Positive energy representations of double extensions of Hilbert loop algebras

Abstract

A real Lie algebra with a compatible Hilbert space structure (in the sense that the scalar product is invariant) is called a Hilbert–Lie algebra. Such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras; in particular, any infinite-dimensional simple Hilbert–Lie algebra 𝔨 is of one of the four classical types A_J, B_J, C_J or D_J for some infinite set J. Imitating the construction of affine Kac–Moody algebras, one can then consider affinisations of 𝔨, that is, double extensions of (twisted) loop algebras over 𝔨. Such an affinisation 𝔤 of 𝔨 possesses a root space decomposition with respect to some Cartan subalgebra 𝔥, whose corresponding root system yields one of the seven locally affine root systems (LARS) of type A_J⁽¹⁾, B_J⁽¹⁾, C_J⁽¹⁾, D_J⁽¹⁾, B_J⁽²⁾, C_J⁽²⁾ or BC_J⁽²⁾.

Let D ∈ der(𝔤) with 𝔥 ⊆ ker D (a diagonal derivation of 𝔤). Then every highest weight representation (ρ_λ, L(λ)) of 𝔤 with highest weight λ can be extended to a representation $\widetilde{\rho}_{\lambda}$ of the semi-direct product 𝔤 ⋊ ℝ D. In this paper, we characterise all pairs (λ,D) for which the representation $\widetilde{\rho}_{\lambda}$ is of positive energy, namely, for which the spectrum of the operator $-i \widetilde{\rho}_{\lambda}(D)$ is bounded from below.