Non-abelian extensions of infinite-dimensional Lie groups
Annales de l'Institut Fourier, Volume 57 (2007) no. 1, pp. 209-271.

In this article we study non-abelian extensions of a Lie group G modeled on a locally convex space by a Lie group N. The equivalence classes of such extension are grouped into those corresponding to a class of so-called smooth outer actions S of G on N. If S is given, we show that the corresponding set Ext (G,N) S of extension classes is a principal homogeneous space of the locally smooth cohomology group H ss 2 (G,Z(N)) S . To each S a locally smooth obstruction class χ(S) in a suitably defined cohomology group H ss 3 (G,Z(N)) S is defined. It vanishes if and only if there is a corresponding extension of G by N. A central point is that we reduce many problems concerning extensions by non-abelian groups to questions on extensions by abelian groups, which have been dealt with in previous work. An important tool is a Lie theoretic concept of a smooth crossed module α:HG, which we view as a central extension of a normal subgroup of G.

Dans cet article nous étudions les extensions non abéliennes d’un groupe de Lie G modelé sur un espace localement convexe par un groupe de Lie N. Les classes d’équivalence de telles extensions sont groupées en celles qui correspondent à la classe des actions dites des actions extérieures S de G sur N. Si S est donné, nous montrons que l’ensemble correspondant Ext(G,N) S des classes d’extensions est un espace homogène principal du groupe de cohomologie localement lisse H ss 2 (G,Z(N)) S . Pour chaque S une obstruction localement lisse χ(S) dans un groupe de cohomologie H ss 3 (G,Z(N)) S est définie. Elle s’annule si et seulement si il existe une extension correspondante de G par N. Un point central est que nous ramenons plusieurs problèmes concernant des extensions par des groupes non abéliens à des questions sur des extensions par des groupes abéliens, qui ont été étudiées dans des travaux antérieurs. Un outil important est une notion de module croisé lisse, relevant de la théorie de Lie, α:HG, que nous voyons comme une extension centrale d’un sous-groupe normal de G.

DOI: 10.5802/aif.2257
Classification: 22E65, 57T10, 22E15
Keywords: Lie group extension, smooth outer action, crossed module, Lie group cohomology, automorphisms of group extension
Mot clés : extension de groupes de Lie, action extérieure libre, module croisé, cohomologie des groupes de Lie, extension des automorphismes de groupes
Neeb, Karl-Hermann 1

1 Technische Universität Darmstadt Schlossgartenstrasse 7 64289 Darmstadt(Deutschland)
@article{AIF_2007__57_1_209_0,
     author = {Neeb, Karl-Hermann},
     title = {Non-abelian extensions of infinite-dimensional {Lie} groups},
     journal = {Annales de l'Institut Fourier},
     pages = {209--271},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {1},
     year = {2007},
     doi = {10.5802/aif.2257},
     zbl = {1127.22008},
     mrnumber = {2316238},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2257/}
}
TY  - JOUR
AU  - Neeb, Karl-Hermann
TI  - Non-abelian extensions of infinite-dimensional Lie groups
JO  - Annales de l'Institut Fourier
PY  - 2007
SP  - 209
EP  - 271
VL  - 57
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2257/
DO  - 10.5802/aif.2257
LA  - en
ID  - AIF_2007__57_1_209_0
ER  - 
%0 Journal Article
%A Neeb, Karl-Hermann
%T Non-abelian extensions of infinite-dimensional Lie groups
%J Annales de l'Institut Fourier
%D 2007
%P 209-271
%V 57
%N 1
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2257/
%R 10.5802/aif.2257
%G en
%F AIF_2007__57_1_209_0
Neeb, Karl-Hermann. Non-abelian extensions of infinite-dimensional Lie groups. Annales de l'Institut Fourier, Volume 57 (2007) no. 1, pp. 209-271. doi : 10.5802/aif.2257. http://archive.numdam.org/articles/10.5802/aif.2257/

[1] Baer, R. Erweiterungen von Gruppen und ihren Isomorphismen, Math. Zeit., Volume 38 (1934), pp. 375-416 | DOI | MR | Zbl

[2] Borovoi, Mikhail V. Abelianization of the second nonabelian Galois cohomology, Duke Math. J., Volume 72 (1993) no. 1, pp. 217-239 | DOI | MR | Zbl

[3] Brown, Lawrence G. Extensions of topological groups, Pacific J. Math., Volume 39 (1971), pp. 71-78 | MR | Zbl

[4] Calabi, Lorenzo Sur les extensions des groupes topologiques, Ann. Mat. Pura Appl. (4), Volume 32 (1951), pp. 295-370 | DOI | MR | Zbl

