Yang-Baxter \(R\)-operators for \(osp\) superalgebras. (English) Zbl 1489.81035
Summary: We study Yang-Baxter equations with orthosymplectic supersymmetry. We extend a new approach of the construction of the spinor and metaplectic \(\hat{\mathcal{R}}\)-operators with orthogonal and symplectic symmetries to the supersymmetric case of orthosymplectic symmetry. In this approach the orthosymplectic \(\hat{\mathcal{R}} \)-operator is given by the ratio of two operator valued Euler Gamma-functions. We illustrate this approach by calculating such \(\hat{\mathcal{R}}\) operators in explicit form for special cases of the \(osp(n | 2 m)\) algebra, in particular for a few low-rank cases. We also propose a novel, simpler and more elegant, derivation of the Shankar-Witten type formula for the osp invariant \(\hat{\mathcal{R}}\)-operator and demonstrate the equivalence of the previous approach to the new one in the general case of the \(\hat{\mathcal{R}}\)-operator invariant under the action of the \(osp(n | 2 m)\) algebra.
MSC:
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
References:
| [1] | Isaev, A. P.; Karakhanyan, D.; Kirschner, R., Orthogonal and symplectic Yangians and Yang-Baxter R-operators, Nucl. Phys. B, 904, 124-147 (2016) · Zbl 1332.81096 |
| [2] | Fuksa, J.; Isaev, A. P.; Karakhanyan, D.; Kirschner, R., Yangians and Yang-Baxter R-operators for ortho-symplectic superalgebras, Nucl. Phys. B, 917, 44 (2017) · Zbl 1371.17003 |
| [3] | Shankar, R.; Witten, E., The S-matrix of the kinks of the \(( \overline{\psi} \psi )^2\) model, Nucl. Phys. B, 141, 349-363 (1978) |
| [4] | Chicherin, D.; Derkachov, S.; Isaev, A. P., Conformal group: R-matrix and star-triangle relation, J. High Energy Phys., 04, Article 020 pp. (2013) · Zbl 1342.81204 |
| [5] | Tarasov, V. O.; Takhtajan, L. A.; Faddeev, L. D., Local Hamiltonians for integrable quantum models on a lattice, Theor. Math. Phys., 57, 163-181 (1983) |
| [6] | Faddeev, L. D., How algebraic Bethe ansatz works for integrable model, (Connes, A.; Kawedzki, K.; Zinn-Justin, J., Quantum Symmetries/Symetries Quantiques, Proc. Les-Houches Summer School, LXIV (1998), North-Holland), 149-211 · Zbl 0934.35170 |
| [7] | Karakhanyan, D.; Kirschner, R., Spinorial RRR operator and algebraic Bethe ansatz, Nucl. Phys. B, 951, Article 114905 pp. (2020) · Zbl 1472.81120 |
| [8] | Chicherin, D.; Derkachov, S.; Isaev, A. P., Spinorial R-matrix, J. Phys. A, 46, 48, Article 485201 pp. (2013) · Zbl 1280.81143 |
| [9] | Karowski, M.; Thun, H. J., Complete S-matrix of the \(O(2 N)\) Gross-Neveu model, Nucl. Phys. B, 190, FS3, 61-92 (1981) |
| [10] | Zamolodchikov, Al. B., Factorizable Scattering in Asymptotically Free 2-dimensional Models of Quantum Field Theory (1979), unpublished |
| [11] | Reshetikhin, N. Yu., Algebraic Bethe-ansatz for \(S O(N)\) invariant transfer-matrices, Zap. Nauč. Semin. LOMI. Zap. Nauč. Semin. LOMI, J. Math. Sci., 54, 3, 940-951 (1991) · Zbl 0723.22026 |
| [12] | Ogievetsky, E.; Wiegmann, P.; Reshetikhin, N., The principal chiral field in two-dimensions on classical Lie algebras: the Bethe ansatz solution and factorized theory of scattering, Nucl. Phys. B, 280, 45 (1987) |
