Riemannian geometry of bicovariant group lattices

Journal of Mathematical Physics

Volumn 44, Issue 9, 2003, Pages 4220-4259

  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: 0242552073     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1594820     Document Type: Article

References (42)

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22. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
23. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
24. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
25. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
26. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
27. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
28. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
29. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
30. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
31. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
32. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
33. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
34. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
35. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
36. A. Das, M. Kaku, and P. K. Townsend, Phys. Lett. 81B, 11 (1979); L. Smolin, Nucl. Phys. B 148, 333 (1979); K. I. Macrea, Phys. Rev. D 23, 900 (1981); C. L. T. Mannion and J. G. Taylor, Phys. Lett. 100B, 261 (1981); K. Kondo, Prog. Theor. Phys. 72, 841 (1984); G. Friedberg, R. Friedberg, T. D. Lee, and H. G. Ren, Nucl. Phys. B 245, 145 (1984); H. Römer and M. Zähringer, Class. Quantum Grav. 3, 897 (1986); M. Caselle, A. D'Adda, and L. Magnea, Phys. Lett. B 192, 406 (1987); 192, 411 (1987); P. Renteln and L. Smolin, Class. Quantum Grav. 6, 275 (1989); N. Kawamoto and H. B. Nielsen, Phys. Rev. D 43, 1150 (1991); O. Boström, M. Miller, and L. Smolin, "Anew discretization of classical and quantum general relativity," preprint CGPG 94-3-3; L. Brewin, Class. Quantum Grav. 15, 3085 (1998); 15, 2427 (1998); R. Loll, Living Rev. Relativ. 1, 13 (1998); G. Gionti, "Discrete approaches towards the definition of a quantum theory of gravity," gr-qc/9812080.
37. R. Sauer, Differenzengeometrie (Springer, Berlin, 1970).
38. h,h′ denoted the quadrangle coefficients of a 2-form ψ with respect to the original product in Ψ. Here it refers to the ∩-product. Of course, they are different in general.
39. 2=0.
40. A corresponding Mathematica notebook is available at the authors' homepages. Mathematica is a registered trademark of Wolfram, Research. See S. Wolfram, The Mathematica Book, (Cambridge University Press, Cambridge, 1999).
41. Distances in communication networks are measured by sending a signal from one point to another and reflecting it back to the point of origin. The elapsed time read off from a clock at the point of emission is then a measure of the distance between the two points.
42. M. Dinneen, M. R. Fellows, and V. Faber, "Algebraic constructions of efficient broadcast networks," in Lecture Notes in Computer Science 539 (Springer, New York, 1991), p. 152.