Research On Two Classes Of Algebraic Structures And Representation Theory Based On 3-lie Algebras

Lie algebra is a branch of non-associative algebra.It is the source of many important problems in finite group theory.It has a good algebraic structure.Its method and theory have been applied in many fields of mathematics and theoretical physics.3-Lie algebra is a generalization of Lie algebra,Hom-3-Lie algebra is a deformation of 3-Lie algebra.This paper connects 3-Lie algebra with the stability of functional equations,and combines Hom-3-Lie algebra with product,complex and symplectic structures.The generalized Ulam-Hyers stability of homomorphisms,derivations and generalized derivations on Banach 3-Lie algebras and the product,complex and symplectic structures on Hom-3-Lie algebras are studied.Firstly,the generalized Ulam-Hyers stability of homomorphisms,derivations and generalized derivations on Banach 3-Lie algebras are discussed.Secondly,the definitions and basic properties of product,complex,symplectic structure and phase space on Hom-3-Lie algebra are given,then four special product(complex)structures are studied: strict product(complex)structure,abelian product(complex)structure,strong abelian product(complex)structure and perfect product(complex)structure,and then proved that Hom-3-Lie algebra has phase space if and only if it is compatible with a Hom-3-pre-Lie algebra.Finally,add compatibility conditions between product and complex structure,symplectic and paracomplex structure,symplectic and complex structure,and introduce the complex product structure,para-K(?)hler structure and pseudo-K(?)hler on the Hom-3-Lie algebra.On this basis,the relationship between the perfect para-K(?)hler Hom-3-Lie algebra and the pseudoRiemannian Hom-3-Lie algebra is established.