In this thesis, we firstly extend elements and periodicity properties of the theta function theory to functions that represent a wider domain of symmetries and properties, graded with different amounts of p ≥ 1, p ϵ N. Unlike theta functions, these generalised, "higher-level Appell functions" K(_p) satisfy open quasiperiodicity relations, with additive theta function terms emerging as violating terms of open quasiperiodic K(_p)’s. We evaluate the S and T modular transformations of these functions and show that the S-transform of K(_p) does not just give back K(_p), but also includes p additional 0-functions which are precisely those violating the quasiperiodicity of Appell functions. This sets a new pattern of modular group representations on functions that are not double quasiperiodic. While calculating the S-transform of K(_p), a newly arising function, namely ɸ(T, μ) will be also thoroughly analysed. As two interesting applications, we firstly study the modular group action on unitary and on an admissible class of non-unitary N = 2 characters which are not periodic under the spectral flow and cannot therefore be rationally expressed through theta functions. Secondly we continue this study for the admissible representation of the affine Lie superalgebra sl (2|l). We see in the final result for both cases that the functions A(T, V) are the "violating" terms of unitary calculations. We lastly confirm all our results by some sets of consistency checks including an essential residue calculation. We believe this new way of using Appell functions, could be used for any other algebraic structure whose characters can be rewritten in terms of higher-level Appell functions.

Aquesta tesi doctoral tracta sobre C*-àlgebres i els seus invariants de Teoria K. Ens hem centrat principalment en l’estructura d’una classe de C*-àlgebres anomenada camps continus i l’estudi d’un dels seus invariants: el semigrup de Cuntz. Més concretament, analitzem el següent: (1)- Estructura dels camps continus : A la literatura hi ha dos exemples que donen una idea clara sobre la complexitat dels camps continus de C*-àlgebres. El primer va ser construït per M. Dadarlat i G. A. Elliott al 2007 i és un camp continu A sobre l’interval unitat amb fibres mútuament isomorfes, Teoria K no finitament generada i que no és localment trivial enlloc. El segon exemple mostra que, fins i tot quan la Teoria K de les fibres s’anul·la, el camp pot ser no trivial enlloc si l’espai base té dimensió infinita (Dadarlat, 2009). Veient aquests exemples és natural preguntar-se quina és l’estructura dels camps continus d’àlgebres de Kirchberg sobre un espai de dimensió finita, amb fibres mútuament isomorfes i Teoria K finitament generada. Tractem aquesta qüestió al Capítol 2 de la memòria. (2)- El semigrup de Cuntz de camps continus : Per a C*-àlgebres de dimensió baixa sense obstruccions cohomològiques, una descripció del seu semigrup de Cuntz, a través d’avaluació puntual, s’ha obtingut en termes de funcions semicontínues sobre l’expectre que prenen valors en els enters positius estesos (Robert, 2009). Per camps més generals la clau està en descriure l’aplicació següent: _: Cu(A) ! Q x2X Cu(Ax) donada per _hai = (ha(x)i)x2X; on Cu(Ax) és el semigrup de Cuntz de la fibra Ax. En el Capítol 3 de la memòria, l’aplicació _ s’estudia en el cas que X tingui dimensió petita i totes les fibres de la C(X)-àlgebra A no són necessàriament isomorfes entre sí. Més concretament, demostrem que és possible recuperar el semigrup de Cuntz d’una classe adequada de camps continus com el semigrup de seccions globals de tx2XCu(Ax) a X. Això s’utilitza posteriorment per reescriure un resultat de classificació degut a Dadarlat, Elliott i Niu (2012) utilitzant un sol invariant en comptes d’un feix de grups. (3)-Funcions de dimensió en una C*-algebra : L’estudi de funcions de dimensió va ser iniciat per Cuntz a 1978, i desenvolupat posteriorment per Blackadar i Handelman al 1982. En el seu article van aparèixer dues preguntes naturals: decidir si l’espai afí de funcions de dimensió és un símplex, i si també el conjunt de funcions de dimensió semicontínues inferiorment és dens a l’espai de totes les funcions de dimensió. En el Capítol 4 calculem el rang estable d’algunes classes de camps continus i això ens ajuda a provar que les dues conjectures anteriors tenen resposta afirmativa per camps continus A sobre espais de dimensió 1 i amb hipòtesis febles en les seves fibres. / This thesis deals with C*-algebras and their K-theoretical invariants. We have mainly focused on the structure of a class of C*-algebras called continuous fields, and the study of one of its invariants, the Cuntz semigroup. More concretely, we analyse the following: (1)-Structure of Continuous Fields of C*-algebras : In the literature there are two examples which clearly give an idea about the complexity of continuous field C*-algebras. The first one was constructed by M. Dadarlat and G. A. Elliott in 2007, and it is a continuous field C*- algebra A over the unit interval with mutually isomorphic fibers, with non-finitely generated K-theory and such that it is nowhere locally trivial. The second example shows that, even if the K-theory of the fibers vanish, the field can be nowhere locally trivial if the base space is infinite-dimensional (Dadarlat, 2009). From the above examples, it is natural to ask which is the structure of continuous fields of Kirchberg algebras over a finite-dimensional space with mutually isomorphic fibers and finitely generated K-theory. This question has been adressed in Chapter 2 of the memoir. (2)-The Cuntz semigroup of continuous field C*-algebras : For commutative C*-algebras of lower dimension where there are no cohomological obstructions, a description of their Cuntz semigroup via point evaluation has been obtained in terms of (extended) integer valued lower semicontinuous functions on their spectrum (Robert, 2009). For more general continuous fields, the key is to describe the map : Cu(A) ! Q x2X Cu(Ax) given by hai = (ha(x)i)x2X; where Cu(Ax) is the Cuntz semigroup of the fiber Ax. In Chapter 3 of the memoir, the map is studied in the case when X has low dimension and all the fibers of the C(X)-algebra A are not necessarily mutually isomorphic. Concretely, we prove that it is possible to recover the Cuntz semigroup of a suitable class of continuous fields as the semigroup of global sections of tx2XCu(Ax) to X. This is further used to rephrase a classification result by Dadarlat, Elliott and Niu (2012) by using a single invariant instead of a sheaf of groups. (3)-Dimension Functions on a C*-algebra : The study of dimension functions on C -algebras was started by Cuntz in 1978, and further developed by B. Blackadar and D. Handelman in 1982. In the latter article, two natural questions arised: to decide whether the affine space of dimension functions is a simplex, and also whether the set of lower semicontinuous dimension functions is dense in the space of all dimension functions. In Chapter 4 we compute the stable rank of some class of continuous field C*-algebras, which helps us to move on to show that the above two conjectures have affirmative answers for continuous fields A over one-dimensional spaces and with mild assumptions on their fibers.

