Positive energy representations of double extensions of Hilbert loop algebras

Marquis, Timothée [UCL] Neeb, Karl-Hermann

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 $mathfrak{k}$ 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 $mathfrak{k}$, that is, double extensions of (twisted) loop algebras over $mathfrak{k}$. Such an affinisation $mathfrak{g}$ of $mathfrak{k}$ possesses a root space decomposition with respect to some Cartan subalgebra $mathfrak{h}$, whose corresponding root system yields one of the seven locally affine root systems (LARS) of type $A_J^{(1)}$, $B^{(1)}_J$, $C^{(1)}_J$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ or $BC_J^{(2)}$. Let $D$ be a diagonal derivation of $mathfrak{g}$. Then every highest weight representation $(rho_{lambda},L(lambda))$ of $mathfrak{g}$ with highest weight $lambda$ can be extended to a representation $widetilde{rho}_{lambda}$ of the semi-direct product $mathfrak{g} rtimes_{mathbb{R}} D$. In this paper, we characterise all pairs $(lambda,D)$ 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.