Classification of simple transitive 2-representations

Abstract: The representation theory of finitary 2-categories is a generalization of the classical representation theory of finite dimensional associative algebras. A key notion in classical representation theory is the notion of simple modules as those are in some sense the building blocks of all modules. A correct analogue of simple modules in the realm of 2-representations is the notion of simple transitive 2-representations since those also turn out to be building blocks of 2-representations.This thesis is concerned with the classification of simple transitive 2-representations for a number of different interesting 2-categories. In Paper I we study simple transitive 2-representations of Soergel bimodules in Coxeter type I2(4) and show that all simple transitive 2-representations in this case are equivalent to cell 2-representations. In Paper II we classify simple transitive 2-representations for the quotient of the 2-category of Soergel bimodules over the coinvariant algebra which is associated to the two-sided cell that is the closest to the two-sided cell containing the identity element, in all Coxeter types but I2(12), I2(18) and I2(30). It turns out that, in most of the cases, simple transitive 2-representations are exhausted by cell 2-representations. However, in Coxeter types I2(2k), where k ≥ 3, there exist simple transitive 2-representations which are not equivalent to cell 2-representations. In Paper III we show that for any complex polynomial p(X) the set of irreducible, integer matrices which are annihilated by p(X) is finite. Moreover, we study the set of irreducible, integral matrices satisfying X² = nX, for n ≥ 1, and count its elements. In Paper IV we show that every simple transitive 2-representations of the 2-category of projective functors for a certain quotient of the quadratic dual of the preprojective algebra associated with a tree is equivalent to a cell 2-representation. Finally, in Paper V we study simple transitive 2-representations of certain 2-subcategories of the 2-categories of projective functors over star algebras. In the simplest case, which is associated with Dynkin type A2, we show that simple transitive 2-representations are classified by cell 2-representations. However, in the general case we conjecture that there exist many more simple transitive 2-representations.