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, RingelHall 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) 

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Abstract: 
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}^{n1}x_ix_{i+1})+x_1x_n $ and defined by generators and generalized Serre relations, is proved to be a finitedimensional simple Lie algebra of type $\mathbb{D}_n$, and realized by the
RingelHall 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. 
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