Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 027, 23 pages      arXiv:2007.03003      https://doi.org/10.3842/SIGMA.2021.027
Contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi

From Orthocomplementations to Locality

Pierre Clavier a, Li Guo b, Sylvie Paycha a and Bin Zhang c
a) Institute of Mathematics, University of Potsdam, D-14476 Potsdam, Germany
b) Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, USA
c) School of Mathematics, Yangtze Center of Mathematics, Sichuan University, Chengdu, 610064, China

Received July 06, 2020, in final form March 02, 2021; Published online March 22, 2021

Abstract
After some background on lattices, the locality framework introduced in earlier work by the authors is extended to cover posets and lattices. We then extend the correspondence between Euclidean structures on vector spaces and orthogonal complementations to a one-one correspondence between a class of locality structures and orthocomplementations on bounded lattices. This recasts in the context of renormalisation classical results in lattice theory.

Key words: locality; lattice; poset; orthocomplementation; renormalisation.

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References

  1. Berline N., Vergne M., Local Euler-Maclaurin formula for polytopes, Mosc. Math. J. 7 (2007), 355-386, arXiv:math.CO/0507256.
  2. Birkhoff G., Lattice theory, American Mathematical Society Colloquium Publications.
  3. Cattaneo G., Manià A., Abstract orthogonality and orthocomplementation, Proc. Cambridge Philos. Soc. 76 (1974), 115-132.
  4. Cattaneo G., Nisticò G., Orthogonality and orthocomplementations in the axiomatic approach to quantum mechanics: remarks about some critiques, J. Math. Phys. 25 (1984), 513-531.
  5. Chajda I., Länger H., The lattice of subspaces of a vector space over a finite field, Soft Comput. 23 (2019), 3261-3267.
  6. Clavier P., Guo L., Paycha S., Zhang B., An algebraic formulation of the locality principle in renormalisation, Eur. J. Math. 5 (2019), 356-394, arXiv:1711.00884.
  7. Clavier P., Guo L., Paycha S., Zhang B., Renormalisation via locality morphisms, Rev. Colombiana Mat. 53 (2019), 113-141, arXiv:1810.03210.
  8. Clavier P., Guo L., Paycha S., Zhang B., Renormalisation and locality: branched zeta values, in Algebraic Combinatorics, Resurgence, Moulds and Applications (Carma), Irma Lectures in Math. and Theor. Phys., Vol. 32, Editors F. Chapoton, F. Fauvet, C. Malvenuto, J.Y. Thibon, Eur. Math. Soc. Publishing House, Berlin, 2020, 85-132, arXiv:1807.07630.
  9. Clavier P., Guo L., Paycha S., Zhang B., Locality and renormalization: universal properties and integrals on trees, J. Math. Phys. 61 (2020), 022301, 19 pages, arXiv:1811.01215.
  10. Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210 (2000), 249-273, arXiv:hep-th/9912092.
  11. Dang N.V., Zhang B., Renormalization of Feynman amplitudes on manifolds by spectral zeta regularization and blow-ups, J. Eur. Math. Soc. (JEMS) 23 (2021), 503-556, arXiv:1712.03490.
  12. Engesser K., Gabbay D.M., Lehmann D. (Editors), Handbook of quantum logic and quantum structures: quantum logic, Elsevier Science B.V., Amsterdam, 2007.
  13. Faure C.-A., Frölicher A., Modern projective geometry, Mathematics and its Applications, Vol. 521, Kluwer Academic Publishers, Dordrecht, 2000.
  14. Foissy L., Les algèbres de Hopf des arbres enracinés décorés. I, Bull. Sci. Math. 126 (2002), 193-239, arXiv:math.QA/0105212.
  15. Foissy L., Les algèbres de Hopf des arbres enracinés décorés. II, Bull. Sci. Math. 126 (2002), 249-288.
  16. Garoufalidis S., Pommersheim J., Sum-integral interpolators and the Euler-Maclaurin formula for polytopes, Trans. Amer. Math. Soc. 364 (2012), 2933-2958, arXiv:1002.3522.
  17. Grätzer G., Lattice theory. First concepts and distributive lattices, W.H. Freeman and Co., San Francisco, Calif., 1971.
  18. Grätzer G., Lattice theory: foundation, Birkhäuser/Springer Basel AG, Basel, 2011.
  19. Gross H., Quadratic forms in infinite-dimensional vector spaces, Progress in Mathematics, Vol. 1, Birkhäuser, Boston, Mass., 1979.
  20. Guo L., Paycha S., Zhang B., Algebraic Birkhoff factorization and the Euler-Maclaurin formula on cones, Duke Math. J. 166 (2017), 537-571, arXiv:1306.3420.
  21. Guo L., Paycha S., Zhang B., A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles, Pacific J. Math. 307 (2020), 159-196, arXiv:1501.00426.
  22. Hedlíková J., Pulmannová S., Orthogonality spaces and atomistic orthocomplemented lattices, Czechoslovak Math. J. 41(116) (1991), 8-23.
  23. Kalmbach G., Orthomodular lattices, London Mathematical Society Monographs, Vol. 18, Academic Press Inc., London, 1983.
  24. MacLaren M.D., Atomic orthocomplemented lattices, Pacific J. Math. 14 (1964), 597-612.
  25. Maeda F., Maeda S., Theory of symmetric lattices, Die Grundlehren der mathematischen Wissenschaften, Vol. 173, Springer-Verlag, New York - Berlin, 1970.
  26. Manchon D., Hopf algebras, from basics to applications to renormalization, arXiv:math.QA/0408405.
  27. Padmanabhan R., Rudeanu S., Axioms for lattices and Boolean algebras, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.
  28. Rutherford D.E., Introduction to lattice theory, Hafner Publishing Co., New York, 1965.
  29. Szymańska-Bartman M., Weak orthogonalities on a partially ordered set, in Contributions to General Algebra (Proc. Klagenfurt Conf., Klagenfurt, 1978), Heyn, Klagenfurt, 1979, 381-388.

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