@article{CML_2010__2_1_37_0,
author = {Neeb, Karl-Hermann},
title = {Semibounded representations and invariant cones in infinite dimensional lie algebras},
journal = {Confluentes Mathematici},
pages = {37--134},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {2},
number = {1},
year = {2010},
doi = {10.1142/S1793744210000132},
language = {en},
url = {http://www.numdam.org/articles/10.1142/S1793744210000132/}
}
TY - JOUR AU - Neeb, Karl-Hermann TI - Semibounded representations and invariant cones in infinite dimensional lie algebras JO - Confluentes Mathematici PY - 2010 SP - 37 EP - 134 VL - 2 IS - 1 PB - World Scientific Publishing Co Pte Ltd UR - http://www.numdam.org/articles/10.1142/S1793744210000132/ DO - 10.1142/S1793744210000132 LA - en ID - CML_2010__2_1_37_0 ER -
%0 Journal Article %A Neeb, Karl-Hermann %T Semibounded representations and invariant cones in infinite dimensional lie algebras %J Confluentes Mathematici %D 2010 %P 37-134 %V 2 %N 1 %I World Scientific Publishing Co Pte Ltd %U http://www.numdam.org/articles/10.1142/S1793744210000132/ %R 10.1142/S1793744210000132 %G en %F CML_2010__2_1_37_0
Neeb, Karl-Hermann. Semibounded representations and invariant cones in infinite dimensional lie algebras. Confluentes Mathematici, Tome 2 (2010) no. 1, pp. 37-134. doi : 10.1142/S1793744210000132. http://www.numdam.org/articles/10.1142/S1793744210000132/
[1] R. Abraham and J. E. Marsden , Foundations of Mechanics , 2nd edn. ( Benjamin/Cummings , 1978 ) .
[2] H. Araki and W. Wyss, Helv. Phys. Acta 37, 136 (1964).
[3] M. F. Atiyah and A. N. Pressley, Arithmetic and Geometry II (Birkhäuser, 1983) pp. 33-63.
[4] B. Bakalov, Comm. Math. Phys. 271, 223 (2007), DOI: 10.1007/s00220-006-0182-2 .
[5] E. Balslev, J. Manuceau and A. Verbeure, Comm. Math. Phys. 8, 315 (1968), DOI: 10.1007/BF01646271 .
[6] D. Beltita , Smooth Homogeneous Structures in Operator Theory ( Chapman and Hall , 2006 ) .
[7] D. Beltita and K.-H. Neeb, J. Lie Th. 18, 933 (2008).
[8] D. Beltita and K.-H. Neeb, Schur-Weyl Theory for C*-algebras, in preparation .
[9] J. Bochnak and J. Siciak, Stud. Math. 39, 77 (1971).
[10] H.-J. Borchers , Translation Group and Particle Representations in Quantum Field Theory ( Springer , 1996 ) .
[11] N. Bourbaki , General Topology ( Springer , 1989 ) .
[12] N. Bourbaki , Espaces Vectoriels Topologiques ( Springer , 2007 ) .
[13] M. J. Bowick and S. G. Rajeev, Nucl. Phys. B 293, 348 (1987), DOI: 10.1016/0550-3213(87)90076-9 .
[14] O. Bratteli and D. W. Robinson , Operator Algebras and Quantum Statistical Mechanics 2 , 2nd edn. ( Springer , 1997 ) .
[15] A. L. Carey, Act. Appl. Math. 1, 321 (1983), DOI: 10.1007/BF00120480 .
[16] A. Carey and K. C. Hannabus, Comm. Math. Phys. 176, 321 (1996), DOI: 10.1007/BF02099552 .
[17] A. Carey and E. Langmann, Geometric Analysis and Applications to Quantum Field Theory, Progr. in Math. 205, eds. P. Bouwknegt and S. Wu (Birkhäuser, 2002) pp. 45-94.
[18] A. Carey and S. N. M. Ruijsenaars, Acta Appl. Math. 10, 1 (1987), DOI: 10.1007/BF00046582 .
