No description available.

This is the author's PhD thesis. During the course of my studies, I have mastered the basics of the theory of Hopf algebras, and in particular, learned of Drinfel'd's concept of a cocycle twist of a Hopf algebra, and of a module algebra over that Hopf algebra. A module algebra is is an algebra over a field upon which a Hopf algebra acts in a certain way. In particular, I came to focus upon how this concept works in the special case when the Hopf algebra is an algebra over a finite group. In this case, the module algebra is an algebra over the field on which the group acts by homomorphisms. This algebra may be twisted into another module algebra by means of a 2-cocycle on the group. Having learned this, my attention was drawn to the work of Vendramin in [Ven12] in which he examined two module algebras over the symmetric group S_n, called Nichols algebras, defined using what are called rack cocycles. He showed that there is a cocycle twist that transforms one algebra into the other, i.e. that they are twist-equivalent. There are two quadratic algebras associated to the Nichols algebras, called E_n and Lambda_n and first described in [FK99] and [Maj05], which are thought to be isomorphic to the Nichols algebras. It has for some years been conjectured, but not proven, that these two algebras are twist-equivalent. The most important result of this thesis is Theorem 4.6, which proves that E_n and Lambda_n are indeed twist-equivalent. Following this result I sought to see if analogous results could be obtained when considering other finite irreducible Coxeter groups than type A, which is what S_n is. To do this requires understanding of rack cocycles, and of the Schur multiplier of a group, which affects what kind of cocycle twisting is possible. I chose to focus on the case where the Coxeter group is a dihedral group since these groups are often fundamental to determining what happens for Coxeter groups of higher dimension. The last part of this work examines questions on whether the rack cocycles analogous to those that defined E_n and Lambda_n are related to each other by cocycle twisting. The dihedral case, however, turns out to be less straightforward than was the case for Coxeter groups of type A, and it seems that there is scope for continuing research in this direction.

Convolution operators on Lp(0,1) have many similarities with the classical Volterra operator V, but it is not known in general for which convolution kernels the resulting operator behaves like V. It is shown that many convolution operators are cyclic, and the cyclic property is related to the invariant subspace lattice of the operator, and to the behavior of the kernel as an element of the Volterra algebra. The convolution operators induced by kernels satisfying a smoothness condition near the origin are shown to have asymptotic behavior that matches that of powers of V, and a new class of convolution operators that are not nilpotent, but have kernels that are not polynomial generators for L1(0,1), are produced. For kernels that are polynomial generators for L1(0,1), the corresponding convolution operators are shown to have the property that their commutant and the strongly-closed subalgebra of B(Lp(0,1)) they generate are equal.

We investigate the question: ''Can there be a non-continuous isomorphism between two profinite groups which are not topologically isomorphic?" On one end of the spectrum, we show that branch and semisimple profinite groups have no non-continuous automorphisms. On the other, many abelian pro-$p$ groups are abstractly but not topologically isomorphic. The question for countably-based profinite groups was totally answered in a previous publication. There are many examples of such groups which are abstractly but not topologically isomorphic: we give explicit constructions of such non-topological isomorphisms We used Pontryagian duality to reduce the question of classifying countably based abelian pro-$p$ groups to that of countable abelian $p$-groups. In the 1930s Ulm and Zippin classified countable abelian $p$-groups. This work was expanded in the 1970s, to give the theory of totally projective abelian $p$-groups. We survey the structural theory of these groups and construct their duals, the totally injective groups. These provide more positive answers to our question: every dual-reduced totally injective pro-$p$ group is abstractly isomorphic to the closure of its torsion subgroup, although in most cases these groups are not topologically isomorphic. We proceed to give a detailed discussion on the features of the abstract and of the topological subgroup structures of such groups. We introduce a new invariant, unbounded multiplicity, of Cartesian products of finite $p$-groups, in the above proof. This allows us to use infinite combinatorial arguments which give more results. Two of these Cartesian groups are isomorphic modulo their torsion subgroups if and only if they have the same unbounded multiplicity. A totally injective pro-$p$ group will be abstractly isomorphic to its closed torsion subgroup whenever the unbounded multiplicity of this subgroup bounds the dimension of continuous torsion-free quotients. Additionally, we construct a new class of commutative, unital pro-$p$ rings. For each totally injective abelian pro-$p$ group $G$, we construct a pro-$p$ ring $R$ with $(R,+)=G$.

No description available.

