Non-abelian extensions of infinite-dimensional Lie groups
Annales de l'Institut Fourier, Volume 57 (2007) no. 1, p. 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 : https://doi.org/10.5802/aif.2257
Classification:  22E65,  57T10,  22E15
Keywords: Lie group extension, smooth outer action, crossed module, Lie group cohomology, automorphisms of group extension
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {1},
     year = {2007},
     pages = {209-271},
     doi = {10.5802/aif.2257},
     mrnumber = {2316238},
     zbl = {1127.22008},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/AIF_2007__57_1_209_0/

[1] Baer, R. Erweiterungen von Gruppen und ihren Isomorphismen, Math. Zeit., Tome 38 (1934), pp. 375-416 | Article | MR 1545456 | Zbl 0009.01101

[2] Borovoi, Mikhail V. Abelianization of the second nonabelian Galois cohomology, Duke Math. J., Tome 72 (1993) no. 1, pp. 217-239 | Article | MR 1242885 | Zbl 0849.12011

[3] Brown, Lawrence G. Extensions of topological groups, Pacific J. Math., Tome 39 (1971), pp. 71-78 | MR 307264 | Zbl 0241.22004

[4] Calabi, Lorenzo Sur les extensions des groupes topologiques, Ann. Mat. Pura Appl. (4), Tome 32 (1951), pp. 295-370 | Article | MR 49907 | Zbl 0054.01302

[5] Cederwall, Martin; Ferretti, Gabriele; Nilsson, Bengt E. W.; Westerberg, Anders Higher-dimensional loop algebras, non-abelian extensions and p-branes, Nuclear Phys. B., Tome 424 (1994) no. 1, pp. 97-123 | Article | MR 1290024 | Zbl 0990.81527

[6] Eilenberg, Samuel; Maclane, Saunders Group extensions and homology, Ann. of Math. (2), Tome 43 (1942), pp. 757-831 | Article | MR 7108 | Zbl 0061.40602

[7] Eilenberg, Samuel; Maclane, Saunders Cohomology theory in abstract groups. II. Group extensions with a non-Abelian kernel, Ann. of Math. (2), Tome 48 (1947), pp. 326-341 | Article | MR 20996 | Zbl 0029.34101

[8] Glöckner, Helge Infinite-dimensional Lie groups without completeness condition, Geometry and analysis on finite- and infinite-dimensional Lie groups, Banach Center Publications, Warsawa (A. Strasburger et al Eds.) Tome 55 (2002), pp. 53-59 | Zbl 1020.58009

[9] Goto, Morikuni On an arcwise connected subgroup of a Lie group, Proc. Amer. Math. Soc., Tome 20 (1969), pp. 157-162 | Article | MR 233923 | Zbl 0182.04602

[10] Huebschmann, Johannes Automorphisms of group extensions and differentials in the Lyndon-Hochschild-Serre spectral sequence, J. Algebra, Tome 72 (1981) no. 2, pp. 296-334 | Article | MR 641328 | Zbl 0443.18018

[11] Kriegl, Andreas; Michor, Peter W. The convenient setting of global analysis, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 53 (1997) | MR 1471480 | Zbl 0889.58001

[12] Mackey, George W. Les ensembles boréliens et les extensions des groupes, J. Math. Pures Appl. (9), Tome 36 (1957), pp. 171-178 | MR 89998 | Zbl 0080.02303

[13] Maclane, S. Homological Algebra, Springer-Verlag (1963)

[14] Milnor, J. On the existence of a connection with curvature zero, Comment. Math. Helv., Tome 32 (1958), pp. 215-223 | Article | MR 95518 | Zbl 0196.25101

[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 830252 | Zbl 0594.22009

[16] Moore, Calvin C. Extensions and low dimensional cohomology theory of locally compact groups. I, II, Trans. Amer. Math. Soc., Tome 113 (1964), p. 40-63, 63–86 | Zbl 0131.26902

[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), Tome 52 (2002) no. 5, pp. 1365-1442 | Article | Numdam | MR 1935553 | Zbl 1019.22012

[19] Neeb, Karl-Hermann Abelian extensions of infinite-dimensional Lie groups, Travaux mathématiques. Fasc. XV, Univ. Luxemb., Luxembourg (Trav. Math., XV) (2004), pp. 69-194 | MR 2143422 | Zbl 1079.22018

[20] Neeb, Karl-Hermann Non-abelian extensions of topological Lie algebras, Comm. Algebra, Tome 34 (2006) no. 3, pp. 991-1041 | Article | MR 2208114 | Zbl 05018918

[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 1798596 | Zbl 0984.46044

[22] Robinson, Derek J. S. Automorphisms of group extensions, Algebra and its applications (New Delhi, 1981), Dekker, New York (Lecture Notes in Pure and Appl. Math.) Tome 91 (1984), pp. 163-167 | MR 750857 | Zbl 0541.20018

[23] Schreier, O. Über die Erweiterungen von Gruppen I, Monatshefte f. Math., Tome 34 (1926), pp. 165-180 | Article | MR 1549403

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

[25] Turing, A. M. The extensions of a group, Compos. Math., Tome 5 (1938), pp. 357-367 | Numdam | Zbl 0018.39201

[26] Varadarajan, V. S. Geometry of quantum theory, Springer-Verlag, New York (1985) | MR 805158 | Zbl 0581.46061

[27] Weibel, Charles A. An introduction to homological algebra, Cambridge University Press, Cambridge, Cambridge Studies in Advanced Mathematics, Tome 38 (1994) | MR 1269324 | Zbl 0797.18001

[28] Wells, Charles Automorphisms of group extensions, Trans. Amer. Math. Soc., Tome 155 (1971), pp. 189-194 | Article | MR 272898 | Zbl 0221.20054