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

top
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

top

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

top
  1. Nicolás Andruskiewitsch, Matías Graña, From racks to pointed Hopf algebras, Adv. Math. 178 (2003), 177-243 Zbl1032.16028MR1994219
  2. E. Cartan, Le troisième théorème fondamental de Lie, C.R. Acad. Sc. T. 190 (1930), 914-1005 Zbl56.0373.01
  3. S. Covez, L’intégration locale des algèbres de Leibniz, (2010) 
  4. W. T. van Est, A group theoretic interpretation of area in the elementary geometries., Simon Stevin 32 (1958), 29-38 Zbl0139.14406MR97764
  5. W. T. van Est, Local and global groups. I, Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indag. Math. 24 (1962), 391-408 Zbl0105.02405MR144999
  6. W. T. van Est, Local and global groups. II, Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indag. Math. 24 (1962), 409-425 Zbl0109.02003MR145000
  7. Roger Fenn, Colin Rourke, Racks and links in codimension two, J. Knot Theory Ramifications 1 (1992), 343-406 Zbl0787.57003MR1194995
  8. Nicholas Jackson, Extensions of racks and quandles, Homology Homotopy Appl. 7 (2005), 151-167 Zbl1077.18010MR2155522
  9. Michael K. Kinyon, Leibniz algebras, Lie racks, and digroups, J. Lie Theory 17 (2007), 99-114 Zbl1129.17002MR2286884
  10. J. L. Loday, T. Pirashvili, The tensor category of linear maps and Leibniz algebras, Georgian Math. J. 5 (1998), 263-276 Zbl0909.18003MR1618360
  11. Jean-Louis Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, R.C.P. 25, Vol. 44 (French) (Strasbourg, 1992) 1993/41 (1993), 127-151, Univ. Louis Pasteur, Strasbourg Zbl0806.55009MR1331623
  12. Jean-Louis Loday, Cyclic homology, 301 (1998), Springer-Verlag, Berlin Zbl0780.18009MR1600246
  13. Jean-Louis Loday, Teimuraz Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296 (1993), 139-158 Zbl0821.17022MR1213376
  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

NotesEmbed ?

top

You must be logged in to post comments.