Given a finite-dimensional algebra A we may associate to it a special endomorphism algebra, R_A, introduced by Auslander. The algebra R_A is a "Schur-like" algebra for A: it contains A as an idempotent subalgebra (up to Morita equivalence) and it is quasihereditary with respect to a particular heredity chain. The main purpose of this thesis is to describe the quasihereditary structure of R_A which arises from such heredity chain, and to investigate the corresponding Ringel dual of R_A. It turns out that R_A belongs to a certain class of strongly quasihereditary algebras defined axiomatically, which we call ultra strongly quasihereditary algebras. We derive the key properties of ultra strongly quasihereditary algebras, and give examples of other algebras which fit into this setting.

This thesis is comprised of three parts. Chapter one contains background material detailing some important aspects of category theory and homological algebra. Beginning with abelian categories, we introduce triangulated categories, the homotopy category and give the construction of the derived category. Differential graded algebras, basic Auslander-Reiten theory, and the Cluster Category of Dynkin Type An are also introduced, which all play a major role in chapters two and three. In [21], the cluster category D of type A1, with Auslander-Reiten quiver ZA1, is introduced. Slices in the Auslander-Reiten quiver of D give rise to direct systems; the homotopy colimit of such direct systems can be computed and these "Prüfer objects" can be adjoined to form a larger category. It is this larger category, D; which is the main object of study in chapter two. We show that D inherits a nice geometrical structure from D; "arcs" between non-neighbouring integers on the number line correspond to indecomposable objects, and in the case of D we also have arcs to infinity which correspond to the Prüfer objects. During the course of chapter two, we show that D is triangulated, compute homs, investigate the geometric model, and we conclude by computing the cluster tilting subcategories of D. Frieze patterns of integers were studied by Conway and Coxeter, see [13] and [14]. Let C be the cluster category of Dynkin type An. Indecomposables in C correspond to diagonals in an (n + 3)-gon. Work done by Caldero and Chapoton showed that the Caldero-Chapoton map (which is a map dependent on a fixed object R of a category, and which goes from the set of objects of that category to Z), when applied to the objects of C can recover these friezes, see [10]. This happens precisely when R corresponds to a triangulation of the (n + 3)-gon, i.e. when R is basic and cluster tilting. Later work (see [6], [22]) generalised this connection with friezes further, now to d-angulations of the (n + 3)-gon with R basic and rigid. In chapter three, we extend these generalisations further still, to the case where the object R corresponds to a general Ptolemy diagram, i.e. R is basic and add(R) is the most general possible torsion class (where the previous efforts have focused on special cases of torsion classes).

This thesis is concerned with the study of non-abelian finite simple groups and their generation. In particular, we are interested in a class of problems called width questions which measure the rate at which a given collection of elements generates a group. Let G be a finite group and let S ⊂ G be a subset that generates G. Then the width of G with respect to S is defined to be the minimal k ∈ N such that any element of G can be written as a product of at most k elements from S. The focus of this work is the width of non-abelian finite simple groups with respect to the set of elements of order p, where p is a fixed prime. We call this the p-width of the group G. The work of Liebeck and Shalev shows that there exists an absolute constant N > 0 that bounds the p-width of any finite simple group (of order divisible by p), and in this thesis we seek the minimal value of N. Of particular interest is the case where p = 2, as the study of involutions plays a large role in the overall study of finite simple groups. The involution width of finite simple groups has received considerable attention in the literature: notably there exists a classification of finite simple groups of involution width two (the so-called strongly real groups). The first main result of this thesis completes the involution width problem: we show that every non-abelian finite simple group has involution width at most four. Furthermore, this result is sharp, as there exist families with involution width precisely four. The proof of this result makes extensive use of the representation theory of finite groups of Lie type, and in particular we develop the theory of minimal degree characters using dual pairs. In the latter part of the thesis, we extend our methods to consider the p-width for odd primes. We partially resolve this problem, obtaining sharp bounds for the p-width of alternating groups and sporadic groups: for any odd prime p, the p-width of An (n ≥ p) is at most three, whereas the p-width of a sporadic group is two, except for a small number of known exceptions. We also consider the p-width for some groups of Lie type of small rank.

