Skip to main content Accessibility help
×
Home

Isomorphisms of Twisted Hilbert Loop Algebras

  • Timothée Marquis (a1) and Karl-Hermann Neeb

Abstract

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 $\mathfrak{k}$ .

Copyright

References

Hide All
[HN14] Hofmann, G. and Neeb, K.-H.,On convex hulls of orbits of Coxeter groups and Weyl groups. Munster J. Math. 7(2014), 463487.
[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
[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
[MN15b] Marquis, T., Positive energy representations of double extensions of Hilbert loop algebras. Preprint(2015). arxiv:1511.03980
[MY06] Morita, J. and Yoshii, Y., Locally extended affine Lie algebras. J. Algebra 301(2006), no. 1,5981.http://dx.doi.Org/10.1 01 6/j.jalgebra.2OO5.O6.O13
[MY15] Morita, J. , Locally loop algebras and locally affine Lie algebras. J. Algebra 440(2015), 379442.http://dx.doi.Org/10.1016/j.jalgebra.2015.05.018
[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. 227262.http://dx.doi.Org/10.1090/conm/506/09943
[Neel4] Neeb, K.-H., Semibounded unitary representations of double extensions of Hilbert-loop groups. Ann. Inst. Fourier (Grenoble) 64(2014), no. 5,18231892. http://dx.doi.Org/10.58O2/aif.2898
[NSOl] Neeb, K.-H. and Stumme, N., The classification of locally finite split simple Lie algebras. J. Reine Angew. Math. 533(2001), 2553.http://dx.doi.Org/10.1515/crll.2OO1.025
[Sch61] Schue, J. R., Cartan decompositions for L* algebras. Trans. Amer. Math. Soc. 98(1961), 334349.
[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. 285302.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed