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Journal ArticleDOI

Differential geometry of group lattices

20 Mar 2003-Journal of Mathematical Physics (American Institute of Physics)-Vol. 44, Iss: 4, pp 1781-1821
TL;DR: In this article, it was shown that first-order differential calculi over the algebra of functions on a discrete set are in bijective correspondence with digraph structures where the vertices are given by the elements of the set.
Abstract: In a series of publications we developed “differential geometry” on discrete sets based on concepts of noncommutative geometry. In particular, it turned out that first-order differential calculi (over the algebra of functions) on a discrete set are in bijective correspondence with digraph structures where the vertices are given by the elements of the set. A particular class of digraphs are Cayley graphs, also known as group lattices. They are determined by a discrete group G and a finite subset S. There is a distinguished subclass of “bicovariant” Cayley graphs with the property ad(S)S⊂S. We explore the properties of differential calculi which arise from Cayley graphs via the above correspondence. The first-order calculi extend to higher orders and then allow us to introduce further differential geometric structures. Furthermore, we explore the properties of “discrete” vector fields which describe deterministic flows on group lattices. A Lie derivative with respect to a discrete vector field and an inner ...
Citations
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Book ChapterDOI
01 Jan 1989

1,062 citations

Proceedings ArticleDOI
10 Mar 2008
TL;DR: In this paper, the authors briefly review ideas about noncommutativity of space-time and approaches toward a corresponding theory of gravity, and propose a non-commutative theory for gravity.
Abstract: We briefly review ideas about “noncommutativity of space‐time” and approaches toward a corresponding theory of gravity.

24 citations

Posted Content
TL;DR: In this paper, the authors study the discrete analogue of a pseudo-Riemannian metric on topologically hypercubic graphs with respect to which all edges are light-like.
Abstract: Differential calculus on discrete spaces is studied in the manner of non-commutative geometry by representing the differential calculus by an operator algebra on a suitable Krein space. The discrete analogue of a (pseudo-)Riemannian metric is encoded in a deformation of the inner product on that space, which is the crucial technique of this paper. We study the general case but find that drastic and possibly vital simplifications occur when the underlying lattice is topologically hypercubic, in which case we explicitly construct mimetic analogues of the volume form, the Hodge star operator, and the Hodge inner product for arbitrary discrete geometries. It turns out that the formalism singles out a pseudo-Riemannian metric on topologically hypercubic graphs with respect to which all edges are lightlike. We study such causal graph complexes in detail and consider some of their possible physical applications, such as lattice Yang-Mills theory and lattice fermions.

21 citations

Journal ArticleDOI
TL;DR: The difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space and the difference operators as the discrete counterparts of the derivatives based upon the differential calculus on the lattice are defined.
Abstract: In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference operators as the discrete counterparts of the derivatives based upon the differential calculus on the lattice. One of the definitions can be extended to the case over the random lattice. We also discuss the relation between our approach and the lattice gauge theory and apply to the discrete integrable systems.

17 citations


Cites background or result from "Differential geometry of group latt..."

  • ...in a way different from other relevant proposals (see, e.g. [19], [20], [21], [22],[ 23 ])....

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  • ...[21], [22],[ 23 ]) the key point is different....

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  • ...[18] and others (see, e.g. [20], [21], [22], [ 23 ], [15], [16])....

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01 Jan 2006
TL;DR: In this paper, the difference discrete connection and curvature on discrete vector bundle over the regular lattice are defined in different but equivalent manners, and the relation between their approach and the lattice gauge theory is discussed.
Abstract: In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference operators as the discrete counterparts of the derivatives based upon the differential calculus on the lattice. One of the definitions can be extended to the case over the random lattice. We also discuss the relation between our approach and the lattice gauge theory and apply to the discrete integrable systems.

15 citations

References
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Book
01 Jan 1998
TL;DR: This book presents an account of newer topics, including Szemer'edi's Regularity Lemma and its use; Shelah's extension of the Hales-Jewett Theorem; the precise nature of the phase transition in a random graph process; the connection between electrical networks and random walks on graphs; and the Tutte polynomial and its cousins in knot theory.
Abstract: The time has now come when graph theory should be part of the education of every serious student of mathematics and computer science, both for its own sake and to enhance the appreciation of mathematics as a whole. This book is an in-depth account of graph theory, written with such a student in mind; it reflects the current state of the subject and emphasizes connections with other branches of pure mathematics. The volume grew out of the author's earlier book, Graph Theory -- An Introductory Course, but its length is well over twice that of its predecessor, allowing it to reveal many exciting new developments in the subject. Recognizing that graph theory is one of several courses competing for the attention of a student, the book contains extensive descriptive passages designed to convey the flavor of the subject and to arouse interest. In addition to a modern treatment of the classical areas of graph theory such as coloring, matching, extremal theory, and algebraic graph theory, the book presents a detailed account of newer topics, including Szemer\'edi's Regularity Lemma and its use, Shelah's extension of the Hales-Jewett Theorem, the precise nature of the phase transition in a random graph process, the connection between electrical networks and random walks on graphs, and the Tutte polynomial and its cousins in knot theory. In no other branch of mathematics is it as vital to tackle and solve challenging exercises in order to master the subject. To this end, the book contains an unusually large number of well thought-out exercises: over 600 in total. Although some are straightforward, most of them are substantial, and others will stretch even the most able reader.

3,751 citations


"Differential geometry of group latt..." refers background in this paper

  • ... of 1-forms is, in general, larger than the A-bimodule generated by the image of the space of functions under the action of the exterior derivative. There is a generalized digraph (“Schreier diagram” [16]) associated with such a first order differential calculus which in general has multiple links and also loops. Some further remarks are collected in section 9. 2 First order differential calculus associa...

