Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups  [ Représentations unitaires délimitées ci-dessous de double-extensions des groupes de lacettes hilbertiens ]
Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 1823-1892.

Une représentation unitaire π d’un groupe de Lie G est dite semi-borné, si les opérateurs idπ(x) de la représentation derivée sont semi-bornés uniformément sur une partie ouverte de l’algèbre de Lie 𝔤 de G. Nous déterminons toutes les représentations irréductibles semi-bornées des groupes ^ φ (K) qui sont extensions doubles du groupe φ (K), où K est un groupe de Lie hilbertien et φ est une automorphisme de K d’ordre fini qui mène à l’un des 7 systèmes de racines affines irréductibles localement finis. Pour atteindre cet objectif, nous étendons la méthode d’induction holomorphe aux certaines classes de groupes de Lie-Fréchet.

Il s’agit du premier papier traitant des aspects globaux des groupes de Lie dont l’algèbre de Lie est une algèbre de Kac–Moody à rang infini.

A unitary representation π of a, possibly infinite dimensional, Lie group G is called semibounded if the corresponding operators idπ(x) from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra 𝔤 of G. We classify all irreducible semibounded representations of the groups ^ φ (K) which are double extensions of the twisted loop group φ (K), where K is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant) and φ is a finite order automorphism of K which leads to one of the 7 irreducible locally affine root systems with their canonical -grading. To achieve this goal, we extend the method of holomorphic induction to certain classes of Fréchet–Lie groups and prove an infinitesimal characterization of analytic operator-valued positive definite functions on Fréchet–BCH–Lie groups.

This is the first paper dealing with global aspects of Lie groups whose Lie algebra is an infinite rank analog of an affine Kac–Moody algebra. That positive energy representations are semibounded is a new insight, even for loops in compact Lie groups.

DOI : https://doi.org/10.5802/aif.2898
Classification : 22E65,  22E45
Mots clés : groupe de Lie de dimension infinie, représentation unitaire, algèbre de Lie hilbertienne, groupe de Lie hilbertien, groupe de Kac–Moody, groupe de lacettes, double-extension, fonction de type positif
@article{AIF_2014__64_5_1823_0,
     author = {Neeb, K. H.},
     title = {Semibounded Unitary Representations of Double Extensions of Hilbert--Loop Groups},
     journal = {Annales de l'Institut Fourier},
     pages = {1823--1892},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {5},
     year = {2014},
     doi = {10.5802/aif.2898},
     mrnumber = {3330925},
     zbl = {06387325},
     language = {en},
     url = {archive.numdam.org/item/AIF_2014__64_5_1823_0/}
}
Neeb, K. H. Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups. Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 1823-1892. doi : 10.5802/aif.2898. http://archive.numdam.org/item/AIF_2014__64_5_1823_0/

[1] Albeverio, S.; Høegh-Krohn, R. J. The energy representation of Sobolev–Lie groups, Composition Math., Volume 36 (1978) no. 1, pp. 37-51 | Numdam | MR 515036 | Zbl 0393.22013

[2] Albeverio, Sergio A.; Høegh-Krohn, Raphael J.; Marion, Jean A.; Testard, Daniel H.; Torrésani, Bruno S. Noncommutative distributions, Monographs and Textbooks in Pure and Applied Mathematics, Volume 175, Marcel Dekker, Inc., New York, 1993, pp. x+190 (Unitary representation of gauge groups and algebras) | MR 1233384 | Zbl 0791.22010

[3] Atiyah, M. F.; Pressley, A. N. Convexity and loop groups, Arithmetic and geometry, Vol. II (Progr. Math.) Volume 36, Birkhäuser Boston, Boston, MA, 1983, pp. 33-63 | MR 717605 | Zbl 0529.22013

[4] Bakalov, Bojko; Nikolov, Nikolay M.; Rehren, Karl-Henning; Todorov, Ivan Unitary positive-energy representations of scalar bilocal quantum fields, Comm. Math. Phys., Volume 271 (2007) no. 1, pp. 223-246 | Article | MR 2283959 | Zbl 1156.81422

