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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

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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
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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


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


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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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