Startseite Mathematik The gauge action, DG Lie algebras and identities for Bernoulli numbers
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

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

  • Urtzi Buijs , José G. Carrasquel-Vera und Aniceto Murillo EMAIL logo
Veröffentlicht/Copyright: 8. Juni 2017

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.

MSC 2010: 17B01; 11B68; 55U35

Communicated by Frederick R. Cohen


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

Funding statement: 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.

References

[1] Arakawa T., Ibukiyama T. and Kaneko M., Bernoulli Numbers and Zeta Functions, Springer Monogr. Math., Springer, Tokyo, 2014. 10.1007/978-4-431-54919-2Suche in 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. 10.1007/s11856-020-2026-8Suche in Google Scholar

[3] Buijs U. and Murillo A., Algebraic models of non-connected spaces and homotopy theory of L algebras, Adv. Math. 236 (2013), 60–91. 10.1016/j.aim.2012.12.014Suche in 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. 10.2140/agt.2013.13.577Suche in Google Scholar

[5] Crabb M. C., The Miki–Gessel Bernoulli number identity, Glasg. Math. J. 47 (2005), 327–328. 10.1017/S0017089505002545Suche in 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. 10.4310/CNTP.2013.v7.n2.a1Suche in Google Scholar

[7] Faber C. and Pandharipande R., Hodge integrals and Gromov–Witten theory, Invent. Math. 139 (2000), 137–199. 10.1007/s002229900028Suche in 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. 10.1201/9781420034295-8Suche in Google Scholar

[9] Gessel I. M., On Miki’s identity for Bernouli numbers, J. Number Theory 110 (2005), 75–82. 10.1016/j.jnt.2003.08.010Suche in Google Scholar

[10] Kontsevich M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157–216. 10.1023/B:MATH.0000027508.00421.bfSuche in Google Scholar

[11] Lawrence R. and Sullivan D., A formula for topology/deformations and its significance, Fund.Math. 225 (2014), 229–242. 10.4064/fm225-1-10Suche in Google Scholar

[12] Miki H., A relation between Bernoulli numbers, J. Number Theory 10 (1978), 297–302. 10.1016/0022-314X(78)90026-4Suche in Google Scholar

[13] Pan H. and Sun Z. W., Identities concerning Bernoulli and Euler polynomials, Acta Arith. 12 (2006), no. 1, 21–39. 10.4064/aa125-1-3Suche in Google Scholar

[14] Parent P. E. and Tanré D., Lawrence–Sullivan models for the interval, Topology Appl. 159 (2012), no. 1, 371–378. 10.1016/j.topol.2011.10.006Suche in Google Scholar

Received: 2015-12-21
Revised: 2016-3-25
Published Online: 2017-6-8
Published in Print: 2017-3-1

© 2017 by De Gruyter

Heruntergeladen am 4.2.2026 von https://www.degruyterbrill.com/document/doi/10.1515/forum-2015-0257/pdf
Button zum nach oben scrollen