[5] Balachandran, V. K. Simple L * -algebras of classical type, Math. Ann., Volume 180 (1969), pp. 205-219 | Article | MR 243362 | Zbl 0159.42203

[6] Beltiţă, Daniel; Neeb, Karl-Hermann A nonsmooth continuous unitary representation of a Banach-Lie group, J. Lie Theory, Volume 18 (2008) no. 4, pp. 933-936 | MR 2523145 | Zbl 1203.22013

[7] Berezin, F. A. Representations of the continuous direct product of universal coverings of the group of motions of the complex ball, Trans. Moscow Math. Soc., Volume 2 (1979), pp. 281-289 | Zbl 0434.22015

[8] Bochnak, Jacek; Siciak, Józef Analytic functions in topological vector spaces, Studia Math., Volume 39 (1971), pp. 77-112 | MR 313811 | Zbl 0214.37703

[9] Carey, A. L. Infinite-dimensional groups and quantum field theory, Acta Appl. Math., Volume 1 (1983) no. 4, pp. 321-331 | Article | MR 734749 | Zbl 0545.22019

[10] Carey, A. L.; Ruijsenaars, S. N. M. On fermion gauge groups, current algebras and Kac-Moody algebras, Acta Appl. Math., Volume 10 (1987) no. 1, pp. 1-86 | Article | MR 904924 | Zbl 0644.22012

[11] Chari, Vyjayanthi; Pressley, Andrew New unitary representations of loop groups, Math. Ann., Volume 275 (1986) no. 1, pp. 87-104 | Article | MR 849057 | Zbl 0603.17012

[12] Chari, Vyjayanthi; Pressley, Andrew Unitary representations of the maps S 1 su (N,1), Math. Proc. Cambridge Philos. Soc., Volume 102 (1987) no. 2, pp. 259-272 | Article | MR 898146 | Zbl 0662.17008

[13] Cuenca Mira, José Antonio; Amable, García Martín; Martín González, Cándido Structure theory for L * -algebras, Math. Proc. Cambridge Philos. Soc., Volume 107 (1990) no. 2, pp. 361-365 | Article | MR 1027788 | Zbl 0763.46052

[14] Fewster, Christopher J.; Hollands, Stefan Quantum energy inequalities in two-dimensional conformal field theory, Rev. Math. Phys., Volume 17 (2005) no. 5, pp. 577-612 | Article | MR 2153774 | Zbl 1083.81581

[15] Gelʼfand, I. M.; Graev, M. I.; Veršik, A. M. Representations of the group of functions taking values in a compact Lie group, Compositio Math., Volume 42 (1980/81) no. 2, pp. 217-243 | Numdam | MR 596877 | Zbl 0449.22019

[16] Glöckner, Helge; Neeb, K.-H. Infinite dimensional Lie groups, Vol. I, Basic Theory and Main Examples (book in preparation)

[17] Goertsches, O. Variationally complete and hyperpolar actions on compact symmetric spaces (2003) (Ph. D. Thesis)

[18] Goodman, Roe; Wallach, Nolan R. Erratum to the paper: “Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle” [J. Reine Angew. Math. 347 (1984), 69–133], J. Reine Angew. Math., Volume 352 (1984), pp. 220 | Article | MR 733047 | Zbl 0514.22012

[19] Helgason, Sigurdur Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, Volume 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978, pp. xv+628 | MR 514561 | Zbl 0993.53002

[20] Hofmann, G.; Neeb, K.-H. On convex hulls of orbits of Coxeter groups and Weyl groups (arXiv:math.RT.1204.2095, to appear in Muenster Journal of Mathematics)

[21] Hofmann, Karl H.; Morris, Sidney A. The structure of compact groups, de Gruyter Studies in Mathematics, Volume 25, Walter de Gruyter & Co., Berlin, 1998, pp. xviii+835 (A primer for the student—a handbook for the expert) | MR 1646190 | Zbl 0919.22001