[19] R. Delbourgo and J. R. Fox, J. Phys. A: Math. Gen. 10, 233 (1977), DOI: 10.1088/0305-4470/10/12/004 .
[20] J. Dixmier , Les C*-algèbres et Leurs Représentations ( Gauthier-Villars , 1964 ) .
[21] Chr. Fewster and S. Hollands, Rev. Math. Phys. 17, 577 (2005), DOI: 10.1142/S0129055X05002406 . [Abstract]
[22] I. M. Gel’fand , N. Ya Vilenkin and Amiel Feinstein , Generalized Functions, Vol. 4: Applications of Harmonic Analysis ( Academic Press , 1964 ) .
[23] V. Georgescu, On the spectral analysis of quantum field Hamiltonians , arXiv:math-ph/0604072v1 .
[24] H. Glöckner, Stud. Math. 153, 147 (2002), DOI: 10.4064/sm153-2-4 .
[25] H. Glöckner and K.-H. Neeb, Infinite Dimensional Lie Groups, Vol. I, Basic Theory and Main Examples, in preparation .
[26] P. Goddard and D. Olive, Internat. J. Mod. Phys. A 1, 303 (1986), DOI: 10.1142/S0217751X86000149 . [Abstract]
[27] R. Goodman and N. R. Wallach, J. Reine Ang. Math. 347, 69 (1984).
[28] R. Goodman and N. R. Wallach, J. Funct. Anal. 63, 299 (1985), DOI: 10.1016/0022-1236(85)90090-4 .
[29] H. Grundling and K.-H. Neeb, Full regularity for a C*-algebra of the canonical commutation relations, Rev. Math. Phys .
[30] L. Guieu and C. Roger , L’Algèbre et le Groupe de Virasoro: Aspects Géométriques et Algébriques, Généralisations , Les Publications ( CRM , 2007 ) .
[31] R. Haag and D. Kastler, J. Math. Phys. 5, 848 (1964), DOI: 10.1063/1.1704187 .
[32] K. C. Hannabus, Quart. J. Math. 33, 91 (1982), DOI: 10.1093/qmath/33.1.91 .
[33] P. de la Harpe, Compos. Math. 25, 245 (1972).
[34] G. C. Hegerfeldt, J. Math. Phys. 13, 821 (1972), DOI: 10.1063/1.1666057 .
[35] J. Hilgert and K. H. Hofmann, Monatsh. Math. 100, 183 (1985), DOI: 10.1007/BF01299267 .
[36] J. Hilgert , K. H. Hofmann and J. D. Lawson , Lie Groups, Convex Cones, and Semigroups ( Oxford Univ. Press , 1989 ) .
[37] J. Hilgert, K.-H. Neeb and W. Plank, Comp. Math. 94, 129 (1994).
[38] J. Hilgert and G. Ólafsson , Causal Symmetric Spaces, Geometry and Harmonic Analysis ( Academy Press , 1996 ) .
[39] K. H. Hofmann and S. A. Morris , The Structure of Compact Groups ( de Gruyter , 1998 ) .
[40] V. G. Kac and D. H. Peterson, Invent. Math. 76, 1 (1984), DOI: 10.1007/BF01388487 .
[41] V. G. Kac and A. K. Raina , Highest Weight Representations of Infinite Dimensional Lie Algebras ( World Scientific , 1987 ) .
[42] W. Kaup, Über die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension I, II, Math. Ann. 257 (1981) 463-486; 262 (1983) 57-75 .
[43] W. Kaup, Manuscripta Math. 92, 191 (1997), DOI: 10.1007/BF02678189 .
[44] A. Kirillov, Funct. Anal. Appl. 21, 122 (1987), DOI: 10.1007/BF01078025 .
[45] A. Kirillov, J. Math. Pures Appl. 77, 735 (1998).
[46] A. A. Kirillov and D. V. Yuriev, Funct. Anal. Appl. 21, 284 (1987), DOI: 10.1007/BF01077802 .
