323 research outputs found

    Cycle and Cocycle Coverings of Graphs

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    Abstract: In this article, we show that for any simple, bridgeless graph G on n vertices, there is a family C of at most n−1 cycles which cover the edges of G at least twice. A similar, dual result is also proven for cocycles namely: for any loopless graph G on n vertices and edges having cogirth g * ≥ 3 and k(G) components, there is a family of at most −n+k(G) cocycles which cover the edges of G at least twice

    C*-algebras associated to coverings of k-graphs

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    A covering of k-graphs (in the sense of Pask-Quigg-Raeburn) induces an embedding of universal C*-algebras. We show how to build a (k+1)-graph whose universal algebra encodes this embedding. More generally we show how to realise a direct limit of k-graph algebras under embeddings induced from coverings as the universal algebra of a (k+1)-graph. Our main focus is on computing the K-theory of the (k+1)-graph algebra from that of the component k-graph algebras. Examples of our construction include a realisation of the Kirchberg algebra \mathcal{P}_n whose K-theory is opposite to that of \mathcal{O}_n, and a class of AT-algebras that can naturally be regarded as higher-rank Bunce-Deddens algebras.Comment: 44 pages, 2 figures, some diagrams drawn using picTeX. v2. A number of typos corrected, some references updated. The statements of Theorem 6.7(2) and Corollary 6.8 slightly reworded for clarity. v3. Some references updated; in particular, theorem numbering of references to Evans updated to match published versio

    Coverings of skew-products and crossed products by coactions

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    Consider a projective limit G of finite groups G_n. Fix a compatible family \delta^n of coactions of the G_n on a C*-algebra A. From this data we obtain a coaction \delta of G on A. We show that the coaction crossed product of A by \delta is isomorphic to a direct limit of the coaction crossed products of A by the \delta^n. If A = C*(\Lambda) for some k-graph \Lambda, and if the coactions \delta^n correspond to skew-products of \Lambda, then we can say more. We prove that the coaction crossed-product of C*(\Lambda) by \delta may be realised as a full corner of the C*-algebra of a (k+1)-graph. We then explore connections with Yeend's topological higher-rank graphs and their C*-algebras.Comment: 19 pages, laTeX. v2: Minor modifications to version 1. This version to appear in the Journal of the Australian Mathematical Society v3: some potentially confusing typos corrected in the proof of Theorem~3.1, as well as a few others. References update

    Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

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    Let S be a flat surface of genus g with cone type singularities. Given a bipartite graph G isoradially embedded in S, we define discrete analogs of the 2^{2g} Dirac operators on S. These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair (S,G) for these discrete Dirac operators to be Kasteleyn matrices of the graph G. As a consequence, if these conditions are met, the partition function of the dimer model on G can be explicitly written as an alternating sum of the determinants of these 2^{2g} discrete Dirac operators.Comment: 39 pages, minor change

    Surface bundles over surfaces: new inequalities between signature, simplicial volume and Euler characteristic

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    We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through a ramified covering. These are the main known examples of bundles with non-zero signature.Comment: 14 pages. Simplified the proof of Proposition 1.2. This is the final version, accepted in Geometriae Dedicat

    Homology for higher-rank graphs and twisted C*-algebras

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    We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C*-algebra of a k-graph by a T-valued 2-cocycle and demonstrate that examples include all noncommutative tori. In the appendices, we construct a cubical set \tilde{Q}(\Lambda) from a k-graph {\Lambda} and demonstrate that the homology and topological realisation of {\Lambda} coincide with those of \tilde{Q}(\Lambda) as defined by Grandis.Comment: 33 pages, 9 pictures and one diagram prepared in TiK

    Coverings of graded pointed Hopf algebras

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    We introduce the concept of a covering of a graded pointed Hopf algebra. The theory developed shows that coverings of a bosonized Nichols algebra can be concretely expressed by biproducts using a quotient of the universal coalgebra covering group of the Nichols algebra. If there are enough quadratic relations, the universal coalgebra covering is given by the bosonization by the enveloping group of the underlying rack.Comment: to appear in J. of Algebr
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