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  • ...set Kto a point representing a coset K′ whenever there is an h ∈S such that K′ = Kh, we obtain a digraph. This coset digraph [47] is also known as the Schreier diagram of the triple (G,S,H) (see Ref. [16], for example). In contrast to the digraphs (group lattices) considered in the previous sections, coset digraphs may have multiple arrows between two sites and even loops (i.e. arrows from a site to i...

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  • ...a differential calculus which corresponds to the hypercubic lattice and which leads to an elegant formulation of lattice gauge theory [1]. A special class of digraphs are Cayley graphs [15] (see Refs. [16,17], for example), which are also known as group lattices in the physics literature. These are determined by a discrete group Gand a subset S. The elements of Gare the vertices of the digraph and the ele...

    [...]

  • ...al first order differential calculus on the two disjoint parts of S 3 in this case. Another choice is S= {(12),(123)}. The corresponding digraph is shown on the righthand side of Fig. 2 (see also Refs. [16,17]). We call a group lattice bicovariant if ad(S)S⊂S. The significance of this definition will be made clear in section 3. Our previous examples of group lattices are indeed bicovariant, except for (S 3,S...

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Book
01 Jun 1977
TL;DR: In this article, the authors introduce the concept of Free Products with Amalgamation (FPAM) and Small Cancellation Theory over free products with amalgamation and HNN extensions.
Abstract: Chapter I. Free Groups and Their Subgroups 1. Introduction 2. Nielsen's Method 3. Subgroups of Free Groups 4. Automorphisms of Free Groups 5. Stabilizers in Aut(F) 6. Equations over Groups 7. Quadratic sets of Word 8. Equations in Free Groups 9. Abstract Lenght Functions 10. Representations of Free Groups 11. Free Pruducts with Amalgamation Chapter II. Generators and Relations 1. Introduction 2. Finite Presentations 3. Fox Calculus, Relation Matrices, Connections with Cohomology 4. The Reidemeister-Schreier Method 5. Groups with a Single Defining Relator 6. Magnus' Treatment of One-Relator Groups Chapter III. Geometric Methods 1. Introduction 2. Complexes 3. Covering Maps 4. Cayley Complexes 5. Planar Caley Complexes 6. F-Groups Continued 7. Fuchsian Complexes 8. Planar Groups with Reflections 9. Singular Subcomplexes 10. Sherical Diagrams 11. Aspherical Groups 12. Coset Diagrams and Permutation Representations 13. Behr Graphs Chpter IV. Free Products and HNN Extensions 1. Free Products 2. Higman-Neumann-Neumann Extensions and Free Products with Amalgamation 3. Some Embedding Theorems 4. Some Decision Problems 5. One-Relator Groups 6. Bipolar Structures 7. The Higman Embedding Theorem 8. Algebraically Closed Groups Chapter V. Small Cancellation Theory 1. Diagrams 2. The Small Cancellation Hypotheses 3. The Basic Formulas 4. Dehn's Algorithm and Greendlinger's Lemma 5. The Conjugacy Problem 6. The Word Problem 7. The Cunjugacy Problme 8. Applications to Knot Groups 9. The Theory over Free Products 10. Small Cancellation Products 11. Small Cancellation Theory over free Products with Amalgamation and HNN Extensions Bibliography Index of Names Subject Index

3,454 citations

Journal ArticleDOI
TL;DR: In this paper, a general theory of non-commutative differential geometry on quantum groups is developed, where bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied.
Abstract: The paper deals with non-commutative differential geometry. The general theory of differential calculus on quantum groups is developed. Bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied. Tensor algebra and external algebra constructions are described. It is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra, so the external derivative of higher order differential forms is well defined and obeys the usual properties. The proper form of the Cartan Maurer formula is found. The vector space dual to the space of left-invariant differential forms is endowed with a bilinear operation playing the role of the Lie bracket (commutator). Generalized antisymmetry relation and Jacobi identity are proved.

1,248 citations


"Differential geometry of group latt..." refers background in this paper

  • ...Woronowicz [34] introduced the wedge product θ ∧ θ ′ = 1 2 (id− σ)(θ ⊗A θ ′) ....

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  • ...A differential calculus which is both left and right covariant is called bicovariant [34]....

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  • ...For a bicovariant differential calculus a bimodule isomorphism σ : Ω(1) ⊗A Ω 1 → Ω(1) ⊗A Ω 1 exists [6, 34] such that σ(θ1 ⊗A θ 2) = θ12 ⊗A θ h1 = θ0 ⊗A θ h1 (4....

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Book ChapterDOI
01 Jan 1989

1,062 citations


"Differential geometry of group latt..." refers background in this paper

  • ...a differential calculus which corresponds to the hypercubic lattice and which leads to an elegant formulation of lattice gauge theory [1]. A special class of digraphs are Cayley graphs [15] (see Refs. [16,17], for example), which are also known as group lattices in the physics literature. These are determined by a discrete group Gand a subset S. The elements of Gare the vertices of the digraph and the ele...

    [...]

  • ...al first order differential calculus on the two disjoint parts of S 3 in this case. Another choice is S= {(12),(123)}. The corresponding digraph is shown on the righthand side of Fig. 2 (see also Refs. [16,17]). We call a group lattice bicovariant if ad(S)S⊂S. The significance of this definition will be made clear in section 3. Our previous examples of group lattices are indeed bicovariant, except for (S 3,S...

    [...]

01 Jan 2002
TL;DR: These notes were prepared by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996 and have subsequently been updated for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne.
Abstract: These notes were prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2002 by C. F. Miller.

913 citations