The local integration of Leibniz algebras

Simon Covez[1]

  • [1] Université du Luxembourg Campus Kirchberg Mathematics Research Unit 6, rue Richard Coudenhove-Kalergi L-1359Luxembourg Grand Duché du Luxembourg

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 1, page 1-35
  • ISSN: 0373-0956

Abstract

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This article gives a local answer to the coquecigrue problem for Leibniz algebras, that is, the problem of finding a generalization of the (Lie) group structure such that Leibniz algebras are the corresponding tangent algebra structure. Using links between Leibniz algebra cohomology and Lie rack cohomology, we generalize the integration of a Lie algebra into a Lie group by proving that every Leibniz algebra is isomorphic to the tangent Leibniz algebra of a local Lie rack. This article ends with an example of a Leibniz algebra integration in dimension 5.

How to cite

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Covez, Simon. "The local integration of Leibniz algebras." Annales de l’institut Fourier 63.1 (2013): 1-35. <http://eudml.org/doc/275518>.

@article{Covez2013,
abstract = {This article gives a local answer to the coquecigrue problem for Leibniz algebras, that is, the problem of finding a generalization of the (Lie) group structure such that Leibniz algebras are the corresponding tangent algebra structure. Using links between Leibniz algebra cohomology and Lie rack cohomology, we generalize the integration of a Lie algebra into a Lie group by proving that every Leibniz algebra is isomorphic to the tangent Leibniz algebra of a local Lie rack. This article ends with an example of a Leibniz algebra integration in dimension 5.},
affiliation = {Université du Luxembourg Campus Kirchberg Mathematics Research Unit 6, rue Richard Coudenhove-Kalergi L-1359Luxembourg Grand Duché du Luxembourg},
author = {Covez, Simon},
journal = {Annales de l’institut Fourier},
keywords = {Leibniz algebra; Lie rack; Leibniz algebra cohomology; rack cohomology},
language = {eng},
number = {1},
pages = {1-35},
publisher = {Association des Annales de l’institut Fourier},
title = {The local integration of Leibniz algebras},
url = {http://eudml.org/doc/275518},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Covez, Simon
TI - The local integration of Leibniz algebras
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 1
SP - 1
EP - 35
AB - This article gives a local answer to the coquecigrue problem for Leibniz algebras, that is, the problem of finding a generalization of the (Lie) group structure such that Leibniz algebras are the corresponding tangent algebra structure. Using links between Leibniz algebra cohomology and Lie rack cohomology, we generalize the integration of a Lie algebra into a Lie group by proving that every Leibniz algebra is isomorphic to the tangent Leibniz algebra of a local Lie rack. This article ends with an example of a Leibniz algebra integration in dimension 5.
LA - eng
KW - Leibniz algebra; Lie rack; Leibniz algebra cohomology; rack cohomology
UR - http://eudml.org/doc/275518
ER -

References

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  14. Karl-Hermann Neeb, Central extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier (Grenoble) 52 (2002), 1365-1442 Zbl1019.22012MR1935553
  15. Karl-Hermann Neeb, Abelian extensions of infinite-dimensional Lie groups, Travaux mathématiques. Fasc. XV (2004), 69-194, Univ. Luxemb., Luxembourg Zbl1079.22018MR2143422
  16. P. A. Smith, The complex of a group relative to a set of generators. II, Ann. of Math. (2) 54 (1951), 403-424 Zbl0044.19804MR48462
  17. P. A. Smith, Some topological notions connected with a set of generators, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2 (1952), 436-441, Amer. Math. Soc., Providence, R. I. Zbl0049.12503MR48463

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