323 research outputs found
Cycle and Cocycle Coverings of Graphs
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
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
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
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
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
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
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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