[5] Cederwall, Martin; Ferretti, Gabriele; Nilsson, Bengt E. W.; Westerberg, Anders Higher-dimensional loop algebras, non-abelian extensions and p-branes, Nuclear Phys. B., Volume 424 (1994) no. 1, pp. 97-123 | DOI | MR | Zbl

[6] Eilenberg, Samuel; MacLane, Saunders Group extensions and homology, Ann. of Math. (2), Volume 43 (1942), pp. 757-831 | DOI | MR | Zbl

[7] Eilenberg, Samuel; MacLane, Saunders Cohomology theory in abstract groups. II. Group extensions with a non-Abelian kernel, Ann. of Math. (2), Volume 48 (1947), pp. 326-341 | DOI | MR | Zbl

[8] Glöckner, Helge Infinite-dimensional Lie groups without completeness condition, Geometry and analysis on finite- and infinite-dimensional Lie groups (A. Strasburger et al Eds.), Volume 55, Banach Center Publications, Warsawa, 2002, pp. 53-59 | Zbl

[9] Goto, Morikuni On an arcwise connected subgroup of a Lie group, Proc. Amer. Math. Soc., Volume 20 (1969), pp. 157-162 | DOI | MR | Zbl

[10] Huebschmann, Johannes Automorphisms of group extensions and differentials in the Lyndon-Hochschild-Serre spectral sequence, J. Algebra, Volume 72 (1981) no. 2, pp. 296-334 | DOI | MR | Zbl

[11] Kriegl, Andreas; Michor, Peter W. The convenient setting of global analysis, Mathematical Surveys and Monographs, 53, American Mathematical Society, Providence, RI, 1997 | MR | Zbl

[12] Mackey, George W. Les ensembles boréliens et les extensions des groupes, J. Math. Pures Appl. (9), Volume 36 (1957), pp. 171-178 | MR | Zbl

[13] MacLane, S. Homological Algebra, Springer-Verlag, 1963

[14] Milnor, J. On the existence of a connection with curvature zero, Comment. Math. Helv., Volume 32 (1958), pp. 215-223 | DOI | MR | Zbl

[15] Milnor, J. Remarks on infinite-dimensional Lie groups, Relativity, groups and topology, II (Les Houches, 1983), North-Holland, Amsterdam, 1984, pp. 1007-1057 | MR | Zbl

[16] Moore, Calvin C. Extensions and low dimensional cohomology theory of locally compact groups. I, II, Trans. Amer. Math. Soc., Volume 113 (1964), p. 40-63, 63–86 | Zbl

[17] Neeb, Karl-Hermann Exact sequences for Lie group cohomology with non-abelian coefficients (in preparation)

[18] Neeb, Karl-Hermann Central extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier (Grenoble), Volume 52 (2002) no. 5, pp. 1365-1442 | DOI | Numdam | MR | Zbl

[19] Neeb, Karl-Hermann Abelian extensions of infinite-dimensional Lie groups, Travaux mathématiques. Fasc. XV (Trav. Math., XV), Univ. Luxemb., Luxembourg, 2004, pp. 69-194 | MR | Zbl

[20] Neeb, Karl-Hermann Non-abelian extensions of topological Lie algebras, Comm. Algebra, Volume 34 (2006) no. 3, pp. 991-1041 | DOI | MR | Zbl

[21] Raeburn, Iain; Sims, Aidan; Williams, Dana P. Twisted actions and obstructions in group cohomology, C * -algebras (Münster, 1999), Springer, Berlin, 2000, pp. 161-181 | MR | Zbl

[22] Robinson, Derek J. S. Automorphisms of group extensions, Algebra and its applications (New Delhi, 1981) (Lecture Notes in Pure and Appl. Math.), Volume 91, Dekker, New York, 1984, pp. 163-167 | MR | Zbl

[23] Schreier, O. Über die Erweiterungen von Gruppen I, Monatshefte f. Math., Volume 34 (1926), pp. 165-180 | DOI | MR

[24] Schreier, O. Über die Erweiterungen von Gruppen II, Abhandlungen Hamburg, Volume 4 (1926), pp. 321-346 | DOI

[25] Turing, A. M. The extensions of a group, Compos. Math., Volume 5 (1938), pp. 357-367 | Numdam | Zbl

[26] Varadarajan, V. S. Geometry of quantum theory, Springer-Verlag, New York, 1985 | MR | Zbl

[27] Weibel, Charles A. An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994 | MR | Zbl

[28] Wells, Charles Automorphisms of group extensions, Trans. Amer. Math. Soc., Volume 155 (1971), pp. 189-194 | DOI | MR | Zbl

Cited by Sources: