Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 096, 50 pages      arXiv:1903.10721      https://doi.org/10.3842/SIGMA.2019.096

The Real Jacobi Group Revisited

Stefan Berceanu
National Institute for Physics and Nuclear Engineering, Department of Theoretical Physics, PO BOX MG-6, Bucharest-Magurele, Romania

Received May 09, 2019, in final form November 25, 2019; Published online December 07, 2019

Abstract
The real Jacobi group $G^J_1(\mathbb{R})$, defined as the semi-direct product of the group ${\rm SL}(2,\mathbb{R})$ with the Heisenberg group $H_1$, is embedded in a $4\times 4$ matrix realisation of the group ${\rm Sp}(2,\mathbb{R})$. The left-invariant one-forms on $G^J_1(\mathbb{R})$ and their dual orthogonal left-invariant vector fields are calculated in the S-coordinates $(x,y,\theta,p,q,\kappa)$, and a left-invariant metric depending of 4 parameters $(\alpha,\beta,\gamma,\delta)$ is obtained. An invariant metric depending of $(\alpha,\beta)$ in the variables $(x,y,\theta)$ on the Sasaki manifold ${\rm SL}(2,\mathbb{R})$ is presented. The well known Kähler balanced metric in the variables $(x,y,p,q)$ of the four-dimensional Siegel-Jacobi upper half-plane $\mathcal{X}^J_1=\frac{G^J_1(\mathbb{R})}{{\rm SO}(2) \times\mathbb{R}} \approx\mathcal{X}_1 \times\mathbb{R}^2$ depending of $(\alpha,\gamma)$ is written down as sum of the squares of four invariant one-forms, where $\mathcal{X}_1$ denotes the Siegel upper half-plane. The left-invariant metric in the variables $(x,y,p,q,\kappa)$ depending on $(\alpha,\gamma,\delta)$ of a five-dimensional manifold $\tilde{\mathcal{X}}^J_1= \frac{G^J_1(\mathbb{R})}{{\rm SO}(2)}\approx\mathcal{X}_1\times\mathbb{R}^3$ is determined.

Key words: Jacobi group; invariant metric; Siegel-Jacobi upper half-plane; balanced metric; extended Siegel-Jacobi upper half-plane; naturally reductive manifold.

pdf (857 kb)   tex (131 kb)  

