Positive energy representations for locally finite split Lie algebras

TL;DR

This paper characterizes positive energy highest weight representations of locally finite split simple Lie algebras extended by a derivation, identifying conditions for bounded below spectra and minimal energy configurations.

Contribution

It provides a complete characterization of pairs (λ, D) for which highest weight representations have positive energy, including the minimal energy case, in the context of infinite-dimensional Lie algebras.

Findings

All pairs (λ, D) with bounded λ that yield positive energy representations are characterized.

The minimal eigenvalue of the extended representation can be normalized to zero by adding an inner derivation.

Explicit descriptions of pairs satisfying the minimal energy condition are provided.

Abstract

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 $(\rho_\lambda, V^\lambda)$ of $\mathfrak g$ extends to a representation $\tilde\rho_\lambda$ of the semidirect product $\mathfrak g \rtimes \mathbb C D$ and we say that $\tilde\rho_\lambda$ is a positive energy representation if the spectrum of $-i\tilde\rho_\lambda(D)$ is bounded from below. In the present note we characterise all pairs $(\lambda,D)$ with $\lambda$ 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 $\lambda$ is bounded, then we accordingly obtain a characterisation of those highest weight representations $\pi_\lambda$ satisfying the positive energy condition with respect to the continuous $\mathbb R$-action induced by $D$. In this context the representation $\pi_\lambda$ 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 $\tilde\rho_\lambda(D)$ is $0$ (minimal energy condition). The corresponding pairs $(\lambda,D)$ satisfying the minimal energy condition are rather easy to describe explicitly.