On The Lower Dimensional Cohomology Of Gl₂ With Coefficients In The Modular Lie Superalgebra W(m,3,(?))

In this thesis, we study the zero-dimensional and one-dimensional cohomology from gI2 to a family of Cartan type modular Lie superalgebras, and calculate the dimension of the zero-dimensional cohomology. As is well known, the theory of Lie superalgebras is a natural generalization of the theory of Lie algebras. In last ten years, many important results of Lie superalgebras have been obtained, but the classification problem is still open for the finite-dimensional simple modular Lie superalgebras. Cohomology plays an important role in the research of the classi-fication problem. Let K be an algebraically closed field of characteristic p>2, gl2 be the general linear Lie algebra, W(m,3,1) be a family of the generalized Witt modular Lie superalgebras. Obviously, W(m.,3,1) is a gl2-module, and it can be decomposed into the direct sum of 8 irreducible submodules. The cohomology from gl2 to W(m,3,1) is isomorphic to the direct sum of the cohomology from gl2 to each irreducible submodule of W(m,3,1). Therefore, we only need to obtain the coho-mology from gl2 to each irreducible submodule of W(m,3,1). In Chapter 3 and 4 we calculate all the zero-dimensional and one-dimensional cohomology from gl2 to irre-ducible submodules of W(m,3,1), respectively. Thus we obtain H0(gl2, W(m,3,1)) and H1(gl2,W(m,3,1)), where H0(gl2, W(m,3,1)) and H1(gl2, W(m,3,1)) are the zero-dimensional and one-dimensional cohomology of gl2 with coefficients in modu-lar Lie superalgebra W(m,3,1). We know the construction of H0(gl2, W(m,3,1)), so its dimension is easy to compute. As a vector subspace of linear mappings, H1(gl2, W(m,3,1)) can be represent by matrixes. H1(gl2, W(m,3,1)) is isomorphic to Der(gl2, W(m,3,1)) modulo Ider(gl2, W(m.,3,1)). The results of this thesis lay the foundations of the cohomology of more complex gl2-modules.