References

  1. Agricola I., Ferreira A.C., Friedrich T., The classification of naturally reductive homogeneous spaces in dimensions $n\leq 6$, Differential Geom. Appl. 39 (2015), 59-92, arXiv:1407.4936.
  2. Albert C., Le groupe de Heisenberg et les variétés de contact riemanniennes, Bull. Sci. Math. 126 (2002), 97-113.
  3. Alekseevskii D.V., Reductive space, Encyclopedia of Mathematics, https://www.encyclopediaofmath.org//index.php?title=Reductive_space&oldid=33884.
  4. Ambrose W., Singer I.M., On homogeneous Riemannian manifolds, Duke Math. J. 25 (1958), 647-669.
  5. Arezzo C., Loi A., Moment maps, scalar curvature and quantization of Kähler manifolds, Comm. Math. Phys. 246 (2004), 543-559.
  6. Arvanitoyeorgos A., An introduction to Lie groups and the geometry of homogeneous spaces, Student Mathematical Library, Vol. 22, Amer. Math. Soc., Providence, RI, 2003.
  7. Bargmann V., Group representations on Hilbert spaces of analytic functions, in Analytic Methods in Mathematical Physics (Sympos., Indiana Univ., Bloomington, Ind., 1968), 1970, 27-63.
  8. Berceanu S., Coherent states and geodesics: cut locus and conjugate locus, J. Geom. Phys. 21 (1997), 149-168, arXiv:dg-ga/9502007.
  9. Berceanu S., Coherent states, phases and symplectic areas of geodesic triangles, in Coherent States, Quantization and Gravity, Editors M. Schlichenmaier, A. Strasburger, S.T. Ali, A. Odjziewicz, Warsaw University Press, Warsaw, 2001, 129-137, arXiv:math.DG/9903190.
  10. Berceanu S., Geometrical phases on Hermitian symmetric spaces, in Recent Advances in Geometry and Topology, Cluj University Press, Cluj-Napoca, 2004, 83-98, arXiv:math.DG/0408233.
  11. Berceanu S., A holomorphic representation of Lie algebras semidirect sum of semisimple and Heisenberg algebras, Romanian J. Phys. 50 (2005), 81-94.
  12. Berceanu S., Realization of coherent state Lie algebras by differential operators, in Advances in Operator Algebras and Mathematical Physics, Theta Ser. Adv. Math., Vol. 5, Theta, Bucharest, 2005, 1-24, arXiv:math.DG/0504053.
  13. Berceanu S., A holomorphic representation of the Jacobi algebra, Rev. Math. Phys. 18 (2006), 163-199, Errata, Rev. Math. Phys. 24 (2012), 1292001, 2 pages, arXiv:math.DG/0408219.
  14. Berceanu S., Coherent states associated to the Jacobi group, Romanian Rep. Phys. 59 (2007), 1089-1101.
  15. Berceanu S., Coherent states associated to the Jacobi group - a variation on a theme by Erich Kähler, J. Geom. Symmetry Phys. 9 (2007), 1-8.
  16. Berceanu S., A holomorphic representation of the multidimensional Jacobi algebra, in Perspectives in Operator Algebras and Mathematical Physics, Theta Ser. Adv. Math., Vol. 8, Theta, Bucharest, 2008, 1-25, arXiv:math.DG/0604381.
  17. Berceanu S., A convenient coordinatization of Siegel-Jacobi domains, Rev. Math. Phys. 24 (2012), 1250024, 38 pages, arXiv:1204.5610.
  18. Berceanu S., Consequences of the fundamental conjecture for the motion on the Siegel-Jacobi disk, Int. J. Geom. Methods Mod. Phys. 10 (2013), 1250076, 18 pages, arXiv:1110.5469.
  19. Berceanu S., Coherent states and geometry on the Siegel-Jacobi disk, Int. J. Geom. Methods Mod. Phys. 11 (2014), 1450035, 25 pages, arXiv:1307.4219.
  20. Berceanu S., Bergman representative coordinates on the Siegel-Jacobi disk, Romanian J. Phys. 60 (2015), 867-896, arXiv:1409.0368.
  21. Berceanu S., Balanced metric and Berezin quantization on the Siegel-Jacobi ball, SIGMA 12 (2016), 064, 28 pages, arXiv:1512.00601.
  22. Berceanu S., Geodesics associated to the balanced metric on the Siegel-Jacobi ball, Romanian J. Phys. 61 (2016), 1137-1160, arXiv:1605.02962.
  23. Berceanu S., Boutet de Monvel L., Linear dynamical systems, coherent state manifolds, flows, and matrix Riccati equation, J. Math. Phys. 34 (1993), 2353-2371.
  24. Berceanu S., Gheorghe A., On the construction of perfect Morse functions on compact manifolds of coherent states, J. Math. Phys. 28 (1987), 2899-2907.
  25. Berceanu S., Gheorghe A., On equations of motion on compact Hermitian symmetric spaces, J. Math. Phys. 33 (1992), 998-1007.
