Quantum Klein Space and Superspace
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

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We give an algebraic quantization, in the sense of quantum groups, of the complex Minkowski space, and we examine the real forms corresponding to the signatures $(3,1)$, $(2,2)$, $(4,0)$, constructing the corresponding quantum metrics and providing an explicit presentation of the quantized coordinate algebras. In particular, we focus on the Kleinian signature $(2,2)$. The quantizations of the complex and real spaces come together with a coaction of the quantizations of the respective symmetry groups. We also extend such quantizations to the $\mathcal{N}=1$ supersetting.
Keywords: quantum groups; supersymmetry.
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     author = {Rita Fioresi and Emanuele Latini and Alessio Marrani},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a65/}
}
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Rita Fioresi; Emanuele Latini; Alessio Marrani. Quantum Klein Space and Superspace. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a65/

[1] Ballesteros A., Gutiérrez-Sagredo I., Herranz F. J., Meusburger C., Naranjo P., “Quantum groups and noncommutative spacetimes with cosmological constant”, J. Phys. Conf. Ser., 880 (2017), 012023, 8 pp., arXiv: 1702.04704 | DOI

[2] Ballesteros A., Herranz F. J., del Olmo M. A., Santander M., “Quantum structure of the motion groups of the two-dimensional Cayley–Klein geometries”, J. Phys. A: Math. Gen., 26 (1993), 5801–5823 | DOI | MR | Zbl

[3] Ballesteros A., Herranz F. J., del Olmo M. A., Santander M., “Quantum $(2+1)$ kinematical algebras: a global approach”, J. Phys. A: Math. Gen., 27 (1994), 1283–1297 | DOI | MR | Zbl

[4] Ballesteros A., Herranz F. J., del Olmo M. A., Santander M., “A new “null-plane” quantum Poincaré algebra”, Phys. Lett. B, 351 (1995), 137–145, arXiv: q-alg/9502019 | DOI | MR | Zbl

[5] Ballesteros A., Herranz F. J., Meusburger C., Naranjo P., “Twisted $(2+1)$ $\kappa$-AdS algebra, Drinfel'd doubles and non-commutative spacetimes”, SIGMA, 10 (2014), 052, 26 pp., arXiv: 1403.4773 | DOI | MR | Zbl

[6] Belavin A. A., Drinfel'd V. G., “Solutions of the classical Yang–Baxter equation for simple Lie algebras”, Funct. Anal. Appl., 16 (1982), 159–180 | DOI | MR

[7] Bonechi F., Giachetti R., Sorace E., Tarlini M., “Induced representations of the one-dimensional quantum Galilei group”, J. Math. Sci., 104 (2001), 1105–1110 | DOI | MR | Zbl

[8] Borowiec A., Lukierski J., Tolstoy V. N., “Quantum deformations of $D = 4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak o}^\star(4)$ symmetries in unified ${\mathfrak o}(4,{\mathbb C})$ setting”, Phys. Lett. B, 754 (2016), 176–181, arXiv: 1511.03653 | DOI | MR | Zbl

[9] Borowiec A., Lukierski J., Tolstoy V. N., “Addendum to "Quantum deformations of $D = 4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak o}^\star(4)$ symmetries in unified ${\mathfrak o}(4,{\mathbb C})$ setting"”, Phys. Lett. B, 770 (2017), 426–430, arXiv: 1704.06852 | DOI | MR

[10] Borowiec A., Lukierski J., Tolstoy V. N., “Basic quantizations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak o}^\star(4)$ symmetries”, J. High Energy Phys., 2017:11 (2017), 187, 35 pp., arXiv: 1708.09848 | DOI | MR | Zbl

[11] Bryant R. L., “Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor”, Global Analysis and Harmonic Analysis (Marseille-Luminy, 1999), Sémin. Congr., 4, Soc. Math. France, Paris, 2000, 53–94, arXiv: math.DG/0004073 | MR | Zbl

