Positive energy representations of double extensions of Hilbert loop algebras. (English) Zbl 1430.17026

Summary: 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 \(D\in\operatorname{der}(\mathfrak{g})\) with \(\mathfrak{h}\subseteq\operatorname{ker}D\) (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.


17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B65 Infinite-dimensional Lie (super)algebras
17B70 Graded Lie (super)algebras
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
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