Derivations And Biderivations On Modular Lie(Super)algebras Of Cartan-type

There are three parts in this thesis.In the first part,we investigate the skew-symmetric biderivations on four classes of re-stricted Cartan-type Lie algebras.We give some properties of the skew-symmetric bideriva-tion on the simple modular Lie algebra.By the action of skew-symmetric biderivations on the generators and the simplicity of the restricted Cartan-type Lie algebras,we show their skew-symmetric biderivations are inner,respectively.As an application of the skew-symmetric biderivation,we give a sufficient and necessary condition for a linear mapping to be a commutative mapping on the restricted Cartan-type Lie algebras.In the second part,we study the skew-symmetric super-biderivation of the generalized Witt modular Lie superalgebra W(m,n;t).We give some properties of the skew-symmetric super-biderivation on the simple modular Lie superalgebra.Utilizing the relationship be-tween the super-biderivation and the super-derivation,we construct a set of zero weight derivations by skew-symmetric super-biderivations and toral elements.We prove that the ze-ro weight derivations are inner by the weight space decomposition of W(m,n;t)with respect to the classical torus.Moreover,we prove that every skew-symmetric super-biderivation of W(m,n;t)is inner.In the third part,we investigate the derivations from the even part of Hamilton simple modular Lie superalgebra H(m,n;t)to the odd part of W(m,n;t).We denote the even part of H(m,n;t)and the odd part of W(m,n;t)by H0 and W1,respectively.And we give the weight space decomposition of W1 with respect to the classical torus of H(m,n;t).Then,we character the zero weight derivations and the exceptional derivations from H0 into W1 by a general reduction method.Moreover,we obtain the derivations from H0 into W1.