On The Lower Dimensional Cohomology Of Sl₂ With Coefficients In The Modular Lie Superalgebras Of Witt Type

In this paper, we study the zero-dimensional and one-dimensional cohomologyof sl₂ with coe?cients in the Cartan type modular Lie superalgebras W(m,n,1).Let K be an algebraically close field of prime characteristic p > 2, W(m,n,1) bethe generalized Witt type modular Lie superalgebras. By definition, W(m,n,1) isa sl₂-module. In Chapter 2 give the direct sum decomposition of irreducible sl₂-submodule of W(m,n,1). For the cohomology of sl₂ with coe?cients in W(m,n,1)is isomorphic to the direct sum of the cohomology of sl₂ with coe?cients in eachirreducible submodule of W(m,n,1). Using the definitions of zero-dimensional andone-dimensional cohomology, in Chapter 3 we calculate all the zero-dimensional andone-dimensional cohomology of irreducible submodules of W(m,n,1), respectively.Thus we given structure of H0(sl₂,W(m,n,1)) and H1(sl₂,W(m,n,1)). This papergives the results of the low-dimensional cohomology of sl₂ to a class of complexmodules. The author expects to provide a useful reference of cohomology of sln-module and gln-module for a further investigation.