This thesis is split up into two parts: the first concerns certain applications of the de la Harpe-Skandalis determinant to K-theory of appropriately regular C*-algebras. The second is concerned with (unital) inclusions of C*-algebras which satisfy a strong tensorial absorption condition. The first chapter following the preliminary section is joint work with Aaron Tikuisis [ST23], while the following chapters are independent. The penultimate chapter is [Sar23b] and the last chapter is essentially [Sar23a]. In the first chapter following the preliminaries, we examine the interplay between the algebraic K₁-group and the unitary algebraic K₁-group of a unital C*-algebra. We prove that for an abundance of unital C*-algebras, the algebraic K₁-group splits naturally as a direct sum of the unitary algebraic K₁-group and the space of continuous real-valued affine functions on the trace simplex. We further prove that if one considers Hausdorffized variants, then for any unital C*-algebra, there is a natural splitting of the Hausdorffized algebraic K₁-group in terms of the Hausdorffized unitary algebraic K₁-group and the space of continuous real-valued affine functions on the trace simplex. Moreover, this a splitting of topological groups. The following chapter studies how certain group homomorphisms between unitary groups of C*-algebras induce maps on the trace simplex. In particular, we show that a contractive group homomorphism between unital C*-algebras which sends the circle to the circle, induces a map between their trace simplices. Under mild regularity conditions these further induce maps between Elliott invariants. As a consequence we show that certain inclusions of C*-algebras are in a correspondence with certain inclusions of unitary groups. Finally we investigate what we call "D-stable inclusions" of C*-algebras, where D is strongly self-absorbing. We give a systematic study and prove that such inclusions between unital, separable, D-stable C*-algebras exist, are abundant, and are non-trivial.

No description available.

We initiate the study of a new class of semigroup C*-algebras arising from number-theoretic considerations; namely, we generalize the construction of Cuntz, Deninger, and Laca by considering the left regular C*-algebras of ax+b-semigroups from actions of congruence monoids on rings of algebraic integers in number fields. Our motivation for considering actions of congruence monoids comes from class field theory and work on Bost–Connes type systems. We give two presentations and a groupoid model for these algebras, and establish a faithfulness criterion for their representations. We then explicitly compute the primitive ideal space, give a semigroup crossed product description of the boundary quotient, and prove that the construction is functorial in the appropriate sense. These C*-algebras carry canonical time evolutions, so that our construction also produces a new class of C*-dynamical systems. We classify the KMS (equilibrium) states for this canonical time evolution, and show that there are several phase transitions whose complexity depends on properties of a generalized ideal class group. We compute the type of all high temperature KMS states, and consider several related C*-dynamical systems. / Graduate

Suppose L is a complete separable metric topological group (ring, field, etc.). L is topologically unique if the Polish topology on L is uniquely determined by its underlying algebraic structure. More specifically, L is topologically unique if an algebraic isomorphism of L with any other complete separable metric topological group (ring, field, etc.) induces a topological isomorphism. A local field is a locally compact topological field with non-discrete topology. The only local fields (up to isomorphism) are the real, complex, and p-adic numbers, finite extensions of the p-adic numbers, and fields of formal power series over finite fields. We establish the topological uniqueness of the special linear Lie algebras over local fields other than the complex numbers (for which this result is not true) in the context of complete separable metric Lie rings. Along the way the topological uniqueness of all local fields other than the field of complex numbers is established, which is derived as a corollary to more general principles which can be applied to a larger class of topological fields. Lastly, also in the context of complete separable metric Lie rings, the topological uniqueness of the special linear Lie algebra over the real division algebra of quaternions, the special orthogonal Lie algebras, and the special unitary Lie algebras is proved.

In this thesis we study gradings on blocks of group algebras. The motivation to study gradings on blocks of group algebras and their transfer via derived and stable equivalences originates from some of the most important open conjectures in representation theory, such as Broue’s abelian defect group conjecture. This conjecture predicts the existence of derived equivalences between categories of modules. Some attempts to prove Broue’s conjecture by lifting stable equivalences to derived equivalences highlight the importance of understanding the connection between transferring gradings via stable equivalences and transferring gradings via derived equivalences. The main idea that we use is the following. We start with an algebra which can be easily graded, and transfer this grading via derived or stable equivalence to another algebra which is not easily graded. We investigate the properties of the resulting grading. In the first chapter we list the background results that will be used in this thesis. In the second chapter we study gradings on Brauer tree algebras, a class of algebras that contains blocks of group algebras with cyclic defect groups. We show that there is a unique grading up to graded Morita equivalence and rescaling on an arbitrary basic Brauer tree algebra. The third chapter is devoted to the study of gradings on tame blocks of group algebras. We study extensively the class of blocks with dihedral defect groups. We investigate the existence, positivity and tightness of gradings, and we classify all gradings on these blocks up to graded Morita equivalence. The last chapter deals with the problem of transferring gradings via stable equivalences between blocks of group algebras. We demonstrate on three examples how such a transfer via stable equivalences is achieved between Brauer correspondents, where the group in question is a TI group.

