The Structure Of A Class Of Finite-dimensional Simple Lie Algebras Attached To Unit Forms

In the recent thirty years,it has become a hot topic on studying the relationship between the representation theories of finite-dimensional algebras and Lie algebras.An important tool to study it is the Ringel-Hall algebra.The thesis aims to study the structure of a complex simple Lie algebra via the representation theory of..N.akayama algebras,where the complex simple Lie algebra is attached to a unit form q?x₁,x₂,…,x_n?=???-???+x_ix_n ,n?3.First,we study some properties of the given unit form and the correspondimg finite dimensional simple Lie algebra.According to these properities,we construct a class of Nakayama algebras,and prove the Ringel-Hall Lie algebra of the root category of the Nakayama algebra is isomorphic to the above finite dimensional simple Lie algebra.Furthermore,as its application of the isomorphism theorem,we give the Dynkin type,a root space decomposition and a Chevalley basis of the finite dimen-sional simple Lie algebra.The thesis is divided into the following four chapters.In Chapter One,we will study some properties of the given unit form.We prove that the unit form q is positive definite and connected,and determine its roots.So the complex Lie algebra,which is attached to the unit form,is a finite-dimensional simple Lie algebra.In Chapter Two,we will study the homological properties of a class of Nakayama algebras A.We determine the Coxeter matrix and their powers,and give homological properties of indecomposable objects in their root category.In Chapter Three,we introduce the Ringel-Hall Lie algebra of the above Nakaya-ma algebra,and prove that it is isomorphic to the above simple Lie algebra.In Chapter Four,as an application of the isomorphism theorem,we prove that there is a bijection between the set of the indecomposable objects of the root cate-gory of the Nakayama algebra and the set of the roots of the positive definite unit form.moreover,we determine the Dynkin type,a root space decomposition and a Chevalley basis of the above simple Lie algebra.