Journal of the Mathematical Society of Japan

Positive energy representations of double extensions of Hilbert loop algebras

Timothée MARQUIS and Karl-Hermann NEEB

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Abstract

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.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 4 (2017), 1485-1518.

Dates
First available in Project Euclid: 25 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1508918566

Digital Object Identifier
doi:10.2969/jmsj/06941485

Mathematical Reviews number (MathSciNet)
MR3715813

Zentralblatt MATH identifier
06821649

Subjects
Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 17B65: Infinite-dimensional Lie (super)algebras [See also 22E65] 17B70: Graded Lie (super)algebras 22E65: Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05]

Keywords
locally affine Lie algebras and root systems positive energy representations

Citation

MARQUIS, Timothée; NEEB, Karl-Hermann. Positive energy representations of double extensions of Hilbert loop algebras. J. Math. Soc. Japan 69 (2017), no. 4, 1485--1518. doi:10.2969/jmsj/06941485. https://projecteuclid.org/euclid.jmsj/1508918566


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