Orientador: Waldyr Alves Rodrigues Junior / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T23:09:32Z (GMT). No. of bitstreams: 1 Resende_AdrianaSouza_M.pdf: 17553204 bytes, checksum: a66cefe30e9957cc4351e03d3aec35b2 (MD5) Previous issue date: 2010 / Resumo: O presente trabalho tem a intenção de apresentar por intermédio de uma linguagem unificada alguns conceitos de cálculo vetorial, álgebra linear (matrizes e transformações lineares) e também algumas idéias elementares sobre os grupos de rotações em duas e três dimensões e seus grupos de recobrimento, que geralmente são tratados como "fragmentos" em várias modalidades de cursos no ensino superior. Acreditamos portanto que nosso texto possas ser útil para alunos dos cursos de graduação dos cursos de Engenharia, Física, Matemática e interessados em Matemática em geral. A linguagem unificada à que nos referimos acima é obtida com a introdução do conceitos das álgebras geométricas (ou de Clifford) onde, como veremos, é possível fornecer uma formulação algébrica elegante aos conceitos de vetores, planos e volumes orientados e definir para tais objetos o produto escalar, os produtos contraídos à esquerda e à direita, o produto exterior (associado, como veremos, em casos particulares ao produto vetorial) e finalmente o produto geométrico (Clifford), o que permite o uso desses conceitos para a solução de inúmeros problemas de geometria analítica no R ² e no R ³. Procuramos ilustrar todos estes conceitos com vários exemplos e exercícios com graus variáveis de dificuldades. Nossa apresentação é bem próxima àquela do livro de Lounesto, e de fato muitas seções são traduções (eventualmente seguidas de comentários) de seções daquele livro. Contudo, em muitos lugares, acreditamos que nossa apresentação esclarece e completa as correspondentes do livro de Lounesto / Abstract: This paper aims to present using an unified language a few concepts of vector calculus, linear algebra (matrices and linear transformations) and also some basic ideas about the groups of rotations in two and three dimensions and their covering group, which generally are treated as "fragments" in various types of courses in higher education. We believe therefore that our text should be useful to students of undergraduate courses like Engineering, Physics, Mathematics and people interested in Mathematics in general. The unified language that we refer to above is obtained by introducing the concept of geometric (or Clifford) algebra where, as we shall see, it is possible to give an elegant algebraic formulation to the concepts of vectors, oriented planes and oriented volumes, and to define to those objects the scalar product, the right and left contracted products, the exterior product (associated, as we shall see, in particular cases to the vector product) and finally the geometric (Clifford) product, and moreover, to use those concepts to solve may problems of analytic geometry in R ² and R ³. We illustrated all those concepts with several examples and exercises with variable degrees of difficulties. Our presentation is nearly the one in Lounesto's book, and in fact some sections are no more than translations (eventually with commentaries) from sections of that book. However, in many places, we believe that our presentation clarify nd completement the corresponding ones in Lounesto's book / Mestrado / Ágebra / Mestre em Matemática

