Self-representations of the Möbius group

Monod, Nicolas; Py, Pierre

Annales Henri Lebesgue, Tome 2 (2019), pp. 259-280.

Contrairement au cas usuel de dimension finie, le groupe de Möbius admet des auto-représentations intéressantes lorsqu’il est de dimension infinie. Nous les construisons et classifions toutes.

Les démonstrations sont conduites dans le cadre équivalent des groupes d’isométries des espaces de Lobatchevski et reposent sur le concept de noyau de type hyperbolique, en analogie avec la notion classique de noyau de type positif ou négatif.

Contrary to the finite-dimensional case, the Möbius group admits interesting self-representations when infinite-dimensional. We construct and classify all these self-representations.

The proofs are obtained in the equivalent setting of isometries of Lobachevsky spaces and use kernels of hyperbolic type, in analogy with the classical concepts of kernels of positive and negative type.

Classification : 53A35, 57S25, 53C50
Mots clés : Möbius group, Lobatchevsky space, hyperbolic space, infinite-dimensional space

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