Following the seminal work of Zhuang, connected Hopf algebras of finite GK-dimension over algebraically closed fields of characteristic zero have been the subject of several recent papers. This thesis is concerned with continuing this line of research and promoting connected Hopf algebras as a natural, intricate and interesting class of algebras. We begin by discussing the theory of connected Hopf algebras which are either commutative or cocommutative, and then proceed to review the modern theory of arbitrary connected Hopf algebras of finite GK-dimension initiated by Zhuang. We next focus on the (left) coideal subalgebras of connected Hopf algebras of finite GK-dimension. They are shown to be deformations of commutative polynomial algebras. A number of homological properties follow immediately from this fact. Further properties are described, examples are considered and invariants are constructed. A connected Hopf algebra is said to be "primitively thick" if the difference between its GK-dimension and the vector-space dimension of its primitive space is precisely one . Building on the results of Wang, Zhang and Zhuang, we describe a method of constructing such a Hopf algebra, and as a result obtain a host of new examples of such objects. Moreover, we prove that such a Hopf algebra can never be isomorphic to the enveloping algebra of a semisimple Lie algebra, nor can a semisimple Lie algebra appear as its primitive space. It has been asked in the literature whether connected Hopf algebras of finite GK-dimension are always isomorphic as algebras to enveloping algebras of Lie algebras. We provide a negative answer to this question by constructing a counterexample of GK-dimension 5. Substantial progress was made in determining the order of the antipode of a finite dimensional pointed Hopf algebra by Taft and Wilson in the 1970s. Our final main result is to show that the proof of their result can be generalised to give an analogous result for arbitrary pointed Hopf algebras.

We construct examples of non-semisimple tensor categories using planar algebras, with our main focus being on a construction from the restricted quantum group Ūq(sl_2). We describe the generators and prove a number of relations for the Ūq(sl_2) planar algebra, as well as describing diagrammatically various homomorphisms between modules, and conjecture a formula for projections onto indecomposable modules.

We study the fields of fractions and the Poisson spectra of polynomial Poisson algebras. First we investigate a Poisson birational equivalence problem for polynomial Poisson algebras over a field of arbitrary characteristic. Namely, the quadratic Poisson Gel'fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is isomorphic to the field of fractions of a Poisson affine space, i.e. a polynomial algebra such that the Poisson bracket of two generators is equal to their product (up to a scalar). We answer positively the quadratic Poisson Gel'fand-Kirillov problem for a large class of Poisson algebras arising as semiclassical limits of quantised coordinate rings, as well as for their quotients by Poisson prime ideals that are invariant under the action of a torus. In particular, we show that coordinate rings of determinantal Poisson varieties satisfy the quadratic Poisson Gel'fand-Kirillov problem. Our proof relies on the so-called characteristic-free Poisson deleting derivation homomorphism. Essentially this homomorphism allows us to simplify Poisson brackets of a given polynomial Poisson algebra by localising at a generator. Next we develop a method, the characteristic-free Poisson deleting derivations algorithm, to study the Poisson spectrum of a polynomial Poisson algebra. It is a Poisson version of the deleting derivations algorithm introduced by Cauchon [8] in order to study spectra of some noncommutative noetherian algebras. This algorithm allows us to define a partition of the Poisson spectrum of certain polynomial Poisson algebras, and to prove the Poisson Dixmier-Moeglin equivalence for those Poisson algebras when the base field is of characteristic zero. Finally, using both Cauchon's and our algorithm, we compare combinatorially spectra and Poisson spectra in the framework of (algebraic) deformation theory. In particular we compare spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.

