Generalizations Of Derivations On Two Kinds Of Solvable Lie Algebras

Significant research has been done in studying derivations of Kac-Moody al-gebras and their subalgebras over a field. A well known result in the theory of Lie algebras, due to H. Zassenhaus, states that all derivations of finite-dimensional simple Kac-Moody algebras over a field are inner derivations. A lot of attention has been recently paid to some generalizations of Lie derivations. On one hand, we generalize results about linear conditions to nonlinear cases; on the other hand, we weaken the derivability on all points just to derivability on some special points. For example, near derivations of any Lie algebras in [1], generalized derivations of finite-dimensional Lie algebras in [21], product zero derivations in [39], general-ized Lie triple derivations in [6,22,40,41], nonlinear maps satisfying derivability on standard parabolic subalgebras of finite-dimensional simple Lie algebras in [7], all derivable points in the algebra of all upper triangular matrices in [43], all derivable points in full matrix algebras in [44]. In this thesis, we study some generalizations of derivations on several solvable Lie algebras. We characterize the product zero derivations and the non-linear derivations on strictly upper triangular matrix Lie algebras N(n,F), and subalgebras of finite-dimensional simple Lie algebras. These results generalize classical theories about derivations of simple Lie algebras.In the introduction, we start from backgrounds and recent development related to this thesis, then show our main results.In Chapter1, we introduce some preliminary notations and give some results about the derivations on simple Lie algebra.In Chapter2, we introduce product zero derivations on strictly upper triangular matrix Lie algebra N(n, F). Firstly, we give the concept of product zero derivations. Then, by the analysis of actions of product zero derivations on a fixed base of strictly upper triangular matrix Lie algebra N(n, F), we prove that any product zero derivation is a sum of an inner derivation, a diagonal derivation, an extremal product zero derivation, a central product zero derivation and a scalar multiplication map on N(n,F). This result generalizes the results about linear derivations on N(n,F).In Chapter3, we introduce non-linear derivations on strictly upper triangular matrix Lie algebras N(n,F). We prove that any non-linear derivation on N(n,F) is a sum of an inner derivation, a diagonal derivation, a central derivation, an extremal derivation, an additive quasi-derivation, a central map and an almost zero map on N(n, F). This result generalizes the results about linear derivations to nonlinear derivations.In Chapter4, we introduce non-linear Lie triple derivations on solvable sub-algebras of finite-dimensional simple Lie algebras. At first, we give the notion of non-linear Lie triple derivations. Then applying the action of the non-linear Lie triple derivation on the chevalley basis of b, we prove that any non-linear Lie triple derivation is a sum of a derivation and an additive quasi-derivation on b. This result generalizes the results about linear Lie triple derivations to nonlinear cases.Lastly, we make a conclusion of the work, and point out some problems needed to be solved.