Research On Several Problems About Structure Of Lie Algebras Associated To S-unitary Matrices

The structuaral problem about the Lie algebras is an important part on the research of algebra.In 2018,Caalim J et al.defined and studied the Lie algebra associated to S-unitary matrices.Let Mm(C)be the set of all m×m matrices over the complex number field C,and let S?M_m(C).Then u_S={A?M_m(C)|SA*=-AS}is a Lie algebra over the real number field R,where the bracket operation of u_S is[X,Y]=XY-YX for any X,Y?u_S.In this thesis,I will study the structure of u_S when S is unitary,including charactering the derivations,biderivations,generalized derivations of u_S,and determining whether u_S is zero product determined.This paper consists of five chapters.The introduction mainly recalls the re-search background and progress about the above topics,and introduces some study-ing methods and results of this thesis.In the first chapter,we mainly introduce some basic concepts and symbols about the Lie algebra u_S,and we give some basic structures and important properties of u_S.In the second chapter,we mainly characterize the derivations on u_S.By ana-lyzing actions of derivations on the basis elements,we prove that any derivation of u_S is a sum of an inner derivation and a central derivation.In the third chapter,we mainly characterize the biderivations and commutative post-Lie algebra structures on u_S.At first,we prove that any anti-commuting map of the derived algebra u'_S is a zero map,and then we prove that any biderivation of u_S is a sum of a generalized inner biderivation and a central biderivation.Further,as its application of biderivation,we prove that all commutative post-Lie algebra structures of u_S are central.In the forth chapter,we prove that u_S is zero product determined.Further-more,we determine the product zero derivations,quasi-derivations and generalized derivations if all eigenvalues of S are equal or negative.At first,we prove that u_S is zero product determined,and then we prove that any product zero derivation on u_S is equivalent to the quasi-derivation.Moreover,by analyzing actions of product zero derivations on the basis elements,we prove that any product zero derivation is a sum of an inner derivation,a central quasi-derivation,and a scalar multiplication map.Finally,we prove that product zero derivations,quasi-derivations,and gener-alized derivations of u_S coincide,and any generalized derivation of u_S is a sum of an inner derivation,a central quasi-derivation,and a scalar multiplication map of u_S.