×
Ballesteros, Ángel; Herranz, Francisco J.; Ragnisco, Orlando; Santander, Mariano

Contractions, deformations and curvature. (English) Zbl 1184.81080

Int. J. Theor. Phys. 47, No. 3, 649-663 (2008).
Summary: The role of curvature in relation with Lie algebra contractions of the pseudo-orthogonal algebras \(\text{so}(p,q)\) is fully described by considering some associated symmetrical homogeneous spaces of constant curvature within a Cayley-Klein framework. We show that a given Lie algebra contraction can be interpreted geometrically as the zero-curvature limit of some underlying homogeneous space with constant curvature. In particular, we study in detail the contraction process for the three classical Riemannian spaces (spherical, Euclidean, hyperbolic), three non-relativistic (Newtonian) spacetimes and three relativistic ((anti-)de Sitter and Minkowskian) spacetimes. Next, from a different perspective, we make use of quantum deformations of Lie algebras in order to construct a family of spaces of non-constant curvature that can be interpreted as deformations of the above nine spaces. In this framework, the quantum deformation parameter is identified as the parameter that controls the curvature of such “quantum” spaces.

Cited in 4 Documents

MSC:

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
22E70 Applications of Lie groups to the sciences; explicit representations

Keywords:

Lie algebras; quantum groups; contraction; curvature; deformation; hyperbolic; de Sitter
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Inönü, E., Wigner, E.P.: Proc. Natl. Acad. Sci. 39, 510 (1953); ibid 40, 119 (1954) · Zbl 0050.02601 · doi:10.1073/pnas.39.6.510
[2] Segal, I.E.: Duke Math. J. 18, 221 (1951) · Zbl 0045.38601 · doi:10.1215/S0012-7094-51-01817-0
[3] Saletan, E.J.: J. Math. Phys. 2, 1 (1961) · Zbl 0098.25804 · doi:10.1063/1.1724208
[4] Weimar-Woods, E.: J. Math. Phys. 36, 4519 (1995) · Zbl 0854.17036 · doi:10.1063/1.530905
[5] Izmest’ev, A.A., Pogosyan, G.S., Sissakian, A.N., Winternitz, P.: J. Phys. A: Math. Gen. 29, 5940 (1996)
[6] Fialowski, A., de Montigny, M.: J. Phys. A: Math. Gen. 38, 6335 (2005) · Zbl 1073.22012 · doi:10.1088/0305-4470/38/28/006
[7] Gilmore, R.: Lie Groups, Lie Algebras and Some of Their Applications. Wiley, New York (1974) · Zbl 0279.22001
[8] Abe, E.: Hopf Algebras. Cambridge Tracts in Mathematics, vol. 74. Cambridge University Press, Cambridge (1980) · Zbl 0476.16008
[9] Drinfeld, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematics, p. 798. MRSI, Berkeley (1986)
[10] Tjin, T.: Int. J. Mod. Phys. A 7, 6175 (1992) · Zbl 0972.17501 · doi:10.1142/S0217751X92002805
[11] Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994) · Zbl 0839.17009
[12] Ballesteros, A., Gromov, N.A., Herranz, F.J., del Olmo, M.A., Santander, M.: J. Math. Phys. 36, 5916 (1995) · Zbl 0868.17012 · doi:10.1063/1.531368
[13] Lukierski, J., Nowicki, A.: Int. J. Mod. Phys. A 18, 7 (2003) · Zbl 1021.81024 · doi:10.1142/S0217751X03013600
[14] Ahluwalia-Khalilova, D.V.: Class. Quantum Grav. 22, 1433 (2005) · Zbl 1064.83022 · doi:10.1088/0264-9381/22/7/015