[47] A. A. Kirillov and D. V. Yuriev, J. Geom. Phys. 5, 351 (1988), DOI: 10.1016/0393-0440(88)90029-0 .
[48] B. Kostant and S. Sternberg, Quantum Theories and Geometry, eds. M. Cahen and M. Flato (Kluwer, 1988) pp. 113-1225.
[49] E. Langmann, J. Math. Phys. 35, 96 (1994), DOI: 10.1063/1.530744 .
[50] L. Lempert, Math. Res. Lett. 2, 479 (1995).
[51] G. Mack and M. de Riese, Simple space-time symmetries: Generalized conformal field theory , arXiv:hep-th/0410277v2 .
[52] M. Magyar , Continuous Linear Representations , Math. Studies 168 ( North-Holland , 1992 ) .
[53] J. E. Marsden and T. Ratiu , Introduction to Mechanics and Symmetry , 2nd edn. , Texts in Appl. Math. 17 ( Springer , 1999 ) .
[54] A. Medina and P. Revoy, Ann. scient. Éc. Norm. Sup. 4e série 18, 533 (1985).
[55] J. Mickelsson, Comm. Math. Phys. 110, 173 (1987), DOI: 10.1007/BF01207361 .
[56] J. Mickelsson , Current Algebras and Groups ( Plenum Press , 1989 ) .
[57] J. Milnor, Relativité, Groupes et Topologie II (Les Houches, 1983), eds. B. DeWitt and R. Stora (North-Holland, 1984) pp. 1007-1057.
[58] K.-H. Neeb, J. Lie Th. 4, 1 (1994).
[59] K.-H. Neeb, J. Reine Angew. Math. 497, 171 (1998).
[60] K.-H. Neeb , Holomorphy and Convexity in Lie Theory , Expositions in Mathematics 28 ( de Gruyter , 1999 ) .
[61] K.-H. Neeb, Semigroup Forum 63, 71 (2001).
[62] K.-H. Neeb, Infinite Dimensional Kähler Manifolds, DMV-Seminar 31, eds. A. Huckleberry and T. Wurzbacher (Birkhäuser, 2001) pp. 131-178.
[63] K.-H. Neeb, Geometry and Analysis on Finite- and Infinite-Dimensional Lie Groups, Banach Center Publications 55, eds. A. Strasburger (Warszawa, 2002) pp. 87-151.
[64] K.-H. Neeb, Ann. l’Inst. Fourier 52, 1365 (2002).
[65] K.-H. Neeb, Lie Theory: Lie Algebras and Representations, Progress in Math. 228, eds. J. P. Anker and B. Ørsted (Birkhäuser, 2004) pp. 213-328.
[66] K.-H. Neeb, Jpn. J. Math. 3rd Ser. 1, 291 (2006).
[67] K.-H. Neeb, Semigroup Forum 77, 5 (2008), DOI: 10.1007/s00233-008-9073-5 .
[68] K.-H. Neeb, Infinite Dimensional Harmonic Analysis IV, eds. J. Hilgert (World Scientific, 2009) pp. 209-222.
[69] K.-H. Neeb , Unitary highest weight modules of locally affine Lie algebras , Proc. of the Workshop on Quantum Affine Algebras, Extended Affine Lie Algebras and Applications , eds. Y. Gao ( Amer. Math. Soc. ) .
[70] K.-H. Neeb, Invariant convex cones in infinite dimensional Lie algebras, in preparation .
[71] K.-H. Neeb, On differentiable vectors for representations of infinite dimensional Lie groups, in preparation .
[72] K.-H. Neeb and B. Ørsted, J. Funct. Anal. 156, 263 (1998), DOI: 10.1006/jfan.1997.3233 .
[73] Y. Neretin, Math. USSR Sbor. 67, 75 (1990), DOI: 10.1070/SM1990v067n01ABEH001321 .
[74] A. Neumann, Geom. Dedicata 79, 299 (2000), DOI: 10.1023/A:1005260629187 .
