Infinite dimensional Riemannian symmetric spaces with fixed-sign curvature operator

Duchesne, Bruno

Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 211-244.

Nous associons à tout espace riemannien symétrique (de dimension finie ou non) une $L^{*}$-algèbre dès lors que l’opérateur de courbure est de signe fixe. Les $L^{*}$-algèbres sont des algèbres de Lie avec une structure d’espace de Hilbert compatible. La $L^{*}$-algèbre que nous construisons est un invariant d’isomorphisme local et nous permet de classifier les espaces symétriques riemanniens simplement connexe avec un opérateur de courbure de signe fixe. Le cas de la courbure négative est mis en avant.

We associate to any Riemannian symmetric space (of finite or infinite dimension) a L${}^{*}$-algebra, under the assumption that the curvature operator has a fixed sign. L${}^{*}$-algebras are Lie algebras with a pleasant Hilbert space structure. The L${}^{*}$-algebra that we construct is a complete local isomorphism invariant and allows us to classify simply-connected Riemannian symmetric spaces with fixed-sign curvature operator. The case of nonpositive curvature is emphasized.

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