It has been proposed that the low energy e↵ective action of the theory of strings and branes possesses a large symmetry described by the Kac-Moody algebra E11. The non-linear realisation of this algebra and its vector representation determines the fields and coordinates of the theory, as well as the equations that describe their dynamics. In order to construct the generators of E11 algebra it is split into representations of its GL(d) ⇥ E11−d subalgebra. Here d is an integer that determines the dimension of the corresponding E11 theory. The low levels of the non-linear realisation contain the set of equations of the supergravity theory in corresponding space-time dimension, while the higher levels introduce an infinite number of fields that are connected to the supergravity ones via a chain of duality relations, as well as standalone fields that have no counterparts in standard supergravity theory. In this thesis we derive the set of commutators of E11 algebra and its vector representation up to a certain level in five and ten-dimensional cases. We use the non-linear realisation approach to construct the generalised vielbein and the Cartan forms of the E11 theory in four, five, ten and eleven dimensions. We then build a set of E11 invariant equations in five and eleven-dimensional theories from the non-linear realisation of E11. The low level equations, when appropriately truncated, are shown to perfectly reproduce the dynamics of the standard supergravity theories in corresponding dimensions. The dynamics of certain higher level fields are considered, including the dual graviton field and an eleven-dimensional field that, when reduced to ten dimensions, gives rise to the Romans mass term in type IIA theory. Lastly, we describe the non-linear realisation of very extended A1 algebra, called A+++ 1 , together with its commutators, Cartan forms and generalised vielbein.

In this thesis we formulate a natural non-commutative Iwasawa Main Conjecture for motives which fulfil the Dabrowski-Panchishkin condition on the level of (φ,Γ)-modules. The basic framework we employ is still Fukaya-Kato’s but we work systematically over Schneider-Teitelbaum’s distribution algebras of compact p-adic Lie groups instead of Iwasawa algebras. This allows us to consider as local conditions not just subrepresentations of the p-adic realisation which fulfil the Dabrowski-Panchishkin conditions but also sub-(φ,Γ)-modules which fulfil the analogous Dabrowski-Panchishkin conditions. We then combine this with Pottharst’s Selmer complexes and a generalisation of Nakamura’s Local Epsilon Conjecture for (φ,Γ)-modules to conjecturally define p-adic L-functions. We prove that the validity of our main conjecture for these p-adic L-functions follows from the validity of Fukaya-Kato’s Equivariant Tamagawa Number Conjecture and our generalisation of Nakamura’s Local Epsilon Conjecture. Moreover we are also able to compute the values of these p-adic L-functions at motivic points. Our formalism allows us, for example, to unify the GL2-main conjecture of elliptic curves which have either ordinary or supersingular reduction at p. In addition, we can use our formalism to give a new, and very natural, interpretation of Pollack’s ±-construction in the context of supersingular elliptic curves and we are hopeful that this new interpretation will in the future lead to the construction of natural non-commutative generalizations.

No description available.

No description available.

Throughout this thesis, we study the representation theory of the Fuss-Catalan algebras, $FC_{2,n}(a, b)$. We prove that these algebras are cellular and we define their cellular basis. In addition, we prove that they form a tower of recollement, and hence, they are quasi-hereditary. By calculating the Gram determinants of certain cell modules for the Fuss-Catalan algebras, we determine when these algebras are not semisimple. Finally, we end with defining homomorphisms between specified cell modules.

The theory of group schemes and their liftings to mixed characteristic valuation rings is well-developed. In [Fal02], a new equi-characteristic analogue of group schemes, known as group schemes with strict ๕-action, or strict ๕-modules, was proposed and developed, including Dieudonné theory. In [Abr04], their theory was studied over a complete discrete valuation ring. In this Thesis, a version of Dieudonné theory is developed for the strict ๕-modules of [Abr04] over a perfect field, using constructions of [Fon77], using very different methods from those deployed in [Fal02]

There is a well-developed theory of unique factorization domains in commutative algebra. The generalization of this concept to non-commutative rings has also been extensively studied (e.g. in [19], [16], [1]). This thesis is concerned with classes of non-commutative rings which are generalizations of non-commutative Noetherian unique factorization rings. Noetherian UFR's are maximal orders and every reflexive ideal is invertible in these rings. Clearly maximal orders and reflexive ideals are important concepts and we examine them in this thesis. UFR's can also be considered as rings in which the divisor class group is trivial. This provides the motivation for us to study this group more generally. In Chapter 1, we give the basic material we shall need from the theory of noncommutative rings-and in Chapter 2, we present the known results about certain classes of rings which are crucial for this thesis. Chapters 3, 4 and 5 contain the original work of this thesis. In Chapter 3, we study the prime Noetherian maximal orders with enough invertible ideals. We show that in such rings every height 1 prime ideal is maximal reflexive and we prove results which generalize some of the results of Asano orders. In Chapter 4, we investigate divisor class groups. We also study the relations between these groups and the divisor -class group of the centre of the ring. In Chapter 5, we introduce the Generalized Dedekind prime rings (G-Dedekind prime rings), which are also a generalization of Noetherian UFR's, and we study this class of rings.