| This thesis defines a new class of Lie Algebras--generalized stochastic Lie algebras,and the structure of generalized stochastic Lie algebras.We’re going to study their structure through their linear transformations.These structure include derivations,Lie triple derivations,commutativity-preserving maps,zero product derivations,and then we determine their specific structure.The essay is divided into the following four chapters:In Chapter One,we first recall some basic conceptions and symbols of Lie algebras.Then we introduce the definition of a generalized stochastic Lie algebras and show its space decomposition.In Chapter Two,we study derivations and Lie thripe derivations of a generalized stochastic Lie algebra.Firstly,by analyzing the action if its derivations on its basis elements,we prove that any derivation is an inner derivation.Then it is showed that its center is zero.Thus any generalized stochastic Lie algebra is a complete Lie algebra.Finally it is showed that any Lie triple derivation of a generalized stochastic Lie algebras is a derivation.In Chapter Three,we determine the innertible maps of a generalized stochastic Lie algebra.Firstly,we prove that any generalized stochastic Lie algebra is zero product determined,and so its zero prduct derivation is a quasi-derivation,which is a sum of an inner derivation and a scalar multiplication map.Secondly,we prove that its quasi-isomorphism is a composition of a nonzero scalar multiplication map,a diagonal automorphism and an inner automorphism.Finally,it is showed that its invertible commutativity-preserving maps coincide with its quasi-automorphisms.In Chapter Four,we prove that any skew-symmetric biderivation of a generalized stochastic Lie algebra is an inner biderivation,and its symmetric bidervation is zero.Thus any biderivation of a generalized stochastic Lie algebra is an inner biderivation.Finally,it is showed that any CPA-structure of a generalized stochastic Lie algebra is trivial. |
