Simplicity Of Hom-type Lie (Super) Algebras And Hamiltonian Structures Of Integrable Systems Associated With Two Types Of Lie Algebras

In recent years,with the continuous development of mathematics and physics,people begin to study the structures of Hom-type Lie(super)algebras.It is well known that Hom-type Lie(super)algebras are some kinds of deformation of Lie(super)algebras,when distortion mappings of Hom Lie(super)algebras are identity mappings,Hom Lie(super)algebras will be degraded into the original Lie(super)algebras,so Hom Lie(super)algebras can be regarded as the generalization of Lie(super)algebras.Decomposition and simplicity are two important research topics of Lie theory,for the Hom-type Lie(super)algebras can also be studied in these areas.Soliton theory is one of the main subjects of nonlinear sciences.Integrable systems and Hamilton structures are mainstream of nonlinear scientific researches.Constructions of integrable systems based on the structures of Lie algebras and extending the original integrable systems are all important research topics of soliton theory.In this paper,on the one hand,we will study the decomposition and simplicity of split involutive regular Hom-Lie algebras and three classes of regular Homtype Lie superalgebras.On the other hand,we will extend soliton integrable systems associated with two types of Lie algebras and their Hamiltonian structures are obtained.There are three parts in this thesis:Firstly,we study the decomposition and simplicity of split regular Hom-Lie algebras.To begin with,we give the concept of a split involutive regular Hom-Lie algebra and the connections of roots for this kind of algebra.By the techniques of connections of roots,we obtain a sufficient condition of a split involutive regular Hom-Lie algebra with a symmetric root system,which can be decomposed into the direct sum of its ideals.Next,we also get a sufficient and necessary condition of a simple split involutive regular Hom-Lie algebra with a symmetric root system.At last,we obtain a sufficient condition of a split involutive regular Hom-Lie algebra with a symmetric root system,which can be decomposed into the direct sum of its simple ideals.Secondly,we study the decomposition and simplicity of split Hom-Lie superalgebras,split ?-Hom-Jordan-Lie superalgebras and split BiHom-Lie superalgebras.To begin with,we give the concept of a split regular Hom-Lie superalgebra and the connections of roots for this kind of algebra.By the techniques of connections of roots,we obtain a sufficient condition of a split regular Hom-Lie superalgebra with a symmetric root system,which can be decomposed into the direct sum of its ideals.And we also get a sufficient and necessary condition of a simple split Hom-Lie superalgebra with a symmetric root system.Moreover,we obtain a sufficient condition of a split regular Hom-Lie superalgebra with a symmetric root system,which can be decomposed into the direct sum of its simple ideals.Next,we give the concept of a split regular ?-Hom-Jordan-Lie superalgebra and the connections of roots for this kind of algebra.By the techniques of connections of roots,we obtain a sufficient condition of a split regular ?-Hom-Jordan-Lie superalgebra with a symmetric root system,which can be decomposed into the direct sum of its ideals.And we also get a sufficient and necessary condition of a simple split ?-Hom-Jordan-Lie superalgebra with a symmetric root system.We obtain a sufficient condition of a split regular ?-Hom-Jordan-Lie superalgebra with a symmetric root system,which can be decomposed into the direct sum of its simple ideals.At last,we give the concept of a split regular BiHom-Lie superalgebra and the connections of roots for this kind of algebra.By the techniques of connections of roots,we obtain a sufficient condition of a split regular BiHom-Lie superalgebra with a symmetric root system,which can be decomposed into the direct sum of its ideals.And we also get a sufficient and necessary condition of a simple split ?-Hom-Jordan-Lie superalgebra with a symmetric root system.We obtain a sufficient condition of a split regular BiHom-Lie superalgebra with a symmetric root system,which can be decomposed into the direct sum of its simple ideals.Finally,we study bi-integrable couplings and tri-integrable couplings of a soliton hierarchy associated with Lie algebras S O(3)and S O(4),and Hamiltonian structures of the obtained integrable couplings.To begin with,based on a soliton hierarchy associated with three-dimensional Lie algebra S O(3),we construct bi-integrable and tri-integrable couplings associated with S O(3)for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations.Moreover,Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by the variational identities.Then,based on a soliton hierarchy associated with six-dimensional Lie algebra S O(4),we construct bi-integrable and tri-integrable couplings associated with S O(4)for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations.Moreover,Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by the variational identities.