Séminaires sur les aspects théoriques et expérimentaux de la gravitation
# Generalized spin representations

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Amphithéâtre Léon Motchane (Institut des Hautes Etudes Scientifiques)
### Amphithéâtre Léon Motchane

#### Institut des Hautes Etudes Scientifiques

Bois-Marie
35, route de Chartres
91440 Bures-sur-Yvette

Description

Motivated by the 32-dimensional extension of the spin representation of the compact Lie algebra so(10) to the 'maximal compact' subalgebra of the real Kac-Moody Lie algebra of type E10 described by Damour et al. and Henneaux et al. Hainke and myself introduced the concept of a generalized spin representation that allows similar constructions for the 'maximal compact' subalgebras of real Kac-Moody Lie algebras of arbitrary simply laced type.
By work of Ghatei, Horn, Weiss and myself, integration of these representations leads to two-fold spin covers of the 'maximal compact' subgroups of the corresponding split real Kac-Moody groups. The problem that semisimple elements generally do not have a locally finite action and therefore obstruct integration is circumvented by an amalgamation method using the Iwasawa decomposition and the theory of buildings. The existence of these spin covers has been conjectured by Damour and Hillmann; it contains an extended Weyl group, which in the E10 case is relevant to fermionic billards.

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