A \(\mathrm{Q}\)-operator for open spin chains. II: Boundary factorization. (English) Zbl 07845293
Summary: One of the features of Baxter’s \(\mathrm{Q}\)-operators for many closed spin chain models is that all transfer matrices arise as products of two \(\mathrm{Q}\)-operators with shifts in the spectral parameter. In the representation-theoretical approach to \(\mathrm{Q}\)-operators, underlying this is a factorization formula for \(\mathrm{L}\)-operators (solutions of the Yang-Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang-Baxter equation) associated to these representations. In the case of quantum affine \(\mathfrak{sl}_{2}\) and diagonal \(\mathrm{K}\)-matrices, we derive such an identity using the recently formulated theory of universal \(\mathrm{K}\)-matrices for quantum affine algebras.
For Part I see [B. Vlaar and R. Weston, J. Phys. A, Math. Theor. 53, No. 24, Article ID 245205, 47 p. (2020; Zbl 1519.82035)].
For Part I see [B. Vlaar and R. Weston, J. Phys. A, Math. Theor. 53, No. 24, Article ID 245205, 47 p. (2020; Zbl 1519.82035)].
MSC:
| 82B23 | Exactly solvable models; Bethe ansatz |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 16T15 | Coalgebras and comodules; corings |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 81U15 | Exactly and quasi-solvable systems arising in quantum theory |
| 81Q80 | Special quantum systems, such as solvable systems |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
Citations:
Zbl 1519.82035References:
| [1] | Appel, A.; Vlaar, B., Universal K-matrices for quantum Kac-Moody algebras, Represent. Theory, 26, 764-824, 2022 · Zbl 1522.17017 · doi:10.1090/ert/623 |
| [2] | Appel, A., Vlaar, B.: Trigonometric K-matrices for finite-dimensional representations of quantum affine algebras (2022). Preprint at arXiv:2203.16503 · Zbl 1522.17017 |
| [3] | Baxter, R., Partition function of the eight-vertex lattice model, Ann. Phys., 70, 1, 193-228, 1972 · Zbl 0236.60070 · doi:10.1016/0003-4916(72)90335-1 |
| [4] | Baxter, R., Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. I. Some fundamental eigenvectors, Ann. Phys., 76, 1, 1-24, 1973 · Zbl 1092.82511 · doi:10.1016/0003-4916(73)90439-9 |
| [5] | Baseilhac, P.; Belliard, S., The half-infinite XXZ chain in Onsager’s approach, Nucl. Phys. B, 873, 3, 550-584, 2013 · Zbl 1282.82010 · doi:10.1016/j.nuclphysb.2013.05.003 |
| [6] | Boos, H.; Göhmann, F.; Klümper, A.; Nirov, K.; Razumov, A., Exercises with the universal R-matrix, J. Phys. A Math. Theor., 43, 41, 2010 · Zbl 1200.81085 · doi:10.1088/1751-8113/43/41/415208 |
| [7] | Boos, H.; Göhmann, F.; Klümper, A.; Nirov, K.; Razumov, A., Universal integrability objects, Theor. Math. Phys., 174, 1, 21-39, 2013 · Zbl 1280.81068 · doi:10.1007/s11232-013-0002-8 |
| [8] | Boos, H.; Göhmann, F.; Klümper, A.; Nirov, K.; Razumov, A., Universal R-matrix and functional relations, Rev. Math. Phys., 26, 6, 1430005, 2014 · Zbl 1344.17013 · doi:10.1142/S0129055X14300052 |
| [9] | Boos, H.; Jimbo, M.; Miwa, T.; Smirnov, F.; Takeyama, Y., Hidden Grassmann structure in the XXZ model II: creation operators, Commun. Math. Phys., 286, 3, 875-932, 2009 · Zbl 1173.82003 · doi:10.1007/s00220-008-0617-z |
| [10] | Balagović, M., Kolb, S.: Universal K-matrix for quantum symmetric pairs. Journal für die reine und angewandte Mathematik (Crelles Journal) (2019) · Zbl 1425.81058 |
| [11] | Bazhanov, V.; Łukowski, T.; Meneghelli, C.; Staudacher, M., A Shortcut to the Q-operator, J. Stat. Mech. Theory Exp., 11, P11002, 2010 · doi:10.1088/1742-5468/2010/11/P11002 |
| [12] | Bazhanov, V.; Lukyanov, S.; Zamolodchikov, A., Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Commun. Math. Phys., 177, 2, 381-398, 1996 · Zbl 0851.35113 · doi:10.1007/BF02101898 |
| [13] | Bazhanov, V.; Lukyanov, S.; Zamolodchikov, A., Integrable structure of conformal field theory II. Q-operator and DDV equation, Commun. Math. Phys., 190, 247-278, 1997 · Zbl 0908.35114 · doi:10.1007/s002200050240 |
