Abstract
We construct firstly the complete list of five quantum deformations of D = 4 complex homogeneous orthogonal Lie algebra \( \mathfrak{o}\left(4;\mathbb{C}\right)\cong \mathfrak{o}\left(3;\mathbb{C}\right)\oplus \mathfrak{o}\left(3;\mathbb{C}\right) \), describing quantum rotational symmetries of four-dimensional complex space-time, in particular we provide the corresponding universal quantum R-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of \( \mathfrak{o}\left(4;\mathbb{C}\right) \): Euclidean \( \mathfrak{o}(4) \), Lorentz \( \mathfrak{o}\left(3,\ 1\right) \), Kleinian \( \mathfrak{o}\left(2,\ 2\right) \) and quaternionic \( {\mathfrak{o}}^{\star }(4) \). For \( \mathfrak{o}\left(3,\ 1\right) \) we only recall well-known results obtained previously by the authors, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) as well as for the complex Lie algebra \( \mathfrak{o}\left(4;\mathbb{C}\right) \) we present new results.
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References
S. Majid, Hopf Algebras for Physics at the Planck Scale, Class. Quant. Grav. 5 (1988) 1587 [INSPIRE].
S. Doplicher, K. Fredenhagen and J.E. Roberts, The quantum structure of space-time at the Planck scale and quantum fields, Commun. Math. Phys. 172 (1995) 187 [hep-th/0303037] [INSPIRE].
L.J. Garay, Quantum gravity and minimum length, Int. J. Mod. Phys. A 10 (1995) 145 [gr-qc/9403008] [INSPIRE].
A. Ashtekar and J. Lewandowski, Background independent quantum gravity: A Status report, Class. Quant. Grav. 21 (2004) R53 [gr-qc/0404018] [INSPIRE].
D. Kaminski, Algebras of Quantum Variables for Loop Quantum Gravity, I. Overview, arXiv:1108.4577 [INSPIRE].
T. Thiemann, Modern Canonical Quantum General relativity, Cambridge University Press (2007).
F. Cianfrani, J. Kowalski-Glikman, D. Pranzetti and G. Rosati, Symmetries of quantum spacetime in three dimensions, Phys. Rev. D 94 (2016) 084044 [arXiv:1606.03085] [INSPIRE].
J. Lukierski, H. Ruegg, A. Nowicki and V.N. Tolstoi, Q deformation of Poincaré algebra, Phys. Lett. B 264 (1991) 331 [INSPIRE].
G. Amelino-Camelia, L. Smolin and A. Starodubtsev, Quantum symmetry, the cosmological constant and Planck scale phenomenology, Class. Quant. Grav. 21 (2004) 3095 [hep-th/0306134] [INSPIRE].
V.G. Drinfeld, Quantum Groups, Proceedings of the ICM, A. Gleason ed., publ. AMS, Berkeley (1985), p. 798, Providence, Rhode Island (1987).
V.G. Drinfeld, Quantum groups, Leningrad (1990) 1, Lect. Notes Math., vol. 1510, Springer, Berlin (1992).
V.G. Drinfeld, Quasi-Hopf algebras, Algebra i Analiz 1 (1989) 114 [Leningrad Math. J. 1 (1990)1419].
S.L. Woronowicz, Compact matrix pseudogroups, Commun. Math. Phys. 111 (1987) 613 [INSPIRE].
P.I. Etingof and D.A. Kazhdan, Quantization of Lie bialgebras. I, Selecta Math. (N.S.) 2 (1996) 1 [q-alg/9506005].
P. Etingof and O. Schiffmann, Lectures on quantum groups, Internationa Press (2002).
V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge University Press (1994).
S. Majid, Foundations of Quantum Groups, Cambridge University Press (1995).
C. Kassel, Quantum Groups, Springer (1995).
A. Klimyk and K. Schmüdgen, Quantum Groups and Their Representations, Springer (1997).
A. Borowiec, J. Lukierski and V.N. Tolstoy, Quantum deformations of D = 4 Euclidean, Lorentz, Kleinian and quaternionic \( {\mathfrak{o}}^{\star }(4) \) symmetries in unified \( \mathfrak{o}\left(4;\mathbb{C}\right) \) setting, Phys. Lett. B 754 (2016) 176 [arXiv:1511.03653] [INSPIRE].
J. Lukierski and V.N. Tolstoy, Quantizations of D = 3 Lorentz symmetry, Eur. Phys. J. C 77 (2017) 226 [arXiv:1612.03866] [INSPIRE].
