Character groups of Hopf algebras as infinite-dimensional Lie groups

Bogfjellmo, Geir; Dahmen, Rafael; Schmeding, Alexander

Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 2101-2155.

Dans cet article, nous étudions les groupes de caractères des algèbres de Hopf du point de vue de la théorie de Lie de dimension infinie. Pour une algèbre de Hopf connexe et graduée, nous munissons le groupe de caractères d’une structure de groupe de Lie de dimension infinie, à valeurs dans une algèbre localement convexe. Cette structure permet de voir le groupe de caractères comme un groupe de Lie de Baker–Campbell–Hausdorff, qui est régulier au sens de Milnor. De plus, nous montrons que certains sous-groupes associés aux idéaux de Hopf sont alors des sous-groupes de Lie du groupe de caractères.

Si l’algèbre de Hopf n’est pas graduée, son groupe de caractères ne sera pas un groupe de Lie, en général. Cependant, nous montrons que pour une algèbre de Hopf quelconque, le groupe de caractères à valeurs dans une algèbre faiblement complète est un groupe pro-Lie au sens de Hofmann et Morris.

In this article character groups of Hopf algebras are studied from the perspective of infinite-dimensional Lie theory. For a graded and connected Hopf algebra we obtain an infinite-dimensional Lie group structure on the character group with values in a locally convex algebra. This structure turns the character group into a Baker–Campbell–Hausdorff–Lie group which is regular in the sense of Milnor. Furthermore, we show that certain subgroups associated to Hopf ideals become closed Lie subgroups of the character group.

If the Hopf algebra is not graded, its character group will in general not be a Lie group. However, we show that for any Hopf algebra the character group with values in a weakly complete algebra is a pro-Lie group in the sense of Hofmann and Morris.

Classification : 22E65, 16T05, 43A40, 58B25, 46H30, 22A05
Keywords: real analytic, infinite-dimensional Lie group, Hopf algebra, continuous inverse algebra, Butcher group, weakly complete space, pro-Lie group
Mot clés : réel analytique, groupe de Lie de dimension infinie, algèbres de Hopf, algèbre avec inversion continue, espace faiblement complet, groupe pro-Lie

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