| [14] | Bazhanov, V.; Lukyanov, S.; Zamolodchikov, A., Integrable structure of conformal field theory III. The Yang-Baxter relation, Commun. Math. Phys., 200, 2, 297-324, 1999 · Zbl 1057.81531 · doi:10.1007/s002200050531 |
| [15] | Bazhanov, V.; Stroganov, Yu, Chiral Potts model as a descendant of the six-vertex model, J. Stat. Phys., 59, 3, 799-817, 1990 · Zbl 0718.60123 · doi:10.1007/BF01025851 |
| [16] | Baseilhac, P.; Tsuboi, Z., Asymptotic representations of augmented q-Onsager algebra and boundary K-operators related to Baxter Q-operators, Nucl. Phys. B, 929, 397-437, 2018 · Zbl 1382.81112 · doi:10.1016/j.nuclphysb.2018.02.017 |
| [17] | Bao, H.; Wang, W., A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs, 2018, Paris: Société mathématique de France, Paris · Zbl 1411.17001 · doi:10.24033/ast.1052 |
| [18] | Chari, V., Braid group actions and tensor products, Int. Math. Res. Not., 7, 357-382, 2002 · Zbl 0990.17009 · doi:10.1155/S107379280210612X |
| [19] | Cherednik, I., Factorizing particles on a half-line and root systems, Theor. Math. Phys., 61, 1, 977-983, 1984 · Zbl 0575.22021 · doi:10.1007/BF01038545 |
| [20] | Chari, V., Pressley, A.: Quantum affine algebras and their representations. Representation theory of groups (Banff, AB, 1994), CMS Conference Proceedings, vol. 16, pp. 59-78. American Mathematical Society, Providence (1994) · Zbl 0855.17009 |
| [21] | Chari, V., Pressley, A.: A guide to quantum groups. Cambridge University Press, Cambridge (1995). Corrected reprint of the 1994 original · Zbl 0839.17010 |
| [22] | Derkachov, S.: Factorization of R-matrix and Baxter’s Q-operator (2005). Preprint at arXiv:math/0507252 |
| [23] | Derkachov, S., Factorization of the R-matrix. I., J. Math. Sci., 143, 1, 2773-2790, 2007 · Zbl 0967.81527 · doi:10.1007/s10958-007-0164-8 |
| [24] | Delius, G.; George, A., Quantum affine reflection algebras of type \(d_n^{(1)}\) and reflection matrices, Lett. Math. Phys., 62, 211-217, 2002 · Zbl 1042.17010 · doi:10.1023/A:1022259710600 |
| [25] | Derkachov, S.; Karakhanyan, D.; Kirschner, R., Baxter Q-operators of the XXZ chain and R-matrix factorization, Nucl. Phys. B, 738, 3, 368-390, 2006 · Zbl 1109.82008 · doi:10.1016/j.nuclphysb.2005.12.015 |
| [26] | Delius, G.; Mackay, N., Quantum group symmetry in sine-Gordon and affine Toda field theories on the half-line, Commun. Math. Phys., 233, 173-190, 2003 · Zbl 1040.81042 · doi:10.1007/s00220-002-0758-4 |
| [27] | Drinfeld, V., Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl., 32, 254-258, 1985 · Zbl 0588.17015 |
| [28] | Drinfeld, V.: Quantum groups. In: Gleason, A.M. (ed.) Proceedings of the International Congress of Mathematicians, Berkeley, pp. 798-820. American Mathematical Society, Providence (1986) |
| [29] | Etingof, P.; Moura, A., Elliptic central characters and blocks of finite-dimensional representations of quantum affine algebras, Represent. Theory, 7, 346-373, 2003 · Zbl 1066.17005 · doi:10.1090/S1088-4165-03-00201-2 |
| [30] | Frenkel, E.; Hernandez, D., Baxter’s relations and spectra of quantum integrable models, Duke Math. J., 164, 12, 2407-2460, 2015 · Zbl 1332.82022 · doi:10.1215/00127094-3146282 |
| [31] | Frenkel, I.; Reshetikhin, N., Quantum affine algebras and holonomic difference equations, Commun. Math. Phys., 146, 1, 1-60, 1992 · Zbl 0760.17006 · doi:10.1007/BF02099206 |
| [32] | Frenkel, E.; Reshetikhin, N., The \(q\)-characters of representations of quantum affine algebras and deformations of \(W\)-algebras, in Recent Developments in Quantum Affine Algebras and related topics, Contemp. Math., 248, 163-205, 1999 · Zbl 0973.17015 · doi:10.1090/conm/248/03823 |
| [33] | Frassek, R.; Szécsényi, I., Q-operators for the open Heisenberg spin chain, Nucl. Phys. B, 901, 229-248, 2015 · Zbl 1332.82014 · doi:10.1016/j.nuclphysb.2015.10.010 |