A. Borowiec, J. Lukierski and V.N. Tolstoy, Quantum deformations of D = 4 Euclidean, Lorentz, Kleinian and quaternionic \( {\mathfrak{o}}^{\star }(4) \) symmetries in unified \( \mathfrak{o}\left(4;\mathbb{C}\right) \) setting — Addendum, Phys. Lett. B 770 (2017) 426 [arXiv:1704.06852] [INSPIRE].
A.A. Belavin and V.G. Drinfeld, Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl. 16 (1982) 159.
A.A. Belavin and V.G. Drinfeld, Triangle equations and simple Lie algebras, Soviet Sci. Rev. Sect. C: Math. Phys. Rev. 4 (1984) 93.
S.M. Khoroshkin and V.N. Tolstoy, Universal R-matrix for quantized superalgebras, Commun. Math. Phys. 141 (1991) 599.
J. Lukierski, A. Nowicki and H. Ruegg, Real forms of complex quantum anti-de Sitter algebra U q (Sp(4; C)) and their contraction schemes, Phys. Lett. B 271 (1991) 321 [hep-th/9108018] [INSPIRE].
S.L. Woronowicz, New quantum deformation of SL(2, ℂ). Hopf algebra level (English summary), Rept. Math. Phys. 30 (1991) 259.
A. Borowiec, J. Lukierski and V.N. Tolstoy, Jordanian twist quantization of D = 4 Lorentz and Poincaré algebras and D = 3 contraction limit, Eur. Phys. J. C 48 (2006) 633 [hep-th/0604146] [INSPIRE].
A. Borowiec, J. Lukierski and V.N. Tolstoy, Once again about quantum deformations of D = 4 Lorentz algebra: Twistings of q-deformation, Eur. Phys. J. C 57(2008) 601 [arXiv:0804.3305] [INSPIRE].
E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini, The quantum Heisenberg goup H(1)q, J. Math. Phys. 32 (1991) 1155 [INSPIRE].
C. Klimčík, Yang-Baxter σ-models and dS/AdS T duality, JHEP 12 (2002) 051 [hep-th/0210095] [INSPIRE].
C. Klimčík, On integrability of the Yang-Baxter σ-model, J. Math. Phys. 50 (2009) 043508 [arXiv:0802.3518] [INSPIRE].
B. Vicedo, Deformed integrable σ-models, classical R-matrices and classical exchange algebra on Drinfel’d doubles, J. Phys. A 48 (2015) 355203 [arXiv:1504.06303] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, Jordanian deformations of the AdS 5 × S 5 superstring, JHEP 04 (2014) 153 [arXiv:1401.4855] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, A Jordanian deformation of AdS space in type IIB supergravity, JHEP 06 (2014) 146 [arXiv:1402.6147] [INSPIRE].
T. Matsumoto and K. Yoshida, Integrable deformations of the AdS 5 × S 5 superstring and the classical Yang-Baxter equation — Towards the gravity/CYBE correspondence, J. Phys. Conf. Ser. 563 (2014) 012020 [arXiv:1410.0575] [INSPIRE].
T. Matsumoto and K. Yoshida, Yang-Baxter σ-models based on the CYBE, Nucl. Phys. B 893 (2015) 287 [arXiv:1501.03665] [INSPIRE].
S.J. van Tongeren, On classical Yang-Baxter based deformations of the AdS 5 × S 5 superstring, JHEP 06 (2015) 048 [arXiv:1504.05516] [INSPIRE].
S.J. van Tongeren, Yang-Baxter deformations, AdS/CFT and twist-noncommutative gauge theory, Nucl. Phys. B 904 (2016) 148 [arXiv:1506.01023] [INSPIRE].
A. Pachol and S.J. van Tongeren, Quantum deformations of the flat space superstring, Phys. Rev. D 93 (2016) 026008 [arXiv:1510.02389] [INSPIRE].
A. Borowiec, H. Kyono, J. Lukierski, J.-i. Sakamoto and K. Yoshida, Yang-Baxter σ-models and Lax pairs arising from κ-Poincaré r-matrices, JHEP 04 (2016) 079 [arXiv:1510.03083] [INSPIRE].
V.G. Drinfeld, Hamiltonian structures of lie groups, lie bialgebras and the geometric meaning of the classical Yang-Baxter equations, Sov. Math. Dokl. 27 (1983) 68 [INSPIRE].
M. Blaszak, On a non-standard algebraic description of integrable nonlinear systems, Physica A 198 (1993) 637.
M.A. Semenov-Tyan-Shanski, Integrable Systems and Factorization Problems, nlin/0209057.