Cette thèse est consacrée à l'étude des algèbres de Hecke carquois et de certaines généralisations des algèbres d'Iwahori-Hecke. Dans un premier temps, nous montrons deux résultats concernant les algèbres de Hecke carquois, dans le cas où le carquois possède plusieurs composantes connexes puis lorsqu'il possède un automorphisme d'ordre fini. Ensuite, nous rappelons un isomorphisme de Brundan-Kleshchev et Rouquier entre algèbres d'Ariki-Koike et certaines algèbres de Hecke carquois cyclotomiques. D'une part nous en déduisons qu'une équivalence de Morita importante bien connue entre algèbres d'Ariki-Koike provient d'un isomorphisme, d'autre part nous donnons une présentation de type Hecke carquois cyclotomique pour l'algèbre de Hecke de G(r,p,n). Nous généralisons aussi l'isomorphisme de Brundan-Kleshchev pour montrer que les algèbres de Yokonuma-Hecke cyclotomiques sont des cas particuliers d'algèbres de Hecke carquois cyclotomiques. Finalement, nous nous intéressons à un problème de combinatoire algébrique, relié à la théorie des représentations des algèbres d'Ariki-Koike. En utilisant la représentation des partitions sous forme d'abaque et en résolvant, via un théorème d'existence de matrices binaires, un problème d'optimisation convexe sous contraintes à variables entières, nous montrons qu'un multi-ensemble de résidus qui est bégayant provient nécessairement d'une multi-partition bégayante. / This thesis is devoted to the study of quiver Hecke algebras and some generalisations of Iwahori-Hecke algebras. We begin with two results concerning quiver Hecke algebras, first when the quiver has several connected components and second when the quiver has an automorphism of finite order. We then recall an isomorphism of Brundan-Kleshchev and Rouquier between Ariki-Koike algebras and certain cyclotomic quiver Hecke algebras. From this, on the one hand we deduce that a well-known important Morita equivalence between Ariki--Koike algebras comes from an isomorphism, on the other hand we give a cyclotomic quiver Hecke-like presentation for the Hecke algebra of type G(r,p,n). We also generalise the isomorphism of Brundan-Kleshchev to prove that cyclotomic Yokonuma-Hecke algebras are particular cases of cyclotomic quiver Hecke algebras. Finally, we study a problem of algebraic combinatorics, related to the representation theory of Ariki-Koike algebras. Using the abacus representation of partitions and solving, via an existence theorem for binary matrices, a constrained optimisation problem with integer variables, we prove that a stuttering multiset of residues necessarily comes from a stuttering multipartition.

We begin by giving some theoretical background to the underlying concepts of spin representations and spinors. This is done from the perspective of Lie groups and Lie algebras. In particular, we discuss the functionality of Clifford algebras in the determination of the double-covering spin groups. An introduction to K-algebras and Clifford algebras is then given, focusing on the properties of pseudo-Euclidean spaces <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BR%7D%5E%7Bp,q%7D" data-classname="equation" data-title="" />. Some low-dimensional examples are also included, culminating with a characterisation of some Clifford algebras as matrix algebras. Elementary representation theory is then introduced and quickly followed by the definition of the Clifford-Lipschitz and spin groups. The work of Lundholm and Svensson (2016), Vaz and da Rocha (2016), and Schwichtenberg (2018) is then united to construct a definition of the spin representations. An attempt at formulating a definition of spinors from a mathematical perspective is then given; formed by combining multiple approaches and definitions of the above-mentioned authors, as well as drawing inspiration from important cases in theoretical physics, in particular that of SO(3) and the Lorentz group SO(1,3).

Doctor of Philosophy / Department of Mathematics / Yan S. Soibelman / Recall the classical Riemann-Roch theorem for curves: Given a smooth projective complex curve and two holomorphic vector bundles E, F on it, the Euler can be computed in terms of the ranks and the degrees of the vector bundles. Remarkably, there are a number of similarly looking formulas in algebra. The simplest example is the Ringel formula in the theory of quivers. It expresses the Euler form of two finite-dimensional representations of a quiver algebra in terms of a certain pairing of their dimension vectors. The existence of Riemann-Roch type formulas in these two settings is a consequence of a deeper similarity in the structure of the corresponding derived categories - those of sheaves on curves and of modules over quiver algebras. The thesis is devoted to a version of the Riemann-Roch formula for abstract derived categories. By the latter we understand the derived categories of differential graded (DG) categories. More specifically, we work with the categories of perfect modules over DG algebras. These are a simultaneous generalization of the derived categories of modules over associative algebras and the derived categories of schemes. Given an arbitrary DG algebra A, satisfying a certain finiteness condition, we define and explicitly describe a canonical pairing on its Hochschild homology. Then we give an explicit formula for the Euler character of an arbitrary perfect A-module, the character is an element of the Hochschild homology of A. In this setting, our noncommutative Riemann-Roch formula expresses the Euler characteristic of the Hom-complex between any two perfect A-modules in terms of the pairing of their Euler characters. One of the main applications of our results is a theorem that the aforementioned pairing on the Hochschild homology is non-degenerate when the DG algebra satisfies a smoothness condition. This theorem implies a special case of the well-known noncommutative Hodge-to-de Rham degeneration conjecture. Another application is related to mathematical physics: We explicitly construct an open-closed topological field theory from an arbitrary Frobenius algebra and then, following ideas of physicists, interpret the noncommutative Riemann-Roch formula as a special case of the so-called topological Cardy condition.

