Papers by Timothée Marquis
International Mathematics Research Notices, 2015
Let g be a locally finite split simple complex Lie algebra of type A J , B J , C J or D J and h ⊆... more Let g be a locally finite split simple complex Lie algebra of type A J , B J , C J or D J and h ⊆ g be a splitting Cartan subalgebra. Fix D ∈ der(g) with h ⊆ ker D (a diagonal derivation). Then every unitary highest weight representation (ρ λ , V λ ) of g extends to a representatioñ ρ λ of the semidirect product g ⋊ CD and we say thatρ λ is a positive energy representation if the spectrum of −iρ λ (D) is bounded from below. In the present note we characterise all pairs (λ, D) with λ bounded for which this is the case.
Canadian Journal of Mathematics, 2016
The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e.... more 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) 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 k.
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Papers by Timothée Marquis