  26. Berceanu S., Gheorghe A., Linear Hamiltonians on homogeneous Kähler manifolds of coherent states, An. Univ. Timicsoara Ser. Mat.-Inform. 39 (2001), 31-55, arXiv:math.DG/0408254.
  27. Berceanu S., Gheorghe A., Differential operators on orbits of coherent states, Romanian J. Phys. 48 (2003), 545-556, arXiv:math.DG/0211054.
  28. Berceanu S., Gheorghe A., Applications of the Jacobi group to quantum mechanics, Romanian J. Phys. 53 (2008), 1013-1021, arXiv:0812.0448.
  29. Berceanu S., Gheorghe A., On the geometry of Siegel-Jacobi domains, Int. J. Geom. Methods Mod. Phys. 8 (2011), 1783-1798, arXiv:1011.3317.
  30. Berceanu S., Schlichenmaier M., Coherent state embeddings, polar divisors and Cauchy formulas, J. Geom. Phys. 34 (2000), 336-358, arXiv:math.DG/9903105.
  31. Berezin F.A., Quantization in complex bounded domains, Dokl. Akad. Nauk SSSR 211 (1973), 1263-1266.
  32. Berezin F.A., Quantization, Math. USSR-Izv. 8 (1974), 1109-1165.
  33. Berezin F.A., General concept of quantization, Comm. Math. Phys. 40 (1975), 153-174.
  34. Berezin F.A., Quantization in complex symmetric spaces, Math. USSR-Izv. 9 (1975), 341-379.
  35. Berndt J., Tricerri F., Vanhecke L., Generalized Heisenberg groups and Damek-Ricci harmonic spaces, Lecture Notes in Mathematics, Vol. 1598, Springer-Verlag, Berlin, 1995.
  36. Berndt J., Vanhecke L., Naturally reductive Riemannian homogeneous spaces and real hypersurfaces in complex and quaternionic space forms, in Differential Geometry and its Applications (Opava, 1992), Math. Publ., Vol. 1, Silesian University Opava, Opava, 1993, 353-364.
  37. Berndt R., Some differential operators in the theory of Jacobi forms, preprint IHES/M/84/10, 1984, 31 pages.
  38. Berndt R., Sur l'arithmétique du corps des fonctions elliptiques de niveau $N$, in Seminar on Number Theory, Paris 1982-83 (Paris, 1982/1983), Progr. Math., Vol. 51, Birkhäuser Boston, Boston, MA, 1984, 21-32.
  39. Berndt R., Schmidt R., Elements of the representation theory of the Jacobi group, Progress in Mathematics, Vol. 163, Birkhäuser Verlag, Basel, 1998.
  40. Besse A.L., Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 10, Springer-Verlag, Berlin, 1987.
  41. Bianchi L., Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti, Mem. Mat. Fis. Soc. Ital. Sci. 11 (1898), 267-352, On the three-dimensional spaces which admit a continuous group of motions, Gen. Relativity Gravitation 33 (2001), 2171-2253.
  42. Blair D.E., Contact manifolds in Riemannian geometry, Lecture Notes in Math., Vol. 509, Springer-Verlag, Berlin - New York, 1976.
  43. Boothby W.M., Wang H.C., On contact manifolds, Ann. of Math. 68 (1958), 721-734.
  44. Boyer C.P., The Sasakian geometry of the Heisenberg group, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52 (2009), 251-262.
  45. Boyer C.P., Galicki K., Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008.
  46. Cahen M., Gutt S., Rawnsley J., Quantization of Kähler manifolds. II, Trans. Amer. Math. Soc. 337 (1993), 73-98.
  47. Calin O., Chang D.-C., Sub-Riemannian geometry. General theory and examples, Encyclopedia of Mathematics and its Applications, Vol. 126, Cambridge University Press, Cambridge, 2009.
  48. Cartan E., Leçons sur la Géométrie des Espaces de Riemann, Gauthier-Villars, Paris, 1928.
  49. Cartan E., La méthode du repére mobile, la théorie des groupes continus et les espaces généralisés, Actualités scientifiques et industrielles, Vol. 194, Hermann & Cie., Paris, 1935.
  50. Cartan E., Les espaces à connexion projective, Abh. Sem. Vektor -Tensor analysis, Moskau 4 (1937), 147-173.
  51. D'Atri J.E., Ziller W., Naturally reductive metrics and Einstein metrics on compact Lie groups, Mem. Amer. Math. Soc. 18 (1979), iii+72 pages.
  52. Donaldson S.K., Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), 479-522.
  53. Dorfmeister J., Nakajima K., The fundamental conjecture for homogeneous Kähler manifolds, Acta Math. 161 (1988), 23-70.
