The Simplicity And The Superderivation Algebra Of The Finite-dimensional Modular Lie Superalgebra W(n,m)

In this paper, we construct a new finite-dimensional simple modular Lie Superalgebra W (n,m),study the simplicity and determine the superderivation algebra of W(n,m). The main contents of this paper are as follows.In chapter 1, we first introduce the development of modular Lie superalgebras, and introduce the research status in the simple modular Lie superalgebras in China and abroad, then according to the difference of the underlying algebras, we classify the finite-dimensional nonclassical simple modular Lie superalgebras constructed in China, and list the important research results about their the simplicity and superderivation algebras. At the end of the chapter 1, we introduce the main contents of this paper. In chapter 2, we first describe the theoretical significance to construct the finite-dimensional simple modular Lie superalgebras, then we use the tensor product of exterior algebra ^(n) with n uncertainty elements x1, x2, …, xn and the truncated polynomial algebra T (m)=F [yn+1, yn+2, …, yn+m] to be the underlying algebras to construct the finite-dimensional modular Lie superalgebra W(n,m), after that, we introduce the general methods to study the simplicity and the superderivation algebras of the finite-dimensional modular Lie superalgebras. In Chapter 3, we first prove that the finite-dimensional modular Lie superalgebra W (n,m) is transitive and irreducible, thus we come to a conclusion that the finite-dimensional modular Lie superalgebra W(n,m) is simple. In chapter 4, we first grade the superderivation algebra of W (n,m), then we classified discuss the structure of each graded part of the superderivation algebra, therefore determine the superderivation algebra Der (W)= ad W.At the end of this paper, we prospect the other research questions of W (n, m), such as the restrictive and representation and so on.