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Positive energy representations for locally finite split Lie algebras

Let $\mathfrak g$ be a locally finite split simple complex Lie algebra of type $A_J$, $B_J$, $C_J$ or $D_J$ and $\mathfrak h \subseteq \mathfrak g$ be a splitting Cartan subalgebra. Fix $D \in \mathrm{der}(\mathfrak g)$ with $\mathfrak h \subseteq \ker D$ (a diagonal derivation). Then every unitary highest weight representation $(ρ_λ, V^λ)$ of $\mathfrak g$ extends to a representation $\tildeρ_λ$ of the semidirect product $\mathfrak g \rtimes \mathbb C D$ and we say that $\tildeρ_λ$ is a positive energy representation if the spectrum of $-i\tildeρ_λ(D)$ is bounded from below. In the present note we characterise all pairs $(λ,D)$ with $λ$ bounded for which this is the case. If $U_1(\mathcal H)$ is the unitary group of Schatten class $1$ on an infinite dimensional real, complex or quaternionic Hilbert space and $λ$ is bounded, then we accordingly obtain a characterisation of those highest weight representations $π_λ$ satisfying the positive energy condition with respect to the continuous $\mathbb R$-action induced by $D$. In this context the representation $π_λ$ is norm continuous and our results imply the remarkable result that, for positive energy representations, adding a suitable inner derivation to $D$, we can achieve that the minimal eigenvalue of $\tildeρ_λ(D)$ is $0$ (minimal energy condition). The corresponding pairs $(λ,D)$ satisfying the minimal energy condition are rather easy to describe explicitly.