Non-abelian extensions of infinite-dimensional Lie groups

Neeb, Karl-Hermann

Annales de l'Institut Fourier, Tome 57 (2007) no. 1, pp. 209-271.

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 $H_{ss}^2{(G,Z\left(N\right))}_S$. Pour chaque $S$ une obstruction localement lisse $\chi \left(S\right)$ dans un groupe de cohomologie $H_{ss}^3{(G,Z\left(N\right))}_S$ 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, $\alpha :H\to G$, que nous voyons comme une extension centrale d’un sous-groupe normal de $G$.

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 $\mathrm{Ext}{(G,N)}_S$ of extension classes is a principal homogeneous space of the locally smooth cohomology group $H_{ss}^2{(G,Z\left(N\right))}_S$. To each $S$ a locally smooth obstruction class $\chi \left(S\right)$ in a suitably defined cohomology group $H_{ss}^3{(G,Z\left(N\right))}_S$ 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 $\alpha :H\to G$, which we view as a central extension of a normal subgroup of $G$.

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

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