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October, 2017 Positive energy representations of double extensions of Hilbert loop algebras
Timothée MARQUIS, Karl-Hermann NEEB
J. Math. Soc. Japan 69(4): 1485-1518 (October, 2017). DOI: 10.2969/jmsj/06941485


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 $D\in\mathrm{der}(\mathfrak{g})$ with $\mathfrak{h}\subseteq\mathrm{ker}D$ (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.


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Timothée MARQUIS. Karl-Hermann NEEB. "Positive energy representations of double extensions of Hilbert loop algebras." J. Math. Soc. Japan 69 (4) 1485 - 1518, October, 2017.


Published: October, 2017
First available in Project Euclid: 25 October 2017

zbMATH: 06821649
MathSciNet: MR3715813
Digital Object Identifier: 10.2969/jmsj/06941485

Primary: 17B10
Secondary: 17B65 , 17B70 , 22E65

Keywords: locally affine Lie algebras and root systems , positive energy representations

Rights: Copyright © 2017 Mathematical Society of Japan


Vol.69 • No. 4 • October, 2017
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