## 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.

### MSC:

 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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