Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T06:04:46.243Z Has data issue: false hasContentIssue false
  • English
  • Français

Integrating morphisms of Lie 2-algebras

Part of: Lie algebras and Lie superalgebras Categories with structure

Published online by Cambridge University Press:  04 February 2013

Behrang Noohi
Behrang Noohi*
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Given two Lie 2-groups, we study the problem of integrating a weak morphism between the corresponding Lie 2-algebras to a weak morphism between the Lie 2-groups. To do so, we develop a theory of butterflies for 2-term L∞-algebras. In particular, we obtain a new description of the bicategory of 2-term L∞-algebras. An interesting observation here is that the role played by 1-connected Lie groups in Lie theory is now played by 2-connected Lie 2-groups. Using butterflies, we also give a functorial construction of 2-connected covers of Lie 2 -groups. Based on our results, we expect that a similar pattern generalizes to Lie n-groups and Lie n-algebras.

Keywords

Lie 2-algebraLie 2-groupLie crossed moduleLie butterflyL∞-algebraconnected cover

MSC classification

Secondary: 17B99: None of the above, but in this section 18D05: Double categories, $2$-categories, bicategories and generalizations
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 2 , February 2013 , pp. 264 - 294
Copyright
© The Author(s) 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[AN]Aldrovandi, E. and Noohi, B., Butterflies III: 2-butterflies and 2-group stacks, in preparation.Google Scholar
[AN09]Aldrovandi, E. and Noohi, B., Butterflies I: morphisms of 2-group stacks, Adv. Math. 221 (2009), 687–773.CrossRefGoogle Scholar
[AKU09]Arvasi, Z., Kuzpinari, T. S. and Uslu, E. Ö., Three crossed modules, Homology, Homotopy Appl. 11 (2009), 161–187.CrossRefGoogle Scholar
[BC04]Baez, J. and Crans, A., Higher-dimensional algebra. VI. Lie 2-algebras, Theory Appl. Categ. 12 (2004), 492–538.Google Scholar
[BN06]Behrend, K. and Noohi, B., Uniformization of Deligne–Mumford analytic curves, J. Reine Angew. Math. 599 (2006), 111–153.Google Scholar
[BG89]Brown, R. and Gilbert, N. D., Algebraic models of 3-types and automorphism structures for crossed modules, Proc. Lond. Math. Soc. (3) 59 (1989), 51–73.CrossRefGoogle Scholar
[Con84]Conduché, D., Modules croisés généralisés de longueur 2, J. Pure Appl. Algebra 34 (1984), 155–178.CrossRefGoogle Scholar
[FG06]Frenkel, E. and Gaitsgory, D., Local geometric Langlands correspondence and affine Kac–Moody algebras, in Algebraic geometry and number theory, Progress in Mathematics, vol. 253 (Birkhäuser, Boston, 2006), 69–260.CrossRefGoogle Scholar
[FZ12]Frenkel, E. and Zhu, X., Gerbal representations of double loop groups, Int. Math. Res. Not. IMRN 2012 (2012), 3929–4013, doi:10.1093/imrn/rnr159.Google Scholar
[Get09]Getzler, E., Lie theory for nilpotent L ∞-algebras, Ann. of Math. (2) 170 (2009), 271–301.CrossRefGoogle Scholar
[Hen08]Henriques, A., Integrating L ∞-algebras, Compositio Math. 144 (2008), 1017–1045.CrossRefGoogle Scholar
[Noo05]Noohi, B., Foundations of topological stacks I, Preprint (2005), arXiv:math/0503247.Google Scholar
[Noo07]Noohi, B., Automorphism 2-group of a weighted projective stack, Preprint (2007), arXiv:0704.1010.Google Scholar
[Noo08]Noohi, B., On weak maps between 2-group, Preprint (2008), arXiv:math/0506313v3.Google Scholar
[Noo10]Noohi, B., Fibrations of topological stacks, Preprint (2010), arXiv:1010.1748.Google Scholar
[Noo12]Noohi, B., Homotopy types of topological stacks, Adv. Math. 230 (2012), 2014–2047.CrossRefGoogle Scholar
[Roy07]Roytenberg, D., On weak Lie 2-algebras, in XXVI Workshop on geometrical methods in physics, AIP Conference Proceedings, vol. 956 (American Institute of Physics, Melville, NY, 2007).Google Scholar
[Zhu09]Zhu, X., The 2-group of linear auto-equivalences of an abelian category and its Lie 2-algebra, Preprint (2009), arXiv:0910.5699.Google Scholar