Infinitesimal Deformation Of The Bimodules Of Alternative Algebras And Compatible Alternative Algebras

In this thesis we mainly study infinitesimal deformation of the bimodules of alternative algebras and compatible alternative algebras.We define the deformations of alternative algebras and the infinitesimal deformation of the bimodules of alternative algebras.The Nijenhuis structure of alternative algebras is given.We give the definition of compatible alternative algebras.We also study matched pairs and Manin triples of compatible alternative algebras.In the first part,we introduce some basic concepts of alternative algebras and the conclusions that need to be used in this thesis,such as the definition of alternative algebras and the bimodulus of them,the matched pairs of alternative algebras and so on.In the second part,firstly,we construct a new left alternative algebra on the semi-direct product of left alternative algebra and its representation space.Then the definition of the H-twisted Rota-Baxter operator of left alternative algebras is given.We give an equivalent condition that a linear map is an H-twisted Rota-Baxter operator and a new structure of left alternative algebras on representation space is constructed by using this operator.In the third part,a special kind of alternative algebras is constructed,and then the infinitesimal deformation of the bimodules of alternative algebras is defined.We discuss the equivalant conditions of two infinitesimal deformations and the conditions that make infinitesimal deformations of bimodules be trivial.The definition of the Nijenhuis operator of alternative algebras and the Nijenhuis operator on the semi-direct product alternative algebras is studied.Finally,we give the relationship between the trivial infinitesimal deformation and the Nijenhuis structure of alternative algebras.In the fourth part,the definition of compatible alternative algebras is given,and the equivalence theorem of the definition is found.Then we give the definition of the bimodules of compatible alternative algebras,and their dual representation is given.At the same time,we study matched pairs of compatible alternative algebras.In the fifth part,firstly we define the Manin triple of compatible alternative algebras,then we give an equivalent condition for judging the Manin triple.We also give the definition of the standard Manin triple of compatible alternative algebras,and discuss its relationship to the matched pairs of compatible alternative algebras.