[12] Carmeli C., Caston L., Fioresi R., Mathematical foundations of supersymmetry, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2011 | DOI | MR | Zbl

[13] Cervantes D., Fioresi R., Lledó M. A., “On chiral quantum superspaces”, Supersymmetry in Mathematics and Physics, Lecture Notes in Math., 2027, Springer, Heidelberg, 2011, 69–99, arXiv: 1109.3632 | DOI | MR | Zbl

[14] Cervantes D., Fioresi R., Lledó M. A., “The quantum chiral Minkowski and conformal superspaces”, Adv. Theor. Math. Phys., 15 (2011), 565–620, arXiv: 1007.4469 | DOI | MR | Zbl

[15] Cervantes D., Fioresi R., Lledó M. A., Nadal F. A., “Quadratic deformation of Minkowski space”, Fortschr. Phys., 60 (2012), 970–976, arXiv: 1207.1316 | DOI | MR | Zbl

[16] Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994 | MR | Zbl

[17] Chen P., Chiang H.-W., Hu Y.-C., “A quantized spacetime based on ${\rm Spin}(3,1)$ symmetry”, Internat. J. Modern Phys. D, 25 (2016), 1645004, 6 pp., arXiv: 1606.01490 | DOI | Zbl

[18] Cianfrani F., Kowalski-Glikman J., Pranzetti D., Rosati G., “Symmetries of quantum spacetime in three dimensions”, Phys. Rev. D, 94 (2016), 084044, 17 pp., arXiv: 1606.03085 | DOI | MR

[19] Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994 | MR | Zbl

[20] Doplicher S., Fredenhagen K., Roberts J. E., “The quantum structure of spacetime at the Planck scale and quantum fields”, Comm. Math. Phys., 172 (1995), 187–220, arXiv: hep-th/0303037 | DOI | MR | Zbl

[21] Douglas M. R., Nekrasov N. A., “Noncommutative field theory”, Rev. Modern Phys., 73 (2001), 977–1029, arXiv: hep-th/0106048 | DOI | MR | Zbl

[22] Drinfel'd V. G., “Quantum groups”, Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986), v. 1, 2, Amer. Math. Soc., Providence, RI, 1987, 798–820 | MR

[23] Dunajski M., “Anti-self-dual four-manifolds with a parallel real spinor”, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458 (2002), 1205–1222, arXiv: math.DG/0102225 | DOI | MR | Zbl

[24] Dunajski M., “Einstein–Maxwell dilaton metrics from three-dimensional Einstein–Weyl structures”, Classical Quantum Gravity, 23 (2006), 2833–2839, arXiv: gr-qc/0601014 | DOI | MR | Zbl

[25] Dunajski M., West S., Anti-self-dual conformal structures in neutral signature, arXiv: math.DG/0610280 | MR

[26] Faddeev L. D., Reshetikhin N. Y., Takhtajan L. A., “Quantization of Lie groups and Lie algebras”, Algebraic Analysis, v. I, Academic Press, Boston, MA, 1988, 129–139 | DOI | MR

[27] Fioresi R., “Quantizations of flag manifolds and conformal space time”, Rev. Math. Phys., 9 (1997), 453–465 | DOI | MR | Zbl

[28] Fioresi R., “Quantum deformation of the flag variety”, Comm. Algebra, 27 (1999), 5669–5685 | DOI | MR | Zbl

[29] Fioresi R., “On algebraic supergroups and quantum deformations”, J. Algebra Appl., 2 (2003), 403–423, arXiv: math.QA/0111113 | DOI | MR | Zbl

[30] Fioresi R., Latini E., “The symplectic origin of conformal and {M}inkowski superspaces”, J. Math. Phys., 57 (2016), 022307, 12 pp., arXiv: 1506.09086 | DOI | MR | Zbl