  54. Dušek Z., Survey on homogeneous geodesics, Note Mat. 28 (2009), suppl. 1, 147-168.
  55. Eichler M., Zagier D., The theory of Jacobi forms, Progress in Mathematics, Vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985.
  56. Evtushik E.L., Moving-frame method, Encyclopedia of Mathematics, http://www.encyclopediaofmath.org/index.php?title=Moving-frame-method&oldid=17828.
  57. Fecko M., Differential geometry and Lie groups for physicists, Cambridge University Press, Cambridge, 2006.
  58. Ferr'andez A., Naveira A.M., Tarrío A.D., Geometry of extended Bianchi-Cartan-Vranceanu spaces, Rev. R. Acad. Cienc. Exactas F'is. Nat. Ser. A Mat. RACSAM 113 (2019), 1515-1532, arXiv:1802.00106.
  59. Folland G.B., Harmonic analysis in phase space, Annals of Mathematics Studies, Vol. 122, Princeton University Press, Princeton, NJ, 1989.
  60. Freitas P.J., On the action of the symplectic group on the Siegel upper half plane, Ph.D. Thesis, University of Illinois at Chicago, 1999.
  61. Gindikin S.G., Pjateckiǐ-Šapiro I.I., Vinberg E.B., Homogeneous Kähler manifolds, in Geometry of Homogeneous Bounded Domains, Springer, Berlin, Heidelberg, 2011, 1-87.
  62. Halverscheid S., Iannuzzi A., On naturally reductive left-invariant metrics of ${\rm SL}(2,{\mathbb R})$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), 171-187.
  63. Helgason S., Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, Vol. 80, Academic Press, Inc., New York - London, 1978.
  64. Hua L.-K., On the theory of automorphic functions of a matrix level. I. Geometrical basis, Amer. J. Math. 66 (1944), 470-488.
  65. Inoguchi J.-I., Van der Veken J., Parallel surfaces in the motion groups $E(1,1)$ and $E(2)$, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), 321-332.
  66. Itzykson C., Remarks on boson commutation rules, Comm. Math. Phys. 4 (1967), 92-122.
  67. Jayne N., Legendre foliations on contact metric manifolds, Ph.D. Thesis, Massey University, 1992.
  68. Kähler E., Raum-Zeit-Individuum, Rend. Accad. Naz. Sci. XL Mem. Mat. 16 (1992), 115-177.
  69. Kähler E., Mathematische Werke/Mathematical works, Walter de Gruyter & Co., Berlin, 2003.
  70. Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. I, Interscience Publishers, New York - London, 1963.
  71. Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. II, Interscience Publishers, New York - London - Sydney, 1969.
  72. Koda T., An introduction to the geometry of homogeneous spaces, in Proceedings of the 13th International Workshop on Differential Geometry and Related Fields, Natl. Inst. Math. Sci. (NIMS), Taejuon, 2009, 121-144.
  73. Kostant B., Quantization and unitary representations. I. Prequantization, in Lectures in Modern Analysis and Applications III, Lecture Notes in Math., Vol. 170, Springer, Berlin - Heidelberg, 1970, 87-208.
  74. Kowalski O., Spaces with volume-preserving symmetries and related classes of Riemannian manifolds, Rend. Sem. Mat. Univ. Politec. Torino (1983), 131-158.
  75. Kowalski O., Szenthe J., On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geom. Dedicata 81 (2000), 209-214.
  76. Kowalski O., Vanhecke L., Four-dimensional naturally reductive homogeneous spaces, Rend. Sem. Mat. Univ. Politec. Torino (1983), 223-232.
  77. Kowalski O., Vanhecke L., Riemannian manifolds with homogeneous geodesics, Boll. Un. Mat. Ital. B 5 (1991), 189-246.
  78. Lang S., ${\rm SL}_2({\mathbb R})$, Graduate Texts in Mathematics, Vol. 105, Springer-Verlag, New York, 1985.
  79. Lisiecki W., A classification of coherent state representations of unimodular Lie groups, Bull. Amer. Math. Soc. (N.S.) 25 (1991), 37-43.
  80. Lisiecki W., Coherent state representations. A survey, Rep. Math. Phys. 35 (1995), 327-358.
  81. Loi A., Mossa R., Berezin quantization of homogeneous bounded domains, Geom. Dedicata 161 (2012), 119-128, arXiv:1106.2510.
  82. Mikeš J., Vanvzurov'a A., Hinterleitner I., Geodesic mappings and some generalizations, Palack'y University Olomouc, Olomouc, 2009.
  83. Milnor J., Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976), 293-329.