Thesis (MSc)--University of Stellenbosch, 2003. / ENGLISH ABSTRACT: Let A and B be unital complex Banach algebras with identities 1 and I' respectively. A linear map T : A -+ B is invertibility preserving if Tx is invertible in B for every invertible x E A. We say that T is unital if Tl = I'. IfTx2 = (TX)2 for all x E A, we call T a Jordan homomorphism. We examine an unsolved problem posed by 1. Kaplansky: Let A and B be unital complex Banach algebras and T : A -+ B a unital invertibility preserving linear map. What conditions on A, Band T imply that T is a Jordan homomorphism? Partial motivation for this problem are the Gleason-Kahane-Zelazko Theorem (1968) and a result of Marcus and Purves (1959), these also being special instances of the problem. We will also look at other special cases answering Kaplansky's problem, the most important being the result stating that if A is a von Neumann algebra, B a semi-simple Banach algebra and T : A -+ B a unital bijective invertibility preserving linear map, then T is a Jordan homomorphism (B. Aupetit, 2000). For a unital complex Banach algebra A, we denote the spectrum of x E A by Sp (x, A). Let a(x, A) denote the union of Sp (x, A) and the bounded components of <C \ Sp (x, A). We denote the spectral radius of x E A by p(x, A). A unital linear map T between unital complex Banach algebras A and B is invertibility preserving if and only if Sp (Tx, B) C Sp (x, A) for all x E A. This leads one to consider the problems that arise when, in turn, we replace the invertibility preservation property of T in Kaplansky's problem with Sp (Tx, B) = Sp (x, A) for all x E A, a(Tx, B) = a(x, A) for all x E A, and p(Tx, B) = p(x, A) for all x E A. We will also investigate some special cases that are solutions to these problems. The most important of these special cases says that if A is a semi-simple Banach algebra, B a primitive Banach algebra with minimal ideals and T : A -+ B a surjective linear map satisfying a(Tx, B) = a(x, A) for all x E A, then T is a Jordan homomorphism (B. Aupetit and H. du T. Mouton, 1994). / AFRIKAANSE OPSOMMING: Gestel A en B is unitale komplekse Banach algebras met identiteite 1 en I' onderskeidelik. 'n Lineêre afbeelding T : A -+ B is omkeerbaar-behoudend as Tx omkeerbaar in B is vir elke omkeerbare element x E A. Ons sê dat T unitaal is as Tl = I'. As Tx2 = (TX)2 vir alle x E A, dan noem ons T 'n Jordan homomorfisme. Ons ondersoek 'n onopgeloste probleem wat deur I. Kaplansky voorgestel is: Gestel A en B is unitale komplekse Banach algebras en T : A -+ B is 'n unitale omkeerbaar-behoudende lineêre afbeelding. Watter voorwaardes op A, B en T impliseer dat T 'n Jordan homomorfisme is? Gedeeltelike motivering vir hierdie probleem is die Gleason-Kahane-Zelazko Stelling (1968) en 'n resultaat van Marcus en Purves (1959), wat terselfdertyd ook spesiale gevalle van die probleem is. Ons salook na ander spesiale gevalle kyk wat antwoorde lewer op Kaplansky se probleem. Die belangrikste van hierdie resultate sê dat as A 'n von Neumann algebra is, B 'n semi-eenvoudige Banach algebra is en T : A -+ B 'n unitale omkeerbaar-behoudende bijektiewe lineêre afbeelding is, dan is T 'n Jordan homomorfisme (B. Aupetit, 2000). Vir 'n unitale komplekse Banach algebra A, dui ons die spektrum van x E A aan met Sp (x, A). Laat cr(x, A) die vereniging van Sp (x, A) en die begrensde komponente van <C \ Sp (x, A) wees. Ons dui die spektraalradius van x E A aan met p(x, A). 'n Unitale lineêre afbeelding T tussen unit ale komplekse Banach algebras A en B is omkeerbaar-behoudend as en slegs as Sp (Tx, B) c Sp (x, A) vir alle x E A. Dit lei ons om die probleme te beskou wat ontstaan as ons die omkeerbaar-behoudende eienskap van T in Kaplansky se probleem vervang met Sp (Tx, B) = Sp (x, A) vir alle x E A, O"(Tx, B) = O"(x, A) vir alle x E A en p(Tx, B) = p(x, A) vir alle x E A, onderskeidelik. Ons salook 'n paar spesiale gevalle van hierdie probleme ondersoek. Die belangrikste van hierdie spesiale gevalle sê dat as A 'n semi-eenvoudige Banach algebra is, B 'n primitiewe Banach algebra met minimale ideale is, en T : A -+ B 'n surjektiewe lineêre afbeelding is sodanig dat O"(Tx, B) = O"(x, A) vir alle x E A, dan is T 'n Jordan homomorfisme (B. Aupetit en H. du T. Mouton, 1994).