Research Doctorate - Doctor of Philosophy (PhD) / There is a strong connection between directed graphs and the shifts of finite type which are an important family of dynamical systems. Higher-rank graphs (or k-graphs) and their C*-algebras were introduced by Kumjian and Pask to generalise directed graphs and their C*-algebras. Kumjian and Pask showed how higher-dimensional shifts of finite type can be associated to k-graphs, but did not discuss how one might associate k-graphs to k-dimensional shifts of finite type. In this thesis we construct a family of 2-graphs A arising from a certain type of algebraic two-dimensional shift of finite type studied by Schmidt, and analyse the structure of their C*-algebras. Graph algebras and k-graph algebras provide a rich source of examples for the classication of simple, purely infinite, nuclear C*-algebras. We give criteria which ensure that the C*-algebra C*(A) is simple, purely infinite, nuclear, and satisfies the hypotheses of the Kirchberg-Phillips Classification Theorem. We perform K-theory calculations for a wide range of our 2-graphs A using the Magma computational algebra system. The results of our calculations lead us to conjecture that the K-groups of C*(A) are finite cyclic groups of the same order. We are able to prove under mild hypotheses that the K-groups have the same order, but we have only numerical evidence to suggest that they are cyclic. In particular, we find several examples for which K1(C*(A)) is nonzero and has torsion, hence these are examples of 2-graph C*-algebras which do not arise as the C*-algebras of directed graphs. Finally, we consider a subfamily of 2-graphs with interesting combinatorial connections. We identify the nonsimple C*-algebras of these 2-graphs and calculate their K-theory.

Research Doctorate - Doctor of Philosophy (PhD) / Directed graphs are combinatorial objects used to model networks like fluid-flow systems in which the direction of movement through the network is important. In 1980, Enomoto and Watatani used finite directed graphs to provide an intuitive framework for the Cuntz-Krieger algebras introduced by Cuntz and Krieger earlier in the same year. The theory of the C*-algebras of directed graphs has since been extended to include infinite graphs, and there is an elegant relationship between connectivity and loops in a graph and the structure theory of the associated C*-algebra. Higher-rank graphs are a higher-dimensional analogue of directed graphs introduced by Kumjian and Pask in 2000 as a model for the higher-rank Cuntz-Krieger algebras introduced by Robertson and Steger in 1999. The theory of the Cuntz-Krieger algebras of higher-rank graphs is relatively new, and a number of questions which have been answered for directed graphs remain open in the higher-rank setting. In particular, for a large class of higher-rank graphs, the gauge-invariant ideal structure of the associated C*-algebra has not yet been identified. This thesis addresses the question of the gauge-invariant ideal structure of the Cuntz-Krieger algebras of higher-rank graphs. To do so, we introduce and analyse the collections of relative Cuntz-Krieger algebras associated to higher-rank graphs. The first two main results of the thesis are versions of the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem which apply to relative Cuntz-Krieger algebras. Using these theorems, we are able to achieve our main goal, producing a classification of the gauge-invariant ideals in the Cuntz-Krieger algebra of a higher-rank graph analogous to that developed for directed graphs by Bates, Hong, Raeburn and Szymañski in 2002. We also demonstrate that relative Cuntz-Krieger algebras associated to higher-rank graphs are always nuclear, and produce conditions on a higher-rank graph under which the associated Cuntz-Krieger algebra is simple and purely infinite.

No description available.

abstract: In this thesis, I investigate the C*-algebras and related constructions that arise from combinatorial structures such as directed graphs and their generalizations. I give a complete characterization of the C*-correspondences associated to directed graphs as well as results about obstructions to a similar characterization of these objects for generalizations of directed graphs. Viewing the higher-dimensional analogues of directed graphs through the lens of product systems, I give a rigorous proof that topological k-graphs are essentially product systems over N^k of topological graphs. I introduce a "compactly aligned" condition for such product systems of graphs and show that this coincides with the similarly-named conditions for topological k-graphs and for the associated product systems over N^k of C*-correspondences. Finally I consider the constructions arising from topological dynamical systems consisting of a locally compact Hausdorff space and k commuting local homeomorphisms. I show that in this case, the associated topological k-graph correspondence is isomorphic to the product system over N^k of C*-correspondences arising from a related Exel-Larsen system. Moreover, I show that the topological k-graph C*-algebra has a crossed product structure in the sense of Larsen. / Dissertation/Thesis / Ph.D. Mathematics 2011

An algebraic structure is finitely related if its clone is determined by a finite set of finitary relations. In this thesis we examine graph algebras in order to determine which of them have this property. We provide a brief sum- mary of a background theory and we present an overview of known results, in particular, we emphasize the relation between finitely related algebras and Mal'cev conditions. Further we present basic results about the structure of graph algebras. The main part of this thesis is a partial classification of finitely related graph algebras. We provide proofs for various classes of graph algebras, for example for algebras defined by connected bipartite graphs or algebras de- fined by graphs containing certain subgraphs, although several cases are missing to complete the classification. 1

No description available.