[15] Herranz, F.J.: Phys. Lett. B 543, 89 (2002) · Zbl 0997.81041 · doi:10.1016/S0370-2693(02)02452-8
[16] Ballesteros, A., Civitarese, O., Herranz, F.J., Reboiro, M.: Phys. Rev. C 66, 064317 (2002) · doi:10.1103/PhysRevC.66.064317
[17] Nijenhuis, A., Richardson, R.W.: J. Math. Mech. 17, 89 (1967)
[18] Herranz, F.J., Santander, M.: math-ph/0612059 (2006)
[19] Ballesteros, A., Herranz, F.J., Ragnisco, O.: Phys. Lett. B 610, 107 (2005) · Zbl 1247.81208 · doi:10.1016/j.physletb.2005.02.007
[20] Ragnisco, O., Ballesteros, A., Herranz, F.J., Musso, F.: SIGMA 3, 026 (2007). math-ph/0611040
[21] Sommerville, D.M.Y.: Proc. Edinburgh Math. Soc. 28, 25 (1910) · JFM 41.0537.01 · doi:10.1017/S0013091500034763
[22] Yaglom, I.M., Rozenfel’d, B.A., Yasinskaya, E.U.: Sov. Math. Surv. 19, 49 (1966)
[23] Yaglom, I.M.: A Simple Non-Euclidean Geometry and Its Physical Basis. Springer, New York (1979) · Zbl 0393.51013
[24] Rozenfel’d, B.A.: A History of Non-Euclidean Geometry. Springer, New York (1988)
[25] Man’ko, V.I., Gromov, N.A.: J. Math. Phys. 33, 1374 (1992) · Zbl 0767.17008 · doi:10.1063/1.529712
[26] Gromov, N.A.: Contractions and Analytical Continuations of the Classical Groups. Unified Approach. Komi Scienfic Center, Syktyvkar (1992) (in Russian)
[27] Santander, M., Herranz, F.J.: Int. J. Mod. Phys. A 12, 99 (1997) · Zbl 0985.81527 · doi:10.1142/S0217751X97000128
[28] Herranz, F.J., Santander, M.: J. Phys. A: Math. Gen. 30, 5411 (1997) · Zbl 0936.81008 · doi:10.1088/0305-4470/30/15/026
[29] Ohn, C.: Lett. Math. Phys. 25, 85 (1992) · Zbl 0766.17016 · doi:10.1007/BF00398304
[30] de Montigny, M., Patera, J.: J. Phys. A: Math. Gen. 24, 525 (1991) · Zbl 0731.17001 · doi:10.1088/0305-4470/24/3/012
[31] Moody, R.V., Patera, J.: J. Phys. A: Math. Gen. 24, 2227 (1991) · Zbl 0751.17025 · doi:10.1088/0305-4470/24/10/014
[32] Herranz, F.J., Santander, M.: J. Phys. A: Math. Gen. 29, 6643 (1996) · Zbl 0905.17037 · doi:10.1088/0305-4470/29/20/018
[33] Herranz, F.J., de Montigny, M., del Olmo, M.A., Santander, M.: J. Phys. A: Math. Gen. 27, 2515 (1994) · Zbl 0834.17004 · doi:10.1088/0305-4470/27/7/027
[34] Wolf, K.B., Boyer, C.B.: J. Math. Phys. 15, 2096 (1974) · Zbl 0293.17008 · doi:10.1063/1.1666589
[35] Bacry, H., Lévy-Leblond, J.M.: J. Math. Phys. 9, 1605 (1968) · Zbl 0162.58606 · doi:10.1063/1.1664490
[36] Bacry, H., Nuyts, J.: J. Math. Phys. 27, 2455 (1986) · Zbl 0602.22009 · doi:10.1063/1.527306
[37] de Montigny, M., Patera, J., Tolar, J.: J. Math. Phys. 35, 405 (1994) · Zbl 0813.17005 · doi:10.1063/1.530893
[38] Jordan, C.: Essai sur la géométrie à n dimensions. In: Oeuvres. Gauthier-Villars, Paris (1961–1964)
[39] Figueroa-O’Farrill, J.M.: J. Math. Phys. 30, 2735 (1989) · Zbl 0699.17008 · doi:10.1063/1.528506
[40] Herranz, F.J., Santander, M.: J. Phys. A: Math. Gen. 35, 6601 (2002) · Zbl 1039.53075 · doi:10.1088/0305-4470/35/31/306
[41] Ballesteros, A., Ragnisco, O.: J. Phys. A: Math. Gen. 31, 3791 (1998) · Zbl 0906.35100 · doi:10.1088/0305-4470/31/16/009
[42] Izmest’ev, A.A., Pogosyan, G.S., Sissakian, A.N., Winternitz, P.: J. Math. Phys. 40, 1549 (1999) · Zbl 1055.17500 · doi:10.1063/1.532820
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.