The Research Of Some Questions In Categorification

This paper is divided into three parts. We first study the theory of2-vertex algebra in categorification, secondly we study the categorification of quantum Heisenberg alge-bra and its representation under a finite subgroup Γ(?) SL3(k), and then we use the diagrammatic theory to categorify the coherent state.In Chapter2, we mainly introduce some related notions and conclusions.In Chapter3, we develop the theory of2-vertex algebra. We give the definition of categorification of vertex algebra by using internal category, i.e. we give the definition of2-vertex algebra. Then we get some important properties that the direct sum and tensor product of two2-vertex algebras are also2-vertex algebras. At the same time we also give the definition of homomorphism between2-vertex algebras and the definition of2-homomorphism between two homomorphisms. Secondly we give the method to construct a2-vertex algebra from a Lie2-algebra.In Chapter4, we give the quantum Heisenberg algebra ηΓ associated to a finite subgroup Γ (?) SL3(k), then categorify it, i.e. we get a2-category HΓ. And then we define an action of HΓ on the derived category (?)nD(AΓn-gmod), i.e. we construct a2-representation of the quantum Heisenberg algebra ηΓIn Chapter5, we mainly study the categorification of the coherent states, which is equivalent to the categorification of the corresponding displacement operators. Based on the categorification of the Heisenberg algebras, we construct some complexes of the following forms in the homotopy category Kom(C) of the2-category C where the elements in complexes are1-morphisms in C, the differential operators are string diagrams which are2-morphisms in C. Then we verify that these complexes satisfy the homotopy relations corresponding those which displacement operators satisfy. So those complexes can be considered as the categorical analogues of the displacement op-erators. Using the diagrammatic calculus, we find that the properties of the categoricaldisplacement operators coincide with those in normal quantum mechanics.