Les algèbres L(g,n,H) ont été introduites par Alekseev-Grosse-Schomerus et Buffenoir-Roche au milieu des années 1990, dans le cadre de la quantification combinatoire de l'espace de modules des G-connexions plates sur la surface S(g,n) de genre g avec n disques ouverts enlevés. L'algèbre de Hopf H, appelée algèbre de jauge, était à l'origine le groupe quantique U_q(g), avec g=Lie(G). Dans cette thèse nous appliquons les algèbres L(g,n,H) à la topologie en basses dimensions (groupe de difféotopie et algèbres d'écheveaux des surfaces), sous l'hypothèse que H est une algèbre de Hopf de dimension finie, factorisable et enrubannée mais pas nécessairement semi-simple, l'exemple phare d'une telle algèbre de Hopf étant le groupe quantique restreint associé à sl(2) (à une racine 2p-ième de l'unité). D'abord, nous construisons en utilisant L(g,n,H) une représentation projective des groupes de difféotopie de S(g,0)D et de S(g,0) (où D est un disque ouvert). Nous donnons des formules pour les représentations d'un ensemble de twists de Dehn qui engendre le groupe de difféotopie; en particulier ces formules nous permettent de montrer que notre représentation est équivalente à celle construite par Lyubashenko-Majid et Lyubashenko via des méthodes catégoriques. Pour le tore S(1,0) avec le groupe quantique restreint associé à sl(2) comme algèbre de jauge, nous calculons explicitement la représentation de SL(2,Z) en utilisant une base convenable de l'espace de représentation et nous en déterminons la structure.Ensuite, nous introduisons une description diagrammatique de L(g,n,H) qui nous permet de définir de façon très naturelle l'application boucle de Wilson W. Cette application associe un élément de L(g,n,H) à chaque entrelac dans (S(g,n)D) x [0,1] qui est parallélisé, orienté et colorié par des H-modules. Quand l'algèbre de jauge est le groupe quantique restreint associé à sl(2), nous utilisons W et les représentations de L(g,n,H) pour construire des représentations des algèbres d'écheveaux S_q(S(g,n)). Pour le tore S(1,0) nous étudions explicitement cette représentation. / The algebras L(g,n,H) have been introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the middle of the 1990's, in the program of combinatorial quantization of the moduli space of flat G-connections over the surface S(g,n) of genus g with n open disks removed. The Hopf algebra H, called gauge algebra, was originally the quantum group U_q(g), with g = Lie(G). In this thesis we apply these algebras L(g,n,H) to low-dimensional topology (mapping class groups and skein algebras of surfaces), under the assumption that H is a finite dimensional factorizable ribbon Hopf algebra which is not necessarily semisimple, the guiding example of such a Hopf algebra being the restricted quantum group associated to sl(2) (at a 2p-th root of unity).First, we construct from L(g,n,H) a projective representation of the mapping class groups of S(g,0)D and of S(g,0) (D being an open disk). We provide formulas for the representations of Dehn twists generating the mapping class group; in particular these formulas allow us to show that our representation is equivalent to the one constructed by Lyubashenko-Majid and Lyubashenko via categorical methods. For the torus S(1,0) with the restricted quantum group associated to sl(2) for the gauge algebra, we compute explicitly the representation of SL(2,Z) using a suitable basis of the representation space and we determine the structure of this representation.Second, we introduce a diagrammatic description of L(g,n,H) which enables us to define in a very natural way the Wilson loop map W. This maps associates an element of L(g,n,H) to any link in (S(g,n)D) x [0,1] which is framed, oriented and colored by H-modules. When the gauge algebra is the restricted quantum group associated to sl(2), we use W and the representations of L(g,n,H) to construct representations of the skein algebras S_q(S(g,n)). For the torus S(1,0) we explicitly study this representation.