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Differential geometry of group lattices

J. Math. Phys. 44, 1781 (2003); doi:10.1063/1.1540713

Issue Date: April 2003

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Aristophanes Dimakis
Department of Financial and Management Engineering, University of the Aegean, 31 Fostini Str., GR-82100 Chios, Greece

Folkert Müller-Hoissen
Max-Planck-Institut für Strömungsforschung, Bunsenstrasse 10, D-37073 Göttingen, Germany
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[subset or is implied by]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 product with forms is defined. The Lie–Cartan identity then holds on all forms for a certain subclass of discrete vector fields. We develop elements of gauge theory and construct an analog of the lattice gauge theory (Yang–Mills) action on an arbitrary group lattice. Also linear connections are considered and a simple geometric interpretation of the torsion is established. By taking a quotient with respect to some subgroup of the discrete group, generalized differential calculi associated with so-called Schreier diagrams are obtained. ©2003 American Institute of Physics.
History: Received 11 July 2002; accepted 22 November 2002
Permalink: http://link.aip.org/link/?JMAPAQ/44/1781/1
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KEYWORDS and PACS

Keywords
PACS
  • 11.15.Ha
    Lattice gauge theory
  • 02.40.Hw
    Classical differential geometry
  • 02.20.Bb
    General structures of groups
  • YEAR: 2003

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ISSN:
0022-2488 (print)   1089-7658 (online)
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REFERENCES (51)
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  39. This means that phi<sub>X</sub><sup>*</sup> intertwines the corresponding maps associated with discrete vector fields. Note that, in general, Z is not unique.
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  41. This expression is not in general a vector field unless S is symmetric, i.e., S = S–1. It is actually a vector field for the opposite group lattice (G,S–1).
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  46. See the discussion of 2-form components at the end of Sec. IV B.
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  49. Note that Hh = H implies (Hg)ad(g–1)h = Hg, so we also have a loop at Hg and then at every coset. If ad(g)h[is-an-element-of]H and thus Hgh = Hg for all g, we can eliminate these loops by reducing S to S\{h}. But if ad(g)h[negated is-an-element-of]H for some g, it will not be possible to get rid of the loops by choosing a smaller set S without simultaneously eliminating some arrows between different points.
  50. Given two spaces with differential calculi (Omegai,di), i = 1,2, the skew tensor product Omega1[direct-product]-hat Omega2 with d(omega1[direct-product]-hat omega2) = (d1omega1)[direct-product]-hat omega2 + (–1)romega1[direct-product]-hat d2omega2 for omega1[is-an-element-of]Omega<sub>1</sub><sup>r</sup> and omega2[is-an-element-of]Omega2 defines a differential calculus on their direct product. See Ref. 4 and D. Kastler, Cyclic Cohomology Within the Differential Envelope (Hermann, Paris, 1988), Appendix A, for example.
  51. It has the properties (fomega,f[prime]omega[prime]) = f[dagger]f[prime](omega,omega[prime]) and (omega,omega[prime])[dagger] = (omega[prime],omega) where [dagger] denotes complex or Hermitian conjugation. Furthermore, (dxµ1[centered ellipsis]dxµr,dxnu1[centered ellipsis]dxnus) = deltar,sgµ1rho1[centered ellipsis]gµrrhordelta<sub>rho[sub 1]</sub><sup>[nu[sub 1]</sup>[centered ellipsis]delta<sub>rho[sub r]</sub><sup>nu[sub r]]</sup>.
  52. Y. Okumura, Prog. Theor. Phys. 96, 1021 (1996).

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