I.Ya. Dorfman and A.S. Fokas, Hamiltonian theory over noncommutative rings and integrability in multidimensions, J. Math. Phys. 33 (1992) 2504 [INSPIRE].
A.S. Fokas and I.M. Gelfand, Algebraic Aspects of Integrable Systems, Birkhauser (1997).
R. Fioresi, E. Latini and A. Marrani, Quantum Klein Space and Superspace, arXiv:1705.01755 [INSPIRE].
N. Beisert, R. Hecht and B. Hoare, Maximally extended \( \mathfrak{s}\mathfrak{l}\left(2\Big|2\right) \) , q-deformed \( \mathfrak{d}\left(2,\ 1;\upepsilon \right) \) and 3D kappa-Poincaré, J. Phys. A 50 (2017) 314003 [arXiv:1704.05093] [INSPIRE].
M.A. Semenov-Tian-Shansky, What is a classical r-matrix?, Funct. Anal. Appl. 17 (1983) 259 [INSPIRE].
P. Kulish, Twist deformations of quantum integrable spin chains, in Noncommutative spacetimes. Symmetries in noncommutative geometry and field theory, P. Aschieri et al. eds., Lect. Notes Phys., vol. 774, Springer-Verlag, Berlin (2009), pg. 165.
E.E. Demidov, Yu.I. Manin, E.E. Mukhin and D.V. Zhdanovich, Nonstandard quantum deformations of GL(n) and constant solutions of the Yang-Baxter equation, Prog. Theor. Phys. Suppl. 102 (1990) 203 [INSPIRE].
O.V. Ogievetsky, Hopf structures on the Borel subalgebra of sl(2), in proceedings of The Winter School Geometry and Physics, Zidkov, January 2013, Czech Republic, Rend. Circ. Math. Palermo Ser. II 37 (1993) 185, Max Planck Int. prepr. MPI-Ph/92-99.
G.W. Delius and A. Huffmann, On quantum Lie algebras and quantum root systems, J. Phys. A 29 (1996) 1703 [q-alg/9506017] [INSPIRE].
P. Aschieri, A. Borowiec and A. Pachol, Observables and dispersion relations in κ-Minkowski spacetime, JHEP 10 (2017) 152 [arXiv:1703.08726] [INSPIRE].
P.P. Kulish and A.I. Mudrov, Twist-related geometries on q-Minkowski space, Proc. Steklov Inst. Math. 226 (1999) 97 [math/9901019] [INSPIRE].
A.P. Isaev and O.V. Ogievetsky, On quantization of r matrices for Belavin-Drinfeld triples, Phys. Atomic Nuclei 64 (2001) 2126 [Yad. Fiz. 64 (2001) 2216].
A. Borowiec, J. Lukierski and V.N. Tolstoy, Basic twist quantization of osp(1|2) and kappa deformation of D = 1 superconformal mechanics, Mod. Phys. Lett. A 18 (2003) 1157 [hep-th/0301033] [INSPIRE].
M. Samsonov, Quantization of semi-classical twists and noncommutative geometry, Lett. Math. Phys. 75 (2006) 63.
M. Samsonov, Semi-classical twists for \( \mathfrak{s}{\mathfrak{l}}_3 \) and \( \mathfrak{s}{\mathfrak{l}}_4 \) boundary r-matrices of Cremmer-Gervais type, Lett. Math. Phys. 72 (2005) 197.
V.V. Lyubashenko, Real and imaginary forms of Quantum groups, Proc. of the Euler Institute, St. Petersburg (1990), Lect. Notes Math., vol. 1510, pg. 67, Springer.
S. Majid and P.K. Osei, Quasitriangular structure and twisting of the 2 + 1 bicrossproduct model, arXiv:1708.07999 [INSPIRE].
P. Podles, Quantization enforces interaction. Quantum mechanics of two particles on quantum sphere, Int. J. Mod. Phys. A7S1B (1992) 805 [INSPIRE].
G. Fiore and J. Wess, On full twisted Poincaré symmetry and QFT on Moyal-Weyl spaces, Phys. Rev. D 75 (2007) 105022 [hep-th/0701078] [INSPIRE].
J. Lukierski and M. Woronowicz, Braided Tensor Products and the Covariance of Quantum Noncommutative Free Fields, J. Phys. A 45 (2012) 215402 [arXiv:1105.3612] [INSPIRE].
J. Lukierski and M. Woronowicz, Braided Field Quantization from Quantum Poincaré Covariance, Int. J. Mod. Phys. A 27 (2012) 1250084 [arXiv:1206.5656] [INSPIRE].