[22] Jakobsen, Hans Plesner; Kac, Victor G. A new class of unitarizable highest weight representations of infinite-dimensional Lie algebras, Nonlinear equations in classical and quantum field theory (Meudon/Paris, 1983/1984) (Lecture Notes in Phys.) Volume 226, Springer, Berlin, 1985, pp. 1-20 | Article | MR 802097 | Zbl 0581.17009

[23] Jakobsen, Hans Plesner; Kac, Victor G. A new class of unitarizable highest weight representations of infinite-dimensional Lie algebras. II, J. Funct. Anal., Volume 82 (1989) no. 1, pp. 69-90 | Article | MR 976313 | Zbl 0688.17007

[24] Janssens, Bas; Neeb, K.-H. Norm continuous unitary representations of Lie algebras of smooth sections (arXiv:math.RT.1302.2535)

[25] Janssens, Bas; Wockel, Christoph Universal central extensions of gauge algebras and groups, J. Reine Angew. Math., Volume 682 (2013), pp. 129-139 | MR 3181501

[26] Kac, Victor G. Infinite-dimensional Lie algebras, Cambridge University Press, Cambridge, 1990, pp. xxii+400 | Article | MR 1104219 | Zbl 0716.17022

[27] Kac, Victor G.; Peterson, D. H. Unitary structure in representations of infinite-dimensional groups and a convexity theorem, Invent. Math., Volume 76 (1984) no. 1, pp. 1-14 | Article | MR 739620 | Zbl 0534.17008

[28] Kühn, Katharina Direct limits of diagonal chains of type O, U, and Sp, and their homotopy groups, Comm. Algebra, Volume 34 (2006) no. 1, pp. 75-87 | Article | MR 2194350 | Zbl 1088.22004

[29] Loos, Ottmar; Neher, Erhard Locally finite root systems, Mem. Amer. Math. Soc., Volume 171 (2004) no. 811, pp. x+214 | Article | MR 2073220 | Zbl 1195.17007

[30] Maier, Peter Central extensions of topological current algebras, Geometry and analysis on finite- and infinite-dimensional Lie groups (Będlewo, 2000) (Banach Center Publ.) Volume 55, Polish Acad. Sci., Warsaw, 2002, pp. 61-76 | MR 1911980 | Zbl 1045.17008

[31] Maier, Peter; Neeb, Karl-Hermann Central extensions of current groups, Math. Ann., Volume 326 (2003) no. 2, pp. 367-415 | Article | MR 1990915 | Zbl 1029.22025

[32] Maissen, Bernhard Lie-Gruppen mit Banachräumen als Parameterräume, Acta Math., Volume 108 (1962), pp. 229-270 | Article | MR 142693 | Zbl 0207.33701

[33] Medina, Alberto; Revoy, Philippe Algèbres de Lie et produit scalaire invariant, Ann. Sci. école Norm. Sup. (4), Volume 18 (1985) no. 3, pp. 553-561 | Numdam | MR 826103 | Zbl 0592.17006

[34] Mickelsson, Jouko Current algebras and groups, Plenum Monographs in Nonlinear Physics, Plenum Press, New York, 1989, pp. xviii+313 | Article | MR 1032521 | Zbl 0726.22015

[35] Morita, Jun; Yoshii, Yoji Locally extended affine Lie algebras, J. Algebra, Volume 301 (2006) no. 1, pp. 59-81 | Article | MR 2230320 | Zbl 1147.17019

[36] Neeb, Karl-Hermann Projective semibounded representations of doubly extended Hilbert–Lie groups (in preparation)

[37] Neeb, Karl-Hermann Holomorphic highest weight representations of infinite-dimensional complex classical groups, J. Reine Angew. Math., Volume 497 (1998), pp. 171-222 | Article | MR 1617431 | Zbl 0894.22007

