Structure And Representation Theory Of Some Finitary 2-categories

We give a description of abstract Duflo involution (uniquely determined for each given left cell) for finitary 2-categories, and study structure and representa-tion theory of some finitary 2-categories associated with finite dimensional alge-bras (over an algebraically closed field).First, in fiat 2-categories. one can define abelian cell 2-representations via Duflo involutions with respect to left cells, which belong to 2-categorical analogues of irreducible representations. In finitary 2-categories, one can define additive cell 2-representations in the form of subquotients of principal 2-representations. Moreover, in fiat setup, for any left cells, the abelianization of such additive cell 2-representations and abelian cell 2-representations obtained via Duflo involution are equivalent to each other. Motivated by the definition of Duflo involution for fiat 2-categories. we define certain analogues of Duflo involution for arbitrary finitary 2-categories and show that such Duflo involution with respect to every left cell does exist for three classes of finitary 2-categories considered in this dissertation. Two of them are associated with the path algebra of finite tree quivers (short for tree path algebras), that is, a class of finitary 2-categories given by dual projective functors and the other given by both dual projective functors and projective bimodules. Clearly, the former contains the latter as 2-subcategories. At the same time, we describe the quiver for the algebra underlying the principal 2-reprosentation for these two classes of finitary 2-categories, which offers some information for the category obtained by acting the abelian principal 2-representation on objects and indeed those categories are equivalent to the module category of the corresponding underlying algebra.Next, in finitary 2-categories, simple transitive 2-representations can be viewed as candidates of "simple" 2-representations. Indeed, for any finitary 2- representations, one can construct a weak Jordan-Holder series, whose corre-sponding weak composition subquotients are simple transitive 2-representations, and obtains the weak Jordan-Holder Theorem. Therefore, it is worthy to clas-sify simple transitive 2-representations for concrete finitary 2-categories. In this dissertation, we classify all simple transitive 2-representations for the first one of those finitary 2-categories associated with tree path algebras in the previous paragraph. Meanwhile, we also consider that class of finitary 2-categories given by projective bimodulcs of finite dimensional algebras among those three above classes and what we concerned is the situation that the involved finite dimension-al algebra is non-injeetive. However we cannot find a general classification, but in two smaller cases, we give a description of the classification result for their sim-ple transitive 2-representations. For the path algebra of a tree quiver, we define its complementary ideals and construct a new finitary 2-category, moreover, we classify all simple transitive 2-representations for the case of uniformly oriented quivers of type An. For these classes of finitary 2-categories, we obtained the following result:every simple transitive 2-representation is equivalent to a cell 2-representa.tion. While, for one class of fiat 2-categories associated with trun-cated polynomial algebras, the statement does not hold and it contains simple transitive 2-representations which are not cell 2-representations.Last, we consider hoAv to compute Drinfeld center for specific finitary 2-categories, which is viewed as the endomorphism categories of the identity 2-functor in 2-categories and is a braided monoidal category. In the last part of this thesis, we compute the Drinfeld center for finitary 2-categories of dual projection functors of tree path algebras, those of complementary ideals of the path algebra of the uniformly oiiented quiver of type An and fiat 2-categories associated with truncated polynomial algebras, among which there exist one’s Drinfeld center biequivaleiit to its morphism category and another’s Drinfeld center consisting of indecomposable objects of pairs determined by identity 1-morphisms.