C. Blohmann, Covariant realization of quantum spaces as star products by Drinfeld twists, J. Math. Phys. 44 (2003) 4736 [math/0209180] [INSPIRE].
P.P. Kulish, Twists of quantum groups and noncommutative field theory, hep-th/0606056 [INSPIRE].
S. Zakrzewski, Poisson structures on the Lorentz group, Lett. Math. Phys. 32 (1994) 11.
S. Zakrzewski, Poisson structures on Poincaré group. Commun. Math. Phys. 185 (1997) 285 [q-alg/9602001].
V.N. Tolstoy, Twisted Quantum Deformations of Lorentz and Poincaré algebras, Bulg. J. Phys. 35 (2008) 441 [arXiv:0712.3962] [INSPIRE].
A. Borowiec and A. Pachol, κ-Deformations and Extended κ-Minkowski Spacetimes, SIGMA 10 (2014) 107 [arXiv:1404.2916] [INSPIRE].
A. Borowiec, J. Lukierski and V.N. Tolstoy, Real and pseudoreal forms of D = 4 complex Euclidean (super)algebras and super-Poincaré/super-Euclidean r-matrices, J. Phys. Conf. Ser. 670 (2016) 012013 [arXiv:1510.09125] [INSPIRE].
E. Witten, (2 + 1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
A. Ballesteros, F.J. Herranz and C. Meusburger, Three-dimensional gravity and Drinfel’d doubles: spacetimes and symmetries from quantum deformations, Phys. Lett. B 687 (2010) 375 [arXiv:1001.4228] [INSPIRE].
A. Ballesteros, F.J. Herranz and C. Meusburger, Drinfel’d doubles for (2 + 1)-gravity, Class. Quant. Grav. 30 (2013) 155012 [arXiv:1303.3080] [INSPIRE].
A. Ballesteros, F.J. Herranz and C. Meusburger, A (2 + 1) non-commutative Drinfel’d double spacetime with cosmological constant, Phys. Lett. B 732 (2014) 201 [arXiv:1402.2884] [INSPIRE].
V.V. Fock and A.A. Rosly, Poisson structure on moduli of flat connections on Riemann surfaces and r matrix, Am. Math. Soc. Transl. 191 (1999) 67 [math/9802054] [INSPIRE].
A.Yu. Alekseev and A.Z. Malkin, Symplectic structure of the moduli space of flat connection on a Riemann surface, Commun. Math. Phys. 169 (1995) 99 [hep-th/9312004] [INSPIRE].
C. Meusburger and B.J. Schroers, Generalised Chern-Simons actions for 3d gravity and kappa-Poincaré symmetry, Nucl. Phys. B 806 (2009) 462 [arXiv:0805.3318] [INSPIRE].
G. Papageorgiou and B.J. Schroers, Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra, JHEP 11 (2010) 020 [arXiv:1008.0279] [INSPIRE].
G. Papageorgiou and B.J. Schroers, A Chern-Simons approach to Galilean quantum gravity in 2 + 1 dimensions, JHEP 11 (2009) 009 [arXiv:0907.2880] [INSPIRE].
P. Stachura, Poisson-Lie structures on Poincaré and Euclidean groups in three dimensions, J. Phys. A 31 (1998) 4555.
A. Borowiec, J. Lukierski, M.n. Mozrzymas and V.N. Tolstoy, N = 1/2 Deformations of Chiral Superspaces from New Twisted Poincaré and Euclidean Superalgebras, JHEP 06 (2012) 154 [arXiv:1112.1936] [INSPIRE].
A. Borowiec, J. Lukierski, M. Mozrzymas and V.N. Tolstoy, New class of quantum deformations of D = 4 Euclidean supersymmetry, arXiv:1211.4546 [INSPIRE].
A.G. Reyman, Poisson structures related to quantum groups, in Quantum Groups and its Applications in Physics, Int. School "Enrico Fermi", Varrena 1994, L. Castellani and J. Wess eds., IOS Press, Amsterdam (1996), pg. 407.
J. de Lucas and D. Wysocki, A Grassmann algebra approach to classifying real coboundary Lie bialgebras, arXiv:1710.05022.
G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge University Press (1990).
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Borowiec, A., Lukierski, J. & Tolstoy, V. Basic quantizations of D = 4 Euclidean, Lorentz, Kleinian and quaternionic \( {\mathfrak{o}}^{\star }(4) \) symmetries. J. High Energ. Phys. 2017, 187 (2017). https://doi.org/10.1007/JHEP11(2017)187
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DOI: https://doi.org/10.1007/JHEP11(2017)187