| [13] | Berezin, F. A., Introduction to Algebra and Analisis with Anticommuting Variables (1983), Moscow State University Press: Moscow State University Press Moscow · Zbl 0527.15020 |
| [14] | Arnaudon, D.; Avan, J.; Crampe, N.; Frappat, L.; Ragoucy, E., R-matrix presentation for (super)-Yangians Y(g), J. Math. Phys., 44, 302 (2003); Arnaudon, D.; Avan, J.; Crampe, N.; Doikou, A.; Frappat, L.; Ragoucy, E., Bethe ansatz equations and exact S matrices for the \(o s p(M | 2 n)\) open super spin chain, Nucl. Phys. B, 687, 3, 257 (2004) · Zbl 1149.82313 |
| [15] | Birman, J.; Wenzl, H., Braids, link polynomials and a new algebra, Trans. Am. Math. Soc., 313, 249-273 (1989) · Zbl 0684.57004 |
| [16] | Brauer, R., On algebras which are connected with the semisimple continuous groups, Ann. Math. (2), 38, 4, 857-872 (1937) · JFM 63.0873.02 |
| [17] | Wenzl, H., On the structure of Brauer’s centralizer algebras, Ann. Math., 128, 173-193 (1988) · Zbl 0656.20040 |
| [18] | Nazarov, M., Young’s orthogonal form for Brauer’s centralizer algebra, J. Algebra, 182, 664-693 (1996) · Zbl 0868.20012 |
| [19] | Isaev, A. P.; Molev, A. I., Fusion procedure for the Brauer algebra, Algebra Anal., 22, 3, 142-154 (2010) · Zbl 1219.81147 |
| [20] | Isaev, A. P.; Molev, A. I.; Ogievetsky, O. V., A new fusion procedure for the Brauer algebra and evaluation homomorphisms, Int. Math. Res. Not., 2571-2606 (2012) · Zbl 1276.20006 |
| [21] | Isaev, A. P.; Podoinitsyn, M. A., D-dimensional spin projection operators for arbitrary type of symmetry via Brauer algebra idempotents (2020), e-print · Zbl 1519.81288 |
| [22] | Kulish, P. P.; Sklyanin, E. K., On solutions of the Yang-Baxter equation, J. Sov. Math.. J. Sov. Math., Zap. Nauč. Semin., 95, 5, 129 (1980) · Zbl 0553.58039 |
| [23] | Kulish, P. P., Integrable graded magnets, J. Sov. Math.. J. Sov. Math., Zap. Nauč. Semin., 145, 140 (1985); Kulish, P. P.; Reshetikhin, N. Yu., Universal R matrix of the quantum superalgebra Osp(2 | 1), Lett. Math. Phys., 18, 143 (1989) · Zbl 0694.17008 |
| [24] | Zamolodchikov, A. B.; Zamolodchikov, Al. B., Relativistic factorized S-matrix in two dimensions having O(N) isotopic symmetry, Nucl. Phys. B, 133, 525 (1978) |
| [25] | Berg, B.; Karowski, M.; Weisz, P.; Kurak, V., Factorized \(U(n)\) symmetric S-matrices in two dimensions, Nucl. Phys. B, 134, 1, 125-132 (1978) |
| [26] | Isaev, A. P., Quantum groups and Yang-Baxter equations, preprint MPIM (Bonn), MPI 2004-132 |
| [27] | Drinfeld, V. G., Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR, 283, 5, 1060 (1985) · Zbl 0588.17015 |
| [28] | Derkachov, S. E., Factorization of R-matrix. I, J. Math. Sci., 143, 1, 2773 (2007) |
| [29] | Derkachov, S. E.; Manashov, A. N., Factorization of R-matrix and Baxter Q-operators for generic \(s \ell(N)\) spin chains, J. Phys. A, Math. Theor., 42, Article 075204 pp. (2009) · Zbl 1157.82011 |
| [30] | Isaev, A. P.; Rubakov, V. A., Theory of Groups and Symmetries. Representations of Groups and Lie Algebras, Applications (2020), World Scientific, 600 pp |
| [31] | Faddeev, L. D.; Reshetikhin, N. Yu.; Takhtajan, L. A., Quantization of Lie groups and Lie algebras, Algebra Anal.. Algebra Anal., Leningr. Math. J., 1, 1, 193-225 (1990), (in Russian). English translation in: · Zbl 0715.17015 |
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