Program of China (973 Program),The National Science Fund for Distinguished Young Scholars (with foreign citizenship)
 Author Name Affiliation E-mail Yalong Yu Fujian Normal University 1670240028@qq.com Zhengxin Chen Fujian Normal University czxing@163.com
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Let $n\geq 4$. The complex Lie algebra, which is attached to a unit form $\mathfrak{q}(x_1,x_2,\cdots, x_n)=\sum\limits_{i=1}^nx_i^2-(\sum\limits_{i=1}^{n-1}x_ix_{i+1})+x_1x_n$ and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type $\mathbb{D}_n$, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.