Differential geometry of group lattices

Journal of Mathematical Physics

Volumn 44, Issue 4, 2003, Pages 1781-1821

  Dimakis, Aristophanes ^a   Müller Hoissen, Folkert ^b  

^a UNIVERSITY OF THE AEGEAN   (Greece)

^b MAX PLANCK INSTITUTE FOR DYNAMICS AND SELF ORGANIZATION   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037398634     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1540713     Document Type: Article

References (64)

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31. The formalism developed in this work may be generalized by replacing ℂ with an arbitrary field double struck K sign.
32. h-1 in the latter work.
33. One could think of extending the formalism to infinite subsets S of an infinite discrete group G. Then one has to find a way to make sense of infinite summations (over the elements of S, at g ∈ G) appearing in the following formulas.
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35. s.
36. g
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44. A Cayley graph without biangles (circuits of length 2) is sometimes called "combinatorial," see Ref. 27.
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47. h) is indeed satisfied as a consequence of (4.4) and (2.10).
48. 36 If there are no triangles, then we have simply dα = θα+ αθ for a 1-form α.
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50. X* intertwines the corresponding maps associated with discrete vector fields. Note that, in general, Z is not unique.
51. These conditions mean that if there is an outgoing X-arrow at g, then there is also precisely one incoming X-arrow. If there is no outgoing arrow, then there is also no incoming arrow.
52. -1).
53. -1hs(gh) (for s with values in S) lies in S, so that condition (2) of Theorem 5.1 holds.
54. X(g).
55. h.
56. Bundles over discrete spaces with varying dimension of the fibers have been considered in particular for particle physics model building. See, for example, A. Connes, "Essay on physics and non-commutative geometry," in The Interface of Mathematics and Particle Physics, edited by D. Quillen, G. Segal and S. Tsou (Oxford University Press, Oxford, 1990), p. 9.
57. See the discussion of 2-form components at the end of Sec. IV B.
58. Such generalized first-order differential calculi have already been considered in H. C. Baehr, A. Dimakis, and F. Müller-Hoissen, "Differential calculi on commutative algebras," preprint MPI-PhT/94-83, hep-th/9412069, Appendix A (1994). They cannot be obtained as a quotient of the universal first order differential calculus.
59. This should not be confused with Cayley coset digraphs as considered, for example, in G. Sabidussi, Duke Math. J. 26, 693 (1959); E. Knill, "Notes on the connectivity of Cayley coset diagraphs," math.CO/9411221.
60. This should not be confused with Cayley coset digraphs as considered, for example, in G. Sabidussi, Duke Math. J. 26, 693 (1959); E. Knill, "Notes on the connectivity of Cayley coset diagraphs," math.CO/9411221.
61. -1)h = Hg, so we also have a loop at Hg and then at every coset. If ad(g)h ∈ H and thus Hgh = Hg for all g, we can eliminate these loops by reducing S to S\{h}. But if ad(g)h ∉ 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.
62. 2 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.
63. νr].
64. Y. Okumura, Prog. Theor. Phys. 96, 1021 (1996).