Typical Simple Lie Algebra Subdivision Times And Limited Root Systems

In this article, we mainly investigate fine gradings on classical simple Lie algebras and finite root system. Our main goal is use finite root system to construct classical simple Lie algebras. By proposition3.8, we know that the Lie algebras constructed by finite root system is semisimple. Therefore, we expect to construct classical simple Lie algebras by finite root system and find out all the classical simple Lie algebras that can be constructed by finite root system. By the particular case in finite root system, we need to focus on fine gradings on classical simple Lie algebras.Firstly, we recall the definitions and results about graded (Lie) algebras in the articles of Bahturin, Kochetov and so on. In order to simplify the process of discussing, we also prove that some gradings are equivalent in the mean of isomorphism, preserving the structures of grading and involution. Secondly, we will give the background of introducing finite root system, the definition of finite root system and some related results in Chapter3. We then give detailed descriptions of fine gradings of classical simple Lie algebras in Chapter4. Finally, using these descriptions, we determine almost all classical simple Lie algebras that can be constructed by finite root system and give specific constructions.