In this thesis we generalise quantum skew Howe duality to Lie superalgebras in type A, and show how this gives a categorification of certain representation categories of $\mathfrak{gl}(m|n)$. In particular, we use skew Howe duality to describe a category of representations generated monoidally by the exterior powers of the fundamental representation. This description is in terms of MOY diagrams, with one additional local relation on $n+1$ strands. This generalises the $n=0$ case from Cautis, Kamnitzer and Morrison. Using this, we give a categorification of this category in terms of foams, which generalises that of Queffelec, Rose and Lauda in the case $n=0$. The Reshetikhin-Turaev procedure gives a knot polynomial associated to $\mathfrak{gl}(m|n)$, which is a specialisation of the HOMFLY polynomial $P(a,q)$ at $a=q^{m-n}$. For the case $n=0$, the polynomial can be described nicely in terms of MOY diagrams, and therefore is related strongly to skew Howe duality. This was used by Queffelec and Rose to define $\mathfrak{sl}(n)$ Khovanov-Rozansky homology by categorified skew Howe duality. For general $n$, the relationship is less nice, and skew Howe duality is not sufficient to describe a homology theory associated with $\mathfrak{gl}(m|n)$ from our approach. Part of the problem is that the representation category no longer contains duals of the fundamental representations, which means that although a braid has an image in this categorified representation category, it is not possible to close this braid in the same way that Queffelec and Rose do. However, the categorified representation category does give partial progress towards the problem of defining a quantum categorification of the Alexander polynomial.

This thesis is about trying to understand various aspects of partial symmetry using ideas from semigroup and category theory. In Chapter 2 it is shown that the left Rees monoids underlying self-similar group actions are precisely monoid HNN-extensions. In particular it is shown that every group HNN-extension arises from a self-similar group action. Examples of these monoids are constructed from fractals. These ideas are generalised in Chapter 3 to a correspondence between left Rees categories, selfsimilar groupoid actions and category HNN-extensions of groupoids, leading to a deeper relationship with Bass-Serre theory. In Chapter 4 of this thesis a functor K between the category of orthogonally complete inverse semigroups and the category of abelian groups is constructed in two ways, one in terms of idempotent matrices and the other in terms of modules over inverse semigroups, and these are shown to be equivalent. It is found that the K-group of a Cuntz-Krieger semigroup of a directed graph G is isomorphic to the operator K0-group of the Cuntz-Krieger algebra of G and the K-group of a Boolean algebra is isomorphic to the topological K0-group of the corresponding Boolean space under Stone duality.

In this thesis we describe the nonassociative geometry probed by closed strings in flat non-geometric R-flux backgrounds, and develop suitable quantization techniques. For this, we propose a Courant sigma-model on an open membrane with target space M, which we regard as a topological sector of closed string dynamics on Rspace. We then reduce it to a twisted Poisson sigma-model on the boundary of the membrane with target space the cotangent bundle T M. The pertinent twisted Poisson structure is provided by a U(1) gerbe in momentum space, which geometrizes R-space. From the membrane perspective, the path integral over multivalued closed string fields in Q-space (i.e. the T-fold endowed with a non-geometric Q- flux which is T-dual to the R-flux), is equivalent to integrating over open strings in R-space. The corresponding boundary correlation functions reproduce Kontsevich's global deformation quantization formula for the twisted Poisson manifolds, which we take as our proposal for quantization. We calculate the corresponding nonassociative star product and its associator, and derive closed formulas for the case of a constant R-flux. We then develop various versions of the Seiberg{Witten map, which relate our nonassociative star products to associative ones and add fluctuations to the R-flux background. We also propose a second quantization method based on quantizing the dual of a Lie 2-algebra via convolution in an integrating Lie 2-group. This formalism provides a categori cation of Weyl's quantization map, and leads to a consistent quantization of Nambu{Poisson 3-brackets. We show that the convolution product coincides with the star product obtained by Kontsevich's formula, and clarify its relation with the twisted convolution products for topological nonassociative torus bundles. As a first step towards formulating quantum gravity on non-geometric spaces, we develop a third quantization method to study nonassociative deformations of geometry in R-space, which is analogous to noncommutative deformations of geometry (i.e. noncommutative gravity). We find that the symmetries underlying these nonassociative deformations generate the non-abelian Lie algebra of translations and Bopp shifts in phase space. Using a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions, and the exterior differential calculus in R-space. We define a suitable integration on these nonassociative spaces, and find that the usual cyclicity of associative noncommutative deformations is replaced by weaker notions of 2-cyclicity and 3-cyclicity. In this setting, we consider extensions to non-constant R-flux backgrounds as well as more generic twisted Poisson structures emerging from non-parabolic monodromies of closed strings. As a first application of our nonassociative star product quantization, we develop nonassociative quantum mechanics based on phase space state functions, wherein 3-cyclicity is instrumental for proving consistency of the formalism. We calculate the expectation values of area and volume operators, and find coarse-graining of the string background due to the R-flux. For a second application, we construct nonassociative deformations of fields, and study perturbative nonassociative scalar field theories on R-space. We nd that nonassociativity induces modi cations to the usual classi cation of Feynman diagrams into planar and non-planar graphs, which are controlled by 3-cyclicity. The example of '4 theory is studied in detail and the one-loop contributions to the two-point function are calculated.

No description available.

No description available.