Simplicity Theorem and Modular Lie Superalgebras of Cartan Type
|School||Harbin Normal University|
|Keywords||Z-graded modular Lie superalgebras Cartan type modular Lie super-algebras simplicity|
The present thesis is devoted to giving the simplicity theorem of Z-graded Lie superalgebras and its proof, and exploiting this theorem to verify the simplicity of Cartan type modular Lie superalgebras. As is well known, theories of modular Lie algebras over a field of prime characteristic and Lie superalgebras over a field of characteristic zero have obtained plentiful fruits. As a natural generalization of Lie algebras, Lie superalgebras become efficient tools for analyzing the properties of physical systems. Lie superalgebras is developed on the basis of Lie algebras, even part of Lie superalgebras exactly is Lie algebras relatively. Therefore, it is usually referred to theory methods and means of Lie algebras. Although it is dif-ferent between modular Lie algebras and modular Lie superalgebras, but theories of modular Lie algebras provide with ways and means to consider the issue for re-searching modular Lie superalgebras. This paper discusses the relationship between filtration and gradations, then induces the judging conditions of simplicity theorem according to some propositions about filtration. So it is generalized that the content of simplicity theorem and the proof is given. Because the difference between non-modular Lie superalgebras and modular Lie superalgebras is Cartan type algebras, hence, many scholars have lots of researches on Cartan type algebras at home and abroad, including simplicity. But unlike previous proof of simplicity, we apply with new ways that it is satisfied with judging conditions of simplicity theorem in four families Cartan type modular Lie superalgebras. And it provides new ideas and methods for proving simplicity of other kinds of Lie superalgebras.