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

SIGMA 11 (2015), 002, 19 pages      arXiv:1408.4842

Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras

Naruhiko Aizawa a, Radhakrishnan Chandrashekar b and Jambulingam Segar c
a) Department of Mathematics and Information Sciences, Osaka Prefecture University, Nakamozu Campus, Sakai, Osaka 599-8531, Japan
b) Department of Physics, National Chung Hsing University, Taichung 40227, Taiwan
c) Department of Physics, Ramakrishna Mission Vivekananda College, Mylapore, Chennai 600 004, India

Received August 22, 2014, in final form December 31, 2014; Published online January 06, 2015

The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters $d$ and $\ell$. The aim of the present work is to investigate the lowest weight representations of CGA with $d = 1$ for any integer value of $\ell$. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if $\ell = 1$ and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when $\ell \neq 1$. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules.

Key words: representation theory; non-semisimple Lie algebra; symmetry of differential equations.

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  1. Aizawa N., Dobrev V.K., Doebner H.-D., Intertwining operators for Schrödinger algebras and hierarchy of invariant equations, in Quantum Theory and Symmetries (Kraków, 2001), Editors E. Kapuścik, A. Horzela, World Sci. Publ., River Edge, NJ, 2002, 222-227.
  2. Aizawa N., Dobrev V.K., Doebner H.-D., Stoimenov S., Intertwining operators for the Schrödinger algebra in $ n \geq 3 $ space dimension, in Proceedings of the VII International Workshop on ''Lie Theory and its Applications in Physics'', Editors H.-D. Doebner, V.K. Dobrev, Heron Press, Sofia, 2008, 372-399.
  3. Aizawa N., Isaac P.S., On irreducible representations of the exotic conformal Galilei algebra, J. Phys. A: Math. Theor. 44 (2011), 035401, 8 pages, arXiv:1010.4075.
  4. Aizawa N., Isaac P.S., Kimura Y., Highest weight representations and Kac determinants for a class of conformal Galilei algebras with central extension, Internat. J. Math. 23 (2012), 1250118, 25 pages, arXiv:1204.2871.
  5. Aizawa N., Kimura Y., Segar J., Intertwining operators for $\ell$-conformal Galilei algebras and hierarchy of invariant equations, J. Phys. A: Math. Theor. 46 (2013), 405204, 14 pages, arXiv:1308.0121.
  6. Alishahiha M., Davody A., Vahedi A., On AdS/CFT of Galilean conformal field theories, J. High Energy Phys. 2009 (2009), no. 8, 022, 16 pages, arXiv:0903.3953.
  7. Andrzejewski K., Galajinsky A., Gonera J., Masterov I., Conformal Newton-Hooke symmetry of Pais-Uhlenbeck oscillator, Nuclear Phys. B 885 (2014), 150-162, arXiv:1402.1297.
  8. Andrzejewski K., Gonera J., Dynamical interpretation of nonrelativistic conformal groups, Phys. Lett. B 721 (2013), 319-322.
  9. Andrzejewski K., Gonera J., Unitary representations of $N$-conformal Galilei group, Phys. Rev. D 88 (2013), 065011, 9 pages, arXiv:1305.4777.
  10. Andrzejewski K., Gonera J., Kijanka-Dec A., Nonrelativistic conformal transformations in Lagrangian formalism, Phys. Rev. D 87 (2013), 065012, 6 pages, arXiv:1301.1531.
  11. Andrzejewski K., Gonera J., Kosiński P., Maślanka P., On dynamical realizations of $l$-conformal Galilei groups, Nuclear Phys. B 876 (2013), 309-321, arXiv:1305.6805.
  12. Andrzejewski K., Gonera J., Maślanka P., Nonrelativistic conformal groups and their dynamical realizations, Phys. Rev. D 86 (2012), 065009, 8 pages, arXiv:1204.5950.
  13. Bagchi A., Gopakumar R., Galilean conformal algebras and AdS/CFT, J. High Energy Phys. 2009 (2009), no. 7, 037, 22 pages, arXiv:0902.1385.
  14. Balasubramanian K., McGreevy J., Gravity duals for nonrelativistic conformal field theories, Phys. Rev. Lett. 101 (2008), 061601, 4 pages, arXiv:0804.4053.