  84. Moscovici H., Coherent state representations of nilpotent Lie groups, Comm. Math. Phys. 54 (1977), 63-68.
  85. Moscovici H., Verona A., Coherent states and square integrable representations, Ann. Inst. H. Poincaré 29 (1978), 139-156.
  86. Neeb K.-H., Coherent states, holomorphic extensions, and highest weight representations, Pacific J. Math. 174 (1996), 497-542.
  87. Neeb K.-H., Holomorphy and convexity in Lie theory, De Gruyter Expositions in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2000.
  88. Nomizu K., Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33-65.
  89. Onishchik A.L., Isotropy group, Encyclopedia of Mathematics, https://www.encyclopediaofmath.org//index.php?title=Isotropy_group&_oldid=33104.
  90. Onishchik A.L., Isotropy representation, Encyclopedia of Mathematics, https://www.encyclopediaofmath.org//index.php?title=Isotropy_representation&oldid=15929.
  91. Patrangenaru V., Classifying $3$- and $4$-dimensional homogeneous Riemannian manifolds by Cartan triples, Pacific J. Math. 173 (1996), 511-532.
  92. Perelomov A., Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986.
  93. Rawnsley J., Coherent states and Kähler manifolds, Quart. J. Math. Oxford Ser. (2) 28 (1977), 403-415.
  94. Rawnsley J., Cahen M., Gutt S., Quantization of Kähler manifolds. I. Geometric interpretation of Berezin's quantization, J. Geom. Phys. 7 (1990), 45-62.
  95. Sasaki S., On almost contact manifolds, Lecture notes, Part 1, Tôhoku University, 1965.
  96. Sasaki S., Hatakeyama Y., On differentiable manifolds with contact metric structures, J. Math. Soc. Japan 14 (1962), 249-271.
  97. Siegel C.L., Symplectic geometry, Amer. J. Math. 65 (1943), 1-86.
  98. Tanno S., The automorphism groups of almost Hermitian manifolds, Trans. Amer. Math. Soc. 137 (1969), 269-275.
  99. Thurston W.P., Three-dimensional geometry and topology, Vol. 1, Princeton Mathematical Series, Vol. 35, Princeton University Press, Princeton, NJ, 1997.
  100. Tricerri F., Vanhecke L., Homogeneous structures on Riemannian manifolds, London Mathematical Society Lecture Note Series, Vol. 83, Cambridge University Press, Cambridge, 1983.
  101. Van der Veken J., Higher order parallel surfaces in Bianchi-Cartan-Vranceanu spaces, Results Math. 51 (2008), 339-359, arXiv:math.DG/0604541.
  102. Vilenkin N.Ja., Special functions and the theory of group representations, Translations of Mathematical Monographs, Vol. 22, Amer. Math. Soc., Providence, R.I., 1968.
  103. Vinberg E.B., Gindikin S.G., Kähler manifolds admitting a transitive solvable group of automorphisms, Math. USSR Sb. 3 (1967), 333-351.
  104. Vranceanu G., Lecţii de Geometrie diferenţială, Vol. I, Editura Didactică şi Pedagogică, Bucureşti, 1962.
  105. Wolf J.A., The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121-1237.
  106. Wolf J.A., Fine structure of Hermitian symmetric spaces, in Symmetric Spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969-1970), Pure and App. Math., Vol. 8, Editors W.M. Boothby, G.I. Weiss, Marcel Dekker and G.L. Weiss, New York, 1972, 271-357.
  107. Woodhouse N., Geometric quantization, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1980.
  108. Yang J.-H., The method of orbits for real Lie groups, Kyungpook Math. J. 42 (2002), 199-272, arXiv:math.RT/0602056.
  109. Yang J.-H., Remark on harmonic analysis on Siegel-Jacobi space, arXiv:math.NT/0612230.
  110. Yang J.-H., Invariant metrics and Laplacians on Siegel-Jacobi space, J. Number Theory 127 (2007), 83-102, arXiv:math.NT/0507215.
  111. Yang J.-H., A partial Cayley transform of Siegel-Jacobi disk, J. Korean Math. Soc. 45 (2008), 781-794, arXiv:math.NT/0507216.
  112. Yang J.-H., Invariant metrics and Laplacians on Siegel-Jacobi disk, Chin. Ann. Math. Ser. B 31 (2010), 85-100, arXiv:math.NT/0507217.
  113. Yano K., The theory of Lie derivatives and its applications, Bibliotheca Mathematica, Vol. 3, North-Holland Publishing Company, Amsterdam, 1957.

Previous article  Next article  Contents of Volume 15 (2019)