Cohomology And Yang-baxter Equation Of Model Filiform Lie Superalgebras

The representation and structure theory of the filiform Lie superalgebras is one of the most important research subjects of the Lie theory.Analogous to what happens in the Lie case,every filiform Lie superalgebra can be obtained by an in-finitesimal deformation of the model filiform Lie superalgebra Ln,m This paper aims to study the cohomology and the Yang-Baxter equation of the model filiform Lie su-peralgebra Ln,m over an algebraically closed field of characteristic zero.At first,The problem of computing the first cohomology of the filiform Lie superalgebras Ln,m with coefficients in adjoint module is converted to the problem of computing the first cohomology of a Abel ideal and a one-dimensional subalgebra of Ln.m by means of spectral sequences,then we characterize the first cohomology of Ln,m by computing the outer derivations;secondly,we make a space direct sum decomposition of Ln,m,so the solutions of the Yang-Baxter equation of the filiform Lie superalgebras Ln,m were divided into several situations,then we solve the Yang-Baxter equation in each case,so we obtain all the solutions of the Yang-Baxter equation of the filiform Lie superalgebras Ln,m in terms of the matrix form.