Isomorphisms of Twisted Hilbert LoopAlgebras

Marquis, Timothée

Canadian journal of mathematics, Tome 69 (2017) no. 2, pp. 453-480

Résumé

The closest infinite-dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e., real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras $\left( \text{LALAs} \right)$ correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$ , also called affinisations of $\mathfrak{k}$ . They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the 7 families $$A_{J}^{\left( 1 \right)},\,B_{J}^{\left( 1 \right)},\,C_{J}^{\left( 1 \right)},\,D_{J}^{\left( 1 \right)},\,B_{J}^{\left( 2 \right)},\,C_{J}^{\left( 2 \right)},\,\,\text{and}\,BC_{J}^{\left( 2 \right)}$$ for some infinite set $J$ . To each of these types corresponds a “minimal ” affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$ , which we call standard.In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$ . The existence of such an isomorphism could also be derived from the classiffication of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitly as a deformation between two twists that is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$ . In subsequent work, this paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$ .

Détail

DOI : 10.4153/CJM-2016-003-x

Mots-clés : 17B65, 17B70, 17B22, 17B10, locally affine Lie algebra, Hilbert-Lie algebra, positive energy representation

Bibliographie

[HN14] [HN14] Hofmann, G. and Neeb, K.-H.,On convex hulls of orbits of Coxeter groups and Weyl groups. Munster J. Math. 7(2014), 463–487. Google Scholar

[LN04] [LN04] Loos, O. and Neher, E., Locally finite root systems. Mem. Amer. Math. Soc. 171(2004), no. 811. http://dx.doi.Org/10.1090/memo/0811 Google Scholar

[MN15a] [MN15a] Marquis, T. and Neeb, K.-H., Positive energy representations for locally finite split Lie algebras. Int. Math. Res. Notices (2015). http://dx.doi.Org/10.1093/imrn/rnv367 Google Scholar

[MN15b] [MN15b] Marquis, T., Positive energy representations of double extensions of Hilbert loop algebras. Preprint(2015). arxiv:1511.03980 Google Scholar

[MY06] [MY06] Morita, J. and Yoshii, Y., Locally extended affine Lie algebras. J. Algebra 301(2006), no. 1,59–81. http://dx.doi.Org/10.1 01 6/j.jalgebra.2OO5.O6.O13 Google Scholar

[MY15] [MY15] Morita, J. , Locally loop algebras and locally affine Lie algebras. J. Algebra 440(2015), 379–442. http://dx.doi.Org/10.1016/j.jalgebra.2015.05.018 Google Scholar

[NeelO] [NeelO] Neeb, K.-H., Unitary highest weight modules of locally affine Lie algebras. In: Quantum affine algebras, extended affine Lie algebras, and their applications, Contemp. Math., 506, American Mathematics Society, Providence, RI, 2010, pp. 227–262. http://dx.doi.Org/10.1090/conm/506/09943 Google Scholar

[Neel4] [Neel4] Neeb, K.-H., Semibounded unitary representations of double extensions of Hilbert-loop groups. Ann. Inst. Fourier (Grenoble) 64(2014), no. 5,1823–1892. http://dx.doi.Org/10.58O2/aif.2898 Google Scholar

[NSOl] [NSOl] Neeb, K.-H. and Stumme, N., The classification of locally finite split simple Lie algebras. J. Reine Angew. Math. 533(2001), 25–53. http://dx.doi.Org/10.1515/crll.2OO1.025 Google Scholar

[Sch61] [Sch61] Schue, J. R., Cartan decompositions for L* algebras. Trans. Amer. Math. Soc. 98(1961), 334–349. Google Scholar

[YoslO] [YoslO] Yoshii, Y., Locally extended affine root systems. In: Quantum affine algebras, extended affine Lie algebras, and their applications, Contemp. Math., 506, American Mathematics Society, Providence, RI, 2010, pp. 285–302. Google Scholar