Show Summary Details
More options …

# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2017: 0.67

Online
ISSN
1435-5337
See all formats and pricing
More options …
Volume 29, Issue 2

# 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
• Email
• Other articles by this author:
/ Aniceto Murillo
• Corresponding author
• Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, Ap. 59, 29080 Málaga, Spain
• Email
• Other articles by this author:
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 $\left(a,b,c\right)$ with $a+b+c=n-1$, $n\ge 4$. 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

## 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 ${L}_{\mathrm{\infty }}$ 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

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,

Export Citation