Given the data of a Coxeter system (W,S), a Coxeter categorification is a 2-category in which the objects are subsets of S, the generating 1-morphisms categorify induction and restriction functors associated to parabolic subgroups, and the generating 2-morphisms impose certain coherence conditions and structural properties among the 1-morphisms. Of particular interest is the structure of the 2-homomorphism spaces of these 1-morphisms. Furthermore, given a connected, reductive, algebraic group G over an algebraically closed field k, a chosen Frobenius endomorphism F on G determines a parameter q in k, and the Weyl group of G gives rise to a Coxeter system. When this system is of rank 1, we construct by generators and relations an extension of the Coxeter categorification, independent of q, where the 2-homomorphism spaces are free modules of finite rank over the ring of Laurent polynomials with integer coefficients. An explicit description of the 2-homomorphism spaces between generating 1-morphisms is given, along with an algorithm lifting these descriptions to the 2-homomorphism spaces of arbitrary 1-morphisms. Then a nontrivial 2-functor from this 2-category is constructed into the 2-category of bimodules. Some conjectural constructions are given in the case that W has arbitrary finite rank, in particular a proposal for the endomorphism ring of the generating 1-morphism from the empty set to S that is an extension of an algebra introduced by Marin.

Let $U$ be the quantum group associated to a Cartan matrix indexed by a set $I$. There is a remarkable family $(\theta_i)_{i \in I}$ of symmetries of $U$ giving rise to the action of a braid group. The goal of this dissertation is to explain how to categorify these symmetries. The quantum group is categorified by a certain 2-category $\mathcal U$ introduced by Khovanov-Lauda and Rouquier. We propose two constructions providing a categorification of the symmetry $\theta_i$ on $\mathcal U$.

First, there is a complex $\Theta_i$ in $\mathcal U$. We prove that $\Theta_i$ is invertible in the homotopy category of $\mathcal U$, and that there are homotopy equivalences $\Theta_i E_i \simeq F_i\Theta_i[-1]$, where $E_i,F_i$ denote the Chevalley generators of $\mathcal U$. Hence, conjugation by $\Theta_i$ provides an auto-equivalence of the homotopy category of $\mathcal U$. This auto-equivalence decategorifies to the symmetry $\theta_i$. To obtain these results, we prove a faithfulness property for the 2-representations of $\mathcal U$.

Second, we construct a monoidal functor $T_i$ by defining it on generators and checking relations. We prove that this functor decategorifies to the symmetry $\theta_i$ on a certain subalgebra of the positive part of $U$. Our approach is based on categorifying the formulas defining the symmetry $\theta_i$ on $U$. These formulas involve the adjoint action of $U$, of which we also construct a categorification.

Part I of this thesis lays the foundations of categorical Demazure operators following the work of Anthony Joseph. In Joseph's work, the Demazure character formula is given a categorification by idempotent functors that also satisfy the braid relations. This thesis defines 2-functors on a category of modules over a half 2-Lie algebra and shows that they indeed categorify Joseph's functors. These categorical Demazure operators are shown to also be idempotent and are conjectured to satisfy the braid relations as well as give a further categorification of the Demazure character formula.

Part II of this thesis gives a presentation of localized affine and degenerate affine Hecke algebras of arbitrary type in terms of weights of the polynomial subalgebra and varied Demazure-BGG type operators. The definition of a graded algebra is given whose category of finite-dimensional ungraded nilpotent modules is equivalent to the category of finite-dimensional modules over an associated degenerate affine Hecke algebra. Moreover, unlike the traditional grading on degenerate affine Hecke algebras, this grading factors through central characters, and thus gives a grading to the irreducible representations of the associated degenerate affine Hecke algebra. This paper extends the results of Rouquier, Brundan and Kleschev on the affine and degenerate affine Hecke algebras for $GL_n$ which are shown to be related to quiver Hecke algebras in type A, and also secretly carry a grading.

The Weyl group used in Lie theory can be generalized into reflection groups in more general division algebras; in particular, to complex reflection groups and quaternionic reflection groups. When dealing with these groups, the concept of an associated braid group can come into play. In the real (Coxeter) case, including Weyl groups, this is often described in terms of generators and relations but in the complex case, usually described topologically.

In this paper, I will describe the fruit of efforts to see whether the concept of a braid group (and pure braid group) can be extended from the complex case to the quaternionic case, in particular the category of representations. In Chapter 2, I prove that the cohomology of a hyperplane complement in a quaternionic module can be treated in some ways like the braid group in the complex case, and in particular to create representations corresponding to the representations of the braid group whose pure braid group action is unipotent.

However, in attempting to extend this further, problems arise. In particular, the quaternionification of a complex reflection group is isomorphic to itself through complex conjugation, producing a permutation of its hyperplanes that can be represented through a linear map in the quaternions, but not in the complex numbers, and in Chapter 3 I prove that there is an invariant that can distinguish between the two, as well as others, through the topology of the complex hyperplane complement, extending work of Guerville-Ballé from a single case to the entire infinite group class G(m, m, 3).

Let A be a finite-dimensional self-injective algebra, graded in non-positive degree. We define A-dgstab, the differential graded stable category of A, to be the quotient of the bounded derived category of dg-modules by the thick subcategory of perfect dg-modules. We prove that A-dgstab is the triangulated hull of the orbit category A-grstab/ Ω(1), which allows computations in the dg-stable category to be performed in the graded stable category. We provide a sufficient condition for the orbit category to be equivalent to A-dgstab and show this condition is satisfied by Nakayama algebras and Brauer tree algebras. When A is a symmetric algebra with socle concentrated in degree -d < 0, we show that A-dgstab has Calabi-Yau dimension -d-1.

Chuang and Rouquier (2017) describe an action by perverse equivalences on the set of bases of a triangulated category of Calabi-Yau dimension -1. We develop an analogue of their theory for Calabi-Yau categories of arbitrary negative dimension and apply this theory to the dg-stable category.

As an example, we analyze the dg-stable category of a Brauer tree algebra, with an arbitrary non-positive grading. We compute the Auslander-Reiten quiver, then develop a combinatorial model for A-dgstab, which we use to describe the action of perverse equivalences. Using our model, we show that perverse equivalences act transitively on the set of bases.