| [34] | Gasper, G., Rahman, M.: Basic hypergeometric series, Encyclopedia of mathematics and its applications, vol. 35. Cambridge University Press, Cambridge (1990) · Zbl 0695.33001 |
| [35] | Hernandez, D.: Avancées concernant les R-matrices et leurs applications [d’après Maulik-Okounkov, Kang-Kashiwara-Kim-Oh, ...] (2019), no. 407, Séminaire Bourbaki, 69ème année, Vol. 2016-2017, no. 1129, pp. 297-331 · Zbl 1483.17011 |
| [36] | Hernandez, D.; Jimbo, M., Asymptotic representations and Drinfeld rational fractions, Compos. Math., 148, 5, 1593-1623, 2012 · Zbl 1266.17010 · doi:10.1112/S0010437X12000267 |
| [37] | Jantzen, J. C.: Lectures on quantum groups. Grad. Stud. Math., vol. 6. American Mathematical Society, Providence (1996) · Zbl 0842.17012 |
| [38] | Jimbo, M., A q-analogue of \(U({\mathfrak{g} \mathfrak{l} }(N+1))\), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys., 11, 3, 247-252, 1986 · Zbl 0602.17005 · doi:10.1007/BF00400222 |
| [39] | Jimbo, M., Quantum R matrix for the generalized Toda system, Commun. Math. Phys., 102, 537-547, 1986 · Zbl 0604.58013 · doi:10.1007/BF01221646 |
| [40] | Kac, V., Infinite-Dimensional Lie Algebras, 1990, Cambridge: Cambridge University Press, Cambridge · Zbl 0716.17022 · doi:10.1017/CBO9780511626234 |
| [41] | Kolb, S., Quantum symmetric Kac-Moody pairs, Adv. Math., 267, 395-469, 2014 · Zbl 1300.17011 · doi:10.1016/j.aim.2014.08.010 |
| [42] | Koornwinder, T.: Special functions and q-commuting variables. In: Special Functions, q-Series and Related Topics, Fields Institute Communications 14, pp. 131-166. American Mathematical Society, Providence (1997) · Zbl 0882.33014 |
| [43] | Kazhdan, D.; Soibelman, Y., Representations of quantum affine algebras, Sel. Math. (N.S.), 1, 3, 537-595, 1995 · Zbl 0842.17021 · doi:10.1007/BF01589498 |
| [44] | Khoroshkin, S.; Tsuboi, Z., The universal R-matrix and factorization of the L-operators related to the Baxter Q-operators, J. Phys. A Math. Theor., 47, 19, 2014 · Zbl 1291.81191 · doi:10.1088/1751-8113/47/19/192003 |
| [45] | Lusztig, G., Introduction to Quantum Groups, 1994, Boston: Birkhäuser, Boston · Zbl 0845.20034 |
| [46] | Mezincescu, L.; Nepomechie, RI, Fractional-spin integrals of motion for the boundary Sine-Gordon model at the free fermion point, Int. J. Mod. Phys. A, 13, 2747-2764, 1998 · Zbl 0931.81006 · doi:10.1142/S0217751X98001402 |
| [47] | Reshetikhin, N., Stokman, J., Vlaar, B.: Boundary quantum Knizhnik-Zamolodchikov equations and fusion. Annales Henri Poincaré 1-41 (2015) · Zbl 1347.81053 |
| [48] | Regelskis, V., Vlaar, B.: Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type · Zbl 1430.39010 |
| [49] | Sklyanin, E., Boundary conditions for integrable quantum systems, J. Phys. A Math. Gen., 21, 2375-2389, 1988 · Zbl 0685.58058 · doi:10.1088/0305-4470/21/10/015 |
| [50] | Tolstoy, V.; Khoroshkin, S., The universal R-matrix for quantum untwisted affine Lie algebras, Funct. Anal. Appl., 26, 69-71, 1992 · Zbl 0758.17011 · doi:10.1007/BF01077085 |
| [51] | Tsuboi, Z., Universal Baxter TQ-relations for open boundary quantum integrable systems, Nucl. Phys. B, 963, 2021 · Zbl 1509.82034 · doi:10.1016/j.nuclphysb.2020.115286 |
| [52] | Vlaar, B.; Weston, R., A Q-operator for open spin chains I: Baxter’s TQ relation, J. Phys. A Math. Theor., 53, 24, 2020 · Zbl 1519.82035 · doi:10.1088/1751-8121/ab8854 |
| [53] | Yang, W-L; Nepomechie, RI; Zhang, Y-Z, Q-operator and T-Q relation from the fusion hierarchy, Phys. Lett. B, 633, 4-5, 664-670, 2006 · Zbl 1247.82019 · doi:10.1016/j.physletb.2005.12.022 |
| [54] | Yang, W-L; Zhang, Y-L, \(T-Q\) relation and exact solution for the XYZ chain with general nondiagonal boundary terms, Nucl. Phys. B, 744, 312-329, 2006 · Zbl 1214.82027 · doi:10.1016/j.nuclphysb.2006.03.025 |
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.