No description available.

Let S and G be finite groups of coprime order such that S acts on G. If S is solvable, Glauberman [11] proves the existence of a bijection between the S-fixed irreducible representations of G and the irreducible representations of Gs. In the case of G solvable, Isaacs [13] uses a totally different method to prove the existence of a bijection between the same two sets of representations. Assuming the existence of the Glauberman correspondence, Boltje [5] uses the method of Explicit Brauer Induction (EBI) to give an explicit version of this correspondence for the case in which S is a p-group. After presenting the above results, we outline a strategy for investigating these correspondences using Explicit Brauer Induction, and we use these ideas to give a new proof for the theorems of Glauberman and Boltje. We move on to suggest some ideas of how this work may extend to Isaacs' correspondence. We also mention a link to Shintani's correspondence [25]. In the final chapter, we look at cryptography, and mention a potential application of some of our techniques (Adams Operations) in this field.

The Morita Frobenius number of an algebra is the number of Morita equivalence classes of its Frobenius twists. Introduced by Kessar in 2004, these numbers are important in the context of Donovan's conjecture for blocks of finite group algebras. Let P be a finite ℓ-group. Donovan's conjecture states that there are finitely many Morita equivalence classes of blocks of finite group algebras with defect groups isomorphic to P. Kessar proved that Donovan's conjecture holds if and only if Weak Donovan's conjecture and the Rationality conjecture hold. Our thesis relates to the Rationality conjecture, which states that there exists a bound on the Morita Frobenius numbers of blocks of finite group algebras with defect groups isomorphic to P, which depends only on SPS. In this thesis we calculate the Morita Frobenius numbers, or produce a bound for the Morita Frobenius numbers, of many of the blocks of quasi-simple finite groups. We also discuss the issues faced in the outstanding blocks and outline some possible approaches to solving these cases.

This thesis is a collection of theorems which say something about the following question: if we know that a bounded operator on a commutative unital Banach algebra is a unital endomorphism, what can we say about its other properties? More specifically, the majority of results say something about how the spectrum of a commutative unital Banach algebra endomorphism is dependent upon the properties of the algebra on which it acts. The main result of the thesis (the subject of Chapter 3) reveals that primary ideals (that is, ideals with single point hulls) can sometimes be particularly important in questions of this type. The thesis also contains some contributions to the Fredholm theory for bounded operators on an arbitrary complex Banach space. The second major result of the thesis is in this direction, and concerns the relationship between the essential spectrum of a bounded operator on a Banach space and those of its restrictions and quotients - `to' and `by' - closed invariant subspaces.

In the first part of this thesis, we give a survey of the conjugacy problem in the braid group, describing the solution provided by Garside theory, and outlining the progress that has been made towards a polynomial time solution in recent years using refinements of Garside's solution, and the Thurston-Nielsen classification of braids, which reduces the problem to the case of pseudo-Anosov braids. Using the faithful Lawrence-Krammer-Bigelow representation of the braid groups, we consider how the eigenspaces of pseudo-Anosov braids can under certain conditions yield invariants of their conjugacy class and thus lead us towards a polynomial time solution of the conjugacy problem. In the second part we introduce Bessis-Garside groups, a generalisation of the methods used by Bessis in his papers on dual braid monoids. We consider the groups given by taking the quotient of the free group by the orbits of its generators under the action of some subgroup of the braid group, and find that in many cases this construction can give us a group with a Garside structure. By means of introduction we review the simple rank 2 case, and summarise examples of such groups already known to admit Garside structures, in particular due to the work of Digne. We then go on to give all those of such groups which can be found as quotients of affine and spherical Artin groups of rank 3. We show that all such groups may be given a cycle presentation, or equivalently may be given as labelled-oriented-graph presented groups, and give conditions on such presentations that are equivalent to the group admitting a `dual' Garside structure. Restricting by the cycle lengths occurring in such presentations we give all Bessis-Garside groups of rank 3 which have all cycles length at most 4, and discuss the case of Bessis-Garside groups with uniform cycle length.