[31] Fioresi R., Latini E., Marrani A., “Klein and conformal superspaces, split algebras and spinor orbits”, Rev. Math. Phys., 29 (2017), 1750011, 37 pp., arXiv: 1603.09063 | DOI | MR | Zbl

[32] Fioresi R., Lledó M. A., The Minkowski and conformal superspaces. The classical and quantum descriptions, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015 | DOI | MR | Zbl

[33] Fioresi R., Lledó M.A., Varadarajan V. S., “The Minkowski and conformal superspaces”, J. Math. Phys., 48 (2007), 113505, 27 pp., arXiv: math.RA/0609813 | DOI | MR | Zbl

[34] Freidel L., Livine E. R., “3D quantum gravity and effective noncommutative quantum field theory”, Phys. Rev. Lett., 96 (2006), 221301, 4 pp., arXiv: hep-th/0512113 | DOI | MR | Zbl

[35] Freidel L., Livine E. R., “Ponzano–Regge model revisited. III Feynman diagrams and effective field theory”, Classical Quantum Gravity, 23 (2006), 2021–2061, arXiv: hep-th/0502106 | DOI | MR | Zbl

[36] Garay L. J., “Quantum gravity and minimum length”, Internat. J. Modern Phys. A, 10 (1995), 145–166, arXiv: gr-qc/9403008 | DOI

[37] Girelli F., Sellaroli G., “${\rm SO}^*(2N)$ coherent states for loop quantum gravity”, J. Math. Phys., 58 (2017), 071708, 31 pp., arXiv: 1701.07519 | DOI | MR | Zbl

[38] Gromov N. A., Man'ko V. I., “Contractions of the irreducible representations of the quantum algebras ${\rm su}_q(2)$ and ${\rm so}_q(3)$”, J. Math. Phys., 33 (1992), 1374–1378 | DOI | MR | Zbl

[39] Heckman J. J., Verlinde H., “Covariant non-commutative space-time”, Nuclear Phys. B, 894 (2015), 58–74, arXiv: 1401.1810 | DOI | MR | Zbl

[40] Hervik S., “Pseudo-Riemannian VSI spaces II”, Classical Quantum Gravity, 29 (2012), 095011, 16 pp., arXiv: 1504.01616 | DOI | MR | Zbl

[41] Hull C., Zwiebach B., “Double field theory”, J. High Energy Phys., 2009:9 (2009), 099, 53 pp., arXiv: 0904.4664 | DOI | MR

[42] Klemm D., Nozawa M., “Geometry of Killing spinors in neutral signature”, Classical Quantum Gravity, 32 (2015), 185012, 36 pp., arXiv: 1504.02710 | DOI | MR | Zbl

[43] Klimyk A., Schmüdgen K., Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997 | DOI | MR | Zbl

[44] Lakshmibai V., Reshetikhin N., “Quantum flag and Schubert schemes”, Deformation Theory and Quantum Groups with Applications to Mathematical Physics (Amherst, MA, 1990), Contemp. Math., 134, Amer. Math. Soc., Providence, RI, 1992, 145–181 | DOI | MR

[45] Lukierski J., Ruegg H., Nowicki A., Tolstoy V. N., “$q$-deformation of Poincaré algebra”, Phys. Lett. B, 264 (1991), 331–338 | DOI | MR

[46] Lukierski J., Ruegg H., Zakrzewski W. J., “Classical and quantum mechanics of free $k$-relativistic systems”, Ann. Physics, 243 (1995), 90–116, arXiv: hep-th/9312153 | DOI | MR | Zbl

[47] Maggiore M., “A generalized uncertainty principle in quantum gravity”, Phys. Lett. B, 204 (1993), 65–69, arXiv: hep-th/9301067 | DOI | MR

[48] Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995 | DOI | MR | Zbl

[49] Majid S., Ruegg H., “Bicrossproduct structure of $\kappa$-Poincaré group and non-commutative geometry”, Phys. Lett. B, 334 (1994), 348–354, arXiv: hep-th/9405107 | DOI | MR | Zbl

