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We study the Hopf-Galois module structure of algebraic integers in some finite extensions of $ p $-adic fields and number fields which are at most tamely ramified. We show that if $ L/K $ is a finite unramified extension of $ p $-adic fields which is Hopf-Galois for some Hopf algebra $ H $ then the ring of algebraic integers $ \OL $ is a free module of rank one over the associated order $ \AH $. If $ H $ is a commutative Hopf algebra, we show that this conclusion remains valid in finite ramified extensions of $ p $-adic fields if $ p $ does not divide the degree of the extension. We prove analogous results for finite abelian Galois extensions of number fields, in particular showing that if $ L/K $ is a finite abelian domestic extension which is Hopf-Galois for some commutative Hopf algebra $ H $ then $ \OL $ is locally free over $ \AH $. We study in greater detail tamely ramified Galois extensions of number fields with Galois group isomorphic to $ C_{p} \times C_{p} $, where $ p $ is a prime number. Byott has enumerated and described all the Hopf-Galois structures admitted by such an extension. We apply the results above to show that $ \OL $ is locally free over $ \AH $ in all of the Hopf-Galois structures, and derive necessary and sufficient conditions for $ \OL $ to be globally free over $ \AH $ in each of the Hopf-Galois structures. In the case $ p = 2 $ we consider the implications of taking $ K = \Q $. In the case that $ p $ is an odd prime we compare the structure of $ \OL $ as a module over $ \AH $ in the various Hopf-Galois structures.

Abstract: $G$-structures on fusion categories have been shown to be an important tool to understand orbifolds of vertex operator algebras \\cite{Kirillov}\\cite{G_crossed_muger}\\cite{Orbifold_Paper}. We continue to develop this idea by generalizing Eilenberg-Maclane's notion of an Abelian $3$-cocycle to describe $G$-structures on fusion categories as $G$-(crossed, ribbon) Abelian $3$-cocycles on an algebra $H$. In particular, we show that a $G$-(crossed, ribbon) Abelian $3$-cocycle on $H$ will induce a $G$-(crossed braided, ribbon) tensor structure on its category of modules $\\mathrm{Mod}(H)$. We then prove that every $G$-(crossed braided, ribbon) fusion category $\\mathcal{C}$ will be equivalent to the category of modules of some finite dimensional algebra $H$ with $G$-structure induced from a $G$-(crossed, ribbon) Abelian $3$-cocycle. We call this $G$-reconstruction. Lastly, we prove that a $G$-ribbon Abelian $3$-cocycle $\\Gamma$ on $H$ allows us to describe the equivariantization $(\\Mod(H))^G$ as the category of modules of a ribbon (weak) quasi Hopf algebra $H\\#_{\\Gamma}\\mathbb{C}[G]$. We call this the Hopf equivariantization theorem. By $G$-reconstruction this shows that if $\\mathcal{V}$ is a strongly rational vertex operator algebra where $G$ acts faithfully on $\\mathcal{V}$ such that $\\mathcal{V}^G$ is also strongly rational, then there is an equivalence of modular fusion categories: \\begin{equation} \\mathrm{Mod}\\ \\mathcal{V}^G \\cong \\mathrm{Mod}(H\\#_{\\Gamma}\\mathbb{C}[G]) \\end{equation} for some finite dimensional algebra $H$ with a $G$-ribbon Abelian $3$-cocycle $\\Gamma$. This provides a proof of the Dijkgraaf-Witten conjecture, and generalizes it as far as possible in the semi-simple setting.

In this dissertation we first introduce an extension of the notion of parking functions to cars of different sizes. We prove a product formula for the number of such sequences and provide a refinement using a multi-parameter extension of the Abel--Rothe polynomial. Next, we study the incidence Hopf algebra on the noncrossing partition lattice. We demonstrate a bijection between the terms in the canceled chain decomposition of its antipode and noncrossing hypertrees. Thirdly, we analyze the sum of the th powers of the descent set statistic on permutations and how many small prime factors occur in these numbers. These results depend upon the base expansion of both the dimension and the power of these statistics. Finally, we inspect the ƒ-vector of the descent polytope DPv, proving a maximization result using an analogue of the boustrophedon transform.

One semi-classical knot invariant involves turning a knot diagram into a curve in ℝ2 which is "decorated" by elements of a ribbon Hopf algebra H. A decorated curve is turned into an element of H using a form of pictoral calculus. The image of this element in a certain quotient space of H defines a framed knot invariant. We generalize this process to define an invariant of links in a thickened surface Σ × [0, 1], where Σ is a connected, oriented surface. In this process, we develop a theory of decorated curves in an arbitrary smooth manifold M using a balanced, flat ribbon Hopf algebra bundle E → M with typical fiber H. The link invariant is defined using decorated curves in the unit tangent bundle of T1Σ, and takes values in the quotient space of the semi-direct product k[ π1(M , b_0) ] ⊗ H.We also define local diagrams to picture the decorated curves. The original pictoral calculus for decorated curves in ℝ2 is recaptured by viewing decorated curves in T1Σ through these local diagrams.

After presenting some preliminary information, this paper presents two proofs regarding group schemes. The first relates the category of affine group schemes to the category of commutative Hopf algebras. The second shows that a commutative group scheme of finite order is in fact killed by its order.

We show that one can extend the definition of Hopf cyclic homology with coefficients such that one can use bialgebras and a larger class of coefficient module/comodules. With the help of this extension, we calculate the bialgebra cyclic homology Uq(햌) of the quantum deformation of an arbitrary semi-simple Lie algebra 햌 and 퓗(N) the Hopf algebra of foliations of codimension N, with several coefficient modules.