[75] A. Neumann, J. Funct. Anal. 189, 80 (2002), DOI: 10.1006/jfan.2000.3739 .
[76] G. I. Ol’shanski, Funct. Anal. Appl. 15, 275 (1982), DOI: 10.1007/BF01106156 .
[77] J. T. Ottesen , Infinite Dimensional Groups and Algebras in Quantum Physics , Lecture Notes in Physics 27 ( Springer , 1995 ) .
[78] V. Yu. Ovsienko, Infinite Dimensional Kähler Manifolds (Oberwolfach, 1995), DMV Sem. 31 (Birkhäuser, 2001) pp. 231-255.
[79] G. K. Pedersen , C*-Algebras and Their Automorphism Groups ( Academic Press , 1989 ) .
[80] R. T. Powers and E. Størmer, Comm. Math. Phys. 16, 1 (1970), DOI: 10.1007/BF01645492 .
[81] A. Pressley and G. Segal , Loop Groups ( Oxford Univ. Press , 1986 ) .
[82] M. C. Reed, Comm. Math. Phys. 14, 336 (1969), DOI: 10.1007/BF01645390 .
[83] S. Reed and B. Simon , Methods of Modern Mathematical Physics I: Functional Analysis ( Academic Press , 1973 ) .
[84] S. N. M. Ruijsenaars, J. Math. Phys. 18, 517 (1977), DOI: 10.1063/1.523295 .
[85] Y. S. Samoilenko , Spectral Theory of Families of Self-Adjoint Operators ( Kluwer , 1991 ) .
[86] M. Schottenloher , A Mathematical Introduction to Conformal Field Theory , Lecture Notes in Physics 43 ( Springer , 1997 ) .
[87] L. Schwartz , Théorie des Distributions ( Hermann , 1973 ) .
[88] G. Segal, Comm. Math. Phys. 80, 301 (1981), DOI: 10.1007/BF01208274 .
[89] I. E. Segal, Proc. Amer. Math. Soc. 81, 197 (1957).
[90] I. E. Segal, Trans. Amer. Math. Soc. 88, 12 (1958), DOI: 10.2307/1993234 .
[91] I. E. Segal, Mat. Fys. Medd. Danske Vid. Selsk. 31, 1 (1959).
[92] I. E. Segal, Proc. Nat. Acad. Sci. USA 57, 194 (1967), DOI: 10.1073/pnas.57.2.194 .
[93] I. E. Segal , Mathematical Cosmology and Extragalactical Astronomy ( Academic Press , 1976 ) .
[94] I. E. Segal, Topics in Funct. Anal., Adv. in Math. Suppl. Studies 3 (1978) pp. 321-343.
[95] I. E. Segal, Proc. Nat. Acad. Sci. USA 75, 1609 (1978), DOI: 10.1073/pnas.75.10.4638 .
[96] D. Shale, Trans. Amer. Math. Soc. 103, 146 (1962), DOI: 10.2307/1993745 .
[97] D. Shale and W. F. Stinespring, J. Math. Mech. 14, 315 (1965).
[98] M. Spera and T. Wurzbacher, Rev. Math. Phys. 10, 705 (1998). [Abstract]
[99] M. Thill, Representations of hermitian commutative *-algebras by unbounded operators , arXiv:0908.3267 .
[100] V. T. Laredo, Comm. Math. Phys. 207, 307 (1999).
[101] V. T. Laredo, J. Funct. Anal. 161, 478 (1999), DOI: 10.1006/jfan.1998.3359 .
[102] H. Upmeier , Symmetric Banach Manifolds and Jordan C*-algebras ( North-Holland , 1985 ) .
[103] M. Vergne, C. R. Acad. Sci. Paris 285, 191 (1977).
[104] A. M. Vershik, Representations of Lie Groups and Related Topics, eds. A. M. Vershik and D. P. Zhelobenko (Gordon and Breach, 1990) pp. 1-37.
[105] E. B. Vinberg, Funct. Anal. Appl. 14, 1 (1980).
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