[38] Neeb, Karl-Hermann Holomorphy and convexity in Lie theory, de Gruyter Expositions in Mathematics, Volume 28, Walter de Gruyter & Co., Berlin, 2000, pp. xxii+778 | Article | MR 1740617 | Zbl 0936.22001

[39] Neeb, Karl-Hermann Central extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier (Grenoble), Volume 52 (2002) no. 5, pp. 1365-1442 http://aif.cedram.org/item?id=AIF_2002__52_5_1365_0 | Article | Numdam | MR 1935553 | Zbl 1019.22012

[40] Neeb, Karl-Hermann Classical Hilbert-Lie groups, their extensions and their homotopy groups, Geometry and analysis on finite- and infinite-dimensional Lie groups (Będlewo, 2000) (Banach Center Publ.) Volume 55, Polish Acad. Sci., Warsaw, 2002, pp. 87-151 | MR 1911982 | Zbl 1010.22024

[41] Neeb, Karl-Hermann Towards a Lie theory of locally convex groups, Jpn. J. Math., Volume 1 (2006) no. 2, pp. 291-468 | Article | MR 2261066 | Zbl 1161.22012

[42] Neeb, Karl-Hermann Semibounded representations and invariant cones in infinite dimensional Lie algebras, Confluentes Math., Volume 2 (2010) no. 1, pp. 37-134 | Article | MR 2649236 | Zbl 1186.22023

[43] Neeb, Karl-Hermann Unitary highest weight modules of locally affine Lie algebras, Quantum affine algebras, extended affine Lie algebras, and their applications (Contemp. Math.) Volume 506, Amer. Math. Soc., Providence, RI, 2010, pp. 227-262 | Article | MR 2642569 | Zbl 1267.17028

[44] Neeb, Karl-Hermann On analytic vectors for unitary representations of infinite dimensional Lie groups, Ann. Inst. Fourier (Grenoble), Volume 61 (2011) no. 5, pp. 1441-1476 | Article | Numdam | MR 2961842 | Zbl 1241.22023

[45] Neeb, Karl-Hermann Semibounded representations of Hermitian Lie groups, Travaux mathématiques. Vol. XXI (Trav. Math.) Volume 21, Fac. Sci. Technol. Commun. Univ. Luxemb., Luxembourg, 2012, pp. 29-109 | MR 2987649

[46] Neeb, Karl-Hermann Holomorphic realization of unitary representations of Banach-Lie groups, Lie groups: structure, actions, and representations (Progr. Math.) Volume 306, Birkhäuser/Springer, New York, 2013, pp. 185-223 | Article | MR 3186693 | Zbl 1282.22012

[47] Neeb, Karl-Hermann; Seppänen, Henrik Borel-Weil theory for groups over commutative Banach algebras, J. Reine Angew. Math., Volume 655 (2011), pp. 165-187 | Article | MR 2806110 | Zbl 1239.22016

[48] Neeb, Karl-Hermann; Wockel, Christoph Central extensions of groups of sections, Ann. Global Anal. Geom., Volume 36 (2009) no. 4, pp. 381-418 | Article | MR 2562922 | Zbl 1215.53031

[49] Neher, E. Generators and relations for 3-graded Lie algebras, J. Algebra, Volume 155 (1993) no. 1, pp. 1-35 | Article | MR 1206620 | Zbl 0769.17019

[50] Ottesen, Johnny T. Infinite-dimensional groups and algebras in quantum physics, Lecture Notes in Physics. New Series m: Monographs, Volume 27, Springer-Verlag, Berlin, 1995, pp. viii+218 | MR 1345151 | Zbl 0848.22026

[51] Palais, Richard S. Foundations of global non-linear analysis, W. A. Benjamin, Inc., New York-Amsterdam, 1968, pp. vii+131 | MR 248880 | Zbl 0164.11102

[52] Pressley, Andrew; Segal, Graeme Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986, pp. viii+318 (Oxford Science Publications) | MR 900587 | Zbl 0638.22009

