AROUND THE LIE CORRESPONDENCE FOR COMPLETE KAC-MOODY GROUPS AND GABBER-KAC SIMPLICITY - Marquis, Timothée (Annales de l'Institut Fourier, 2019)
Let k be a field and A be a generalised Cartan matrix, and let G(A)(k) be the corresponding minimal Kac-Moody group of simply connected type over k. Consider the completion G(A)(pma) (k) of G(A)(k) introduced by O. Mathieu and G. Rousseau, and let U-A(ma)+ (k) denote the unipotent radical of the Borel subgroup of G(A)(pma) (k). In this paper, we exhibit a functorial dependence of the groups U-A(ma+)(k) and G(A)(pma) ( k) ( on their Lie algebra. We also provide several contributions to fundamental questions in the general theory of maximal Kac-Moody groups: (non-)Gabber-Kac simplicity over certain finite fields, (non-)density of a minimal Kac-Moody group in its Mathieu-Rousseau completion, (non-)linearity of maximal pro-p subgroups, and the isomorphism problem.
CRIS-Team WoS Importer:2019-11-15

Half-Lie Groups - Neeb, Karl Hermann; Marquis, Timothée (Transformation Groups, 2018)

Isomorphisms of twisted Hilbert loop algebras - Marquis, Timothée; Neeb, Karl Hermann (Canadian Journal of Mathematics-Journal Canadien De Mathematiques, 2017)
The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras k, also called affinisations of k. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the 7 families A(1)J, B(1)J, C(1)J, D(1)J, B(2)J, C(2)J and BC(2)J for some infinite set J. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra k, which we call standard. In this paper, we give for each affinisation g of a simple Hilbert-Lie algebra k an explicit isomorphism from g to one of the standard affinisations of k. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of g. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of

Positive energy representations for locally finite split Lie algebras - Marquis, Timothée; Neeb, Karl Hermann (International Mathematics Research Notices, 2016)

Positive energy representations of double extensions of Hilbert loop algebras - Neeb, Karl Hermann; Marquis, Timothée (Journal of the Mathematical Society of Japan, 2017)