The Structure of a Lie Algebra Attached to a Unit Form

Abstract:

      Let $n\geq 4$. The complex Lie algebra, which is attached to the unit form $\mathfrak{q}(x_1,x_2,\ldots, x_n)=\sum_{i=1}^nx_i^2-(\sum_{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.