The (Co)Action Of Weak Multiplier Hopf Algebras And Braided Tensor Categories

The aim of this dissertation is to carry out the action theory of weak multiplier Hopf algebras from the following aspects.First,we introduce the definition of module algebra over weak multiplier Hopf algebra,and give some examples.Then we define the smash product and unify the theory of actions of Hopf algebras,weak Hopf algebras and multiplier Hopf algebras to one of actions of weak multiplier Hopf algebras.Moreover,we study the two-sided smash product.One of the main results is the duality theorem for actions and their dual actions on the smash product of weak multiplier Hopf algebras.Second,by constructing the coproduct,counit,canonical idempotent and the antipode,we show that there exists a weak multiplier Hopf algebra structure on the smash product algebra.We also study the integral on the smash product algebra.Further,the fixed point algebra will be discussed.Finally,we will introduce the notion of weak multiplier Hopf algebra dual pairing.It can viewed as a non-trivial generalization of the corresponding notion in multiplier Hopf algebras and weak Hopf algebras.We will obtain some interesting results.With the help of dual pairing,the quantum double of weak multiplier Hopf algebras can be constructed.Therefore,we can obtain more examples of weak multiplier Hopf algebras.