[50] Manin Yu. I., “Gauge fields and holomorphic geometry”, J. Sov. Math., 21 (1983), 465–507 | DOI | MR | Zbl

[51] Manin Yu. I., Topics in noncommutative geometry, M.B. Porter Lectures, Princeton University Press, Princeton, NJ, 1991 | DOI | MR | Zbl

[52] Manin Yu. I., Gauge field theory and complex geometry, Grundlehren der Mathematischen Wissenschaften, 289, 2nd ed., Springer-Verlag, Berlin, 1997 | DOI | MR | Zbl

[53] Maślanka P., “The $n$-dimensional $\kappa$-Poincaré algebra and group”, J. Phys. A: Math. Gen., 26 (1993), L1251–L1253 | DOI | MR | Zbl

[54] Ogievetsky O., Schmidke W. B., Wess J., Zumino B., “$q$-deformed Poincaré algebra”, Comm. Math. Phys., 150 (1992), 495–518 | DOI | MR | Zbl

[55] Ooguri H., Vafa C., “Self-duality and $N=2$ string magic”, Modern Phys. Lett. A, 5 (1990), 1389–1398 | DOI | MR | Zbl

[56] Penrose R., “Twistor algebra”, J. Math. Phys., 8 (1967), 345–366 | DOI | MR | Zbl

[57] Penrose R., “The twistor programme”, Rep. Math. Phys., 12 (1977), 65–76 | DOI | MR

[58] Phung H. H., “On the structure of quantum super groups ${\rm GL}_q(m|n)$”, J. Algebra, 211 (1999), 363–383, arXiv: q-alg/9511023 | DOI | MR | Zbl

[59] Podleś P., Woronowicz S. L., “Quantum deformation of Lorentz group”, Comm. Math. Phys., 130 (1990), 381–431 | DOI | MR | Zbl

[60] Rennecke F., “$O(d,d)$-duality in string theory”, J. High Energy Phys., 2014:10 (2014), 069, 22 pp., arXiv: 1404.0912 | DOI | MR | Zbl

[61] Semenov-Tyan-Shanskii M. A., What is a classical $r$-matrix?, Funct. Anal. Appl., 17 (1983), 259–272 | DOI | MR

[62] Snyder H. S., “Quantized space-time”, Phys. Rev., 71 (1947), 38–41 | DOI | MR | Zbl

[63] Szabo R. J., “Quantum field theory on noncommutative spaces”, Phys. Rep., 378 (2003), 207–299, arXiv: hep-th/0109162 | DOI | MR | Zbl

[64] Taft E., Towber J., “Quantum deformation of flag schemes and Grassmann schemes. I A $q$-deformation of the shape-algebra for ${\rm GL}(n)$”, J. Algebra, 142 (1991), 1–36 | DOI | MR | Zbl

[65] Witten E., “Perturbative gauge theory as a string theory in twistor space”, Comm. Math. Phys., 252 (2004), 189–258, arXiv: hep-th/0312171 | DOI | MR | Zbl

[66] Yaglom I. M., A simple non-Euclidean geometry and its physical basis. An elementary account of Galilean geometry and the Galilean principle of relativity, Heidelberg Science Library, Springer-Verlag, New York–Heidelberg, 1979 | DOI | MR | Zbl

[67] Yang C. N., “On quantized space-time”, Phys. Rev., 72 (1947), 874 | DOI | MR | Zbl

[68] Zakrzewski S., “Quantum Poincaré group related to the $\kappa$-Poincaré algebra”, J. Phys. A: Math. Gen., 27 (1994), 2075–2082 | DOI | MR | Zbl

[69] Zhang H., Zhang R. B., “Dual canonical bases for the quantum general linear supergroup”, J. Algebra, 304 (2006), 1026–1058, arXiv: math.QA/0510186 | DOI | MR | Zbl