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Forum Mathematicum

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Volume 29, Issue 2

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The gauge action, DG Lie algebras and identities for Bernoulli numbers

Urtzi Buijs / José G. Carrasquel-Vera
  • Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, 2 Chemin du Cyclotron B-1348, Louvain-la-Neuve, Belgium
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/ Aniceto Murillo
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  • Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, Ap. 59, 29080 Málaga, Spain
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Published Online: 2017-06-08 | DOI: https://doi.org/10.1515/forum-2015-0257

Abstract

In this paper we prove a family of identities for Bernoulli numbers parameterized by triples of integers (a,b,c) with a+b+c=n-1, n4. These identities are deduced by translating into homotopical terms the gauge action on the Maurer–Cartan set of a differential graded Lie algebra. We show that Euler and Miki’s identities, well-known and apparently non-related formulas, are linear combinations of our family and they satisfy a particular symmetry relation.

Keywords: Gauge action; Bernoulli numbers; homotopy theory of Lie algebras

MSC 2010: 17B01; 11B68; 55U35

References

  • [1]

    Arakawa T., Ibukiyama T. and Kaneko M., Bernoulli Numbers and Zeta Functions, Springer Monogr. Math., Springer, Tokyo, 2014. Google Scholar

  • [2]

    Buijs U., Félix Y., Murillo A. and Tanré D., Lie models of simplicial sets and representability of the Quillen functor, preprint 2015, http://arxiv.org/abs/1508.01442.

  • [3]

    Buijs U. and Murillo A., Algebraic models of non-connected spaces and homotopy theory of L algebras, Adv. Math. 236 (2013), 60–91. Google Scholar

  • [4]

    Buijs U. and Murillo A., The Lawrence–Sullivan construction is the right model of I+, Algebr. Geom. Topol. 13 (2013), no. 1, 577–588. Google Scholar

  • [5]

    Crabb M. C., The Miki–Gessel Bernoulli number identity, Glasg. Math. J. 47 (2005), 327–328. Google Scholar

  • [6]

    Dunne G. V. and Schubert C., Bernoulli number identities from quantum field theory, Commun. Number Theory Phys. 7 (2013), no. 2, 225–249. Google Scholar

  • [7]

    Faber C. and Pandharipande R., Hodge integrals and Gromov–Witten theory, Invent. Math. 139 (2000), 137–199. Google Scholar

  • [8]

    Fukaya K., Deformation theory, homological algebra and mirror symmetry, Geometry and Physics of Branes (Como 2001), Ser. High Energy Phys. Cosmol. Gravit., IOP, Bristol (2003), 121–209. Google Scholar

  • [9]

    Gessel I. M., On Miki’s identity for Bernouli numbers, J. Number Theory 110 (2005), 75–82. Google Scholar

  • [10]

    Kontsevich M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157–216. Google Scholar

  • [11]

    Lawrence R. and Sullivan D., A formula for topology/deformations and its significance, Fund.Math. 225 (2014), 229–242. Google Scholar

  • [12]

    Miki H., A relation between Bernoulli numbers, J. Number Theory 10 (1978), 297–302. Google Scholar

  • [13]

    Pan H. and Sun Z. W., Identities concerning Bernoulli and Euler polynomials, Acta Arith. 12 (2006), no. 1, 21–39. Google Scholar

  • [14]

    Parent P. E. and Tanré D., Lawrence–Sullivan models for the interval, Topology Appl. 159 (2012), no. 1, 371–378. Google Scholar

About the article


Received: 2015-12-21

Revised: 2016-03-25

Published Online: 2017-06-08

Published in Print: 2017-03-01


Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2010-15831

Award identifier / Grant number: MTM2013-41768-P

Award identifier / Grant number: FQM-213

Award identifier / Grant number: MTM2010-18089

Award identifier / Grant number: MTM2013-41768-P

The first author was partially supported by the Ministerio de Economía y Competitividad grants MTM2010-15831, MTM2013-41768-P, by the grants FQM-213, and by the Marie Curie COFUND programme U-mobility, co-financed by the University of Málaga, the European Commision FP7 under GA No. 246550, and Ministerio de Economía y Competitividad (COFUND2013-40259). The second author was partially supported by the Ministerio de Economía y Competitividad grant MTM2010-18089. The third author was partially supported by the Ministerio de Economía y Competitividad grant MTM2013-41768-P and by the Junta de Andalucía grants FQM-213.


Citation Information: Forum Mathematicum, Volume 29, Issue 2, Pages 277–286, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0257.

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