  15. Dixmier J., Enveloping algebras, North-Holland Mathematical Library, Vol. 14, North-Holland Publishing Co., Amsterdam - New York - Oxford, 1977.
  16. Dobrev V.K., Canonical construction of differential operators intertwining representations of real semisimple Lie groups, Rep. Math. Phys. 25 (1988), 159-181.
  17. Dobrev V.K., Subsingular vectors and conditionally invariant ($q$-deformed) equations, J. Phys. A: Math. Gen. 28 (1995), 7135-7155.
  18. Dobrev V.K., Kazhdan-Lusztig polynomials, subsingular vectors and conditionally invariant ($q$-deformed) equations, in Symmetries in Science, IX (Bregenz, 1996), Plenum, New York, 1997, 47-80.
  19. Dobrev V.K., Doebner H.-D., Mrugalla Ch., Lowest weight representations of the Schrödinger algebra and generalized heat Schrödinger equations, Rep. Math. Phys. 39 (1997), 201-218.
  20. Galajinsky A., Masterov I., Remarks on $l$-conformal extension of the Newton-Hooke algebra, Phys. Lett. B 702 (2011), 265-267, arXiv:1104.5115.
  21. Galajinsky A., Masterov I., Dynamical realization of $l$-conformal Galilei algebra and oscillators, Nuclear Phys. B 866 (2013), 212-227, arXiv:1208.1403.
  22. Galajinsky A., Masterov I., Dynamical realizations of $l$-conformal Newton-Hooke group, Phys. Lett. B 723 (2013), 190-195, arXiv:1303.3419.
  23. Hagen C.R., Scale and conformal transformations in Galilean-covariant field theory, Phys. Rev. D 5 (1972), 377-388.
  24. Havas P., Plebański J., Conformal extensions of the Galilei group and their relation to the Schrödinger group, J. Math. Phys. 19 (1978), 482-488.
  25. Henkel M., Schrödinger invariance and strongly anisotropic critical systems, J. Stat. Phys. 75 (1994), 1023-1061, hep-th/9310081.
  26. Hussin V., Jacques M., On nonrelativistic conformal symmetries and invariant tensor fields, J. Phys. A: Math. Gen. 19 (1986), 3471-3485.
  27. Kac V.G., Raina A.K., Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics, Vol. 2, World Scientific Publishing Co., Inc., Teaneck, NJ, 1987.
  28. Lü R., Mazorchuk V., Zhao K., On simple modules over conformal Galilei algebras, J. Pure Appl. Algebra 218 (2014), 1885-1899, arXiv:1310.6284.
  29. Lukierski J., Stichel P.C., Zakrzewski W.J., Exotic Galilean conformal symmetry and its dynamical realisations, Phys. Lett. A 357 (2006), 1-5, hep-th/0511259.
  30. Martelli D., Tachikawa Y., Comments on Galilean conformal field theories and their geometric realization, J. High Energy Phys. 2010 (2010), no. 5, 091, 31 pages, arXiv:0903.5184.
  31. Mrugalla Ch., Quantum mechanical evolution equations based on Lie-algebraic and $q$-deformed symmetries, Ph.D. Thesis, Technischen Universität Clausthal, 1997.
  32. Negro J., del Olmo M.A., Rodríguez-Marco A., Nonrelativistic conformal groups, J. Math. Phys. 38 (1997), 3786-3809.
  33. Niederer U., The maximal kinematical invariance group of the free Schrödinger equation, Helv. Phys. Acta 45 (1972), 802-810.
  34. Nishida Y., Son D.T., Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007), 086004, 14 pages, arXiv:0706.3746.
  35. Shapovalov N.N., A certain bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra, Funct. Anal. Appl. 6 (1972), 307-312.
  36. Son D.T., Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008), 046003, 7 pages, arXiv:0804.3972.
  37. Stichel P.C., Zakrzewski W.J., A new type of conformal dynamics, Ann. Physics 310 (2004), 158-180, hep-th/0309038.
  38. Vinet L., Zhedanov A., Representations of the Schrödinger group and matrix orthogonal polynomials, J. Phys. A: Math. Theor. 44 (2011), 355201, 28 pages, arXiv:1105.0701.
  39. Zhang P., Horváthy P.A., Non-relativistic conformal symmetries in fluid mechanics, Eur. Phys. J. C 65 (2010), 607-614, arXiv:0906.3594.

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