[53] Schmüdgen, Konrad Unbounded operator algebras and representation theory, Operator Theory: Advances and Applications, Volume 37, Birkhäuser Verlag, Basel, 1990, pp. 380 | Article | MR 1056697 | Zbl 0697.47048

[54] Schue, John R. Hilbert space methods in the theory of Lie algebras, Trans. Amer. Math. Soc., Volume 95 (1960), pp. 69-80 | Article | MR 117575 | Zbl 0093.30601

[55] Schue, John R. Cartan decompositions for L * algebras, Trans. Amer. Math. Soc., Volume 98 (1961), pp. 334-349 | MR 133408 | Zbl 0099.10205

[56] Segal, Graeme Unitary representations of some infinite-dimensional groups, Comm. Math. Phys., Volume 80 (1981) no. 3, pp. 301-342 http://projecteuclid.org/euclid.cmp/1103919978 | Article | MR 626704 | Zbl 0495.22017

[57] Segal, I. E. Distributions in Hilbert space and canonical systems of operators, Trans. Amer. Math. Soc., Volume 88 (1958), pp. 12-41 | Article | MR 102759 | Zbl 0099.12104

[58] Segal, I. E. The complex-wave representation of the free boson field, Topics in functional analysis (essays dedicated to M. G. Kreĭn on the occasion of his 70th birthday) (Adv. in Math. Suppl. Stud.) Volume 3, Academic Press, New York-London, 1978, pp. 321-343 | MR 538026 | Zbl 0471.22024

[59] Stumme, Nina Locally finite split Lie algebras (1999) (Ph. D. Thesis) | Zbl 0931.17001

[60] Stumme, Nina Automorphisms and conjugacy of compact real forms of the classical infinite dimensional matrix Lie algebras, Forum Math., Volume 13 (2001) no. 6, pp. 817-851 | Article | MR 1861251 | Zbl 1010.17015

[61] Terng, Chuu-Lian Proper Fredholm submanifolds of Hilbert space, J. Differential Geom., Volume 29 (1989) no. 1, pp. 9-47 http://projecteuclid.org/euclid.jdg/1214442631 | MR 978074 | Zbl 0674.58004

[62] Toledano Laredo, Valerio Positive energy representations of the loop groups of non-simply connected Lie groups, Comm. Math. Phys., Volume 207 (1999) no. 2, pp. 307-339 | Article | MR 1724846 | Zbl 0969.22010

[63] Torrésani, Bruno Unitary positive energy representations of the gauge group, Lett. Math. Phys., Volume 13 (1987) no. 1, pp. 7-15 | Article | MR 878656 | Zbl 0618.22013

[64] Veršik, A. M.; Gelʼfand, I. M.; Graev, M. I. Irreducible representations of the group G X and cohomology, Funkcional. Anal. i Priložen., Volume 8 (1974) no. 2, pp. 67-69 | MR 348032 | Zbl 0299.22004

[65] Wloka, Joseph Partielle Differentialgleichungen, B. G. Teubner, Stuttgart, 1982, pp. 500 (Sobolevräume und Randwertaufgaben. [Sobolev spaces and boundary value problems], Mathematische Leitfäden. [Mathematical Textbooks]) | MR 652934 | Zbl 0482.35001

[66] Wockel, Christoph Smooth extensions and spaces of smooth and holomorphic mappings, J. Geom. Symmetry Phys., Volume 5 (2006), pp. 118-126 | MR 2269885 | Zbl 1108.58006

[67] Yoshii, Yoji Locally extended affine root systems, Quantum affine algebras, extended affine Lie algebras, and their applications (Contemp. Math.) Volume 506, Amer. Math. Soc., Providence, RI, 2010, pp. 285-302 | Article | MR 2642571 | Zbl 1247.17009

[68] Zellner, Ch. Semibounded representations of oscillator groups, 2014 (Dissertation, FAU Erlangen–Nuremberg)