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Freidel-Maillet type presentations of \(U_q(sl_2)\). (English) Zbl 1485.81043

Summary: A unified framework for the Chevalley and equitable presentation of \(U_q(sl_2)\) is introduced. It is given in terms of a system of Freidel-Maillet type equations satisfied by a pair of quantum K-operators \(\mathcal{K}^\pm\), whose entries are expressed in terms of either Chevalley or equitable generators. The Hopf algebra structure is reconsidered in light of this unified framework, and interwining relations for each pair of \(\mathcal{K}^\pm\) are obtained. A K-operator solving a spectral parameter dependent Freidel-Maillet type equation is also considered. Different specializations of this K-operator are shown to admit a decomposition in terms of \(\mathcal{K}^\pm\) of Chevalley or equitable type. Explicit examples of K-matrices without/with spectral parameter are derived by specializing the K-operators previously obtained.

MSC:

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
16T05 Hopf algebras and their applications
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[1] Alnajjar, H., Leonard pairs associated with the equitable generators of the quantum algebra \(U_q(s l_2)\), Linear Multilinear Algebra, 59, 1127-1142 (2011) · Zbl 1260.17011
[2] Appel, A.; Vlaar, B., Universal k-matrices for quantum Kac-Moody algebras
[3] Babelon, O., Liouville theory on the lattice and universal exchange algebra for Bloch waves, (Kulish, P. P., Quantum Groups. Quantum Groups, Lecture Notes in Mathematics, vol. 1510 (1992), Springer: Springer Berlin, Heidelberg) · Zbl 0744.35036
[4] Baxter, R., Exactly Solvable Models in Statistical Mechanics (1982), Academic Press: Academic Press New York
[5] Baseilhac, P., Deformed Dolan-Grady relations in quantum integrable models, Nucl. Phys. B, 709, 491-521 (2005) · Zbl 1160.81392
[6] Baseilhac, P., The alternating presentation of \(U_q( \hat{g l_2})\) from Freidel-Maillet algebras, Nucl. Phys. B, 967, Article 115400 pp. (2021) · Zbl 07408619
[7] Baseilhac, P., On the second realization for the positive part of \(U_q( \hat{s l_2})\) of equitable type
[8] Bockting-Conrad, S.; Terwilliger, P., The algebra \(U_q(s l_2)\) in disguise, Linear Algebra Appl., 459, 548-585 (2014) · Zbl 1348.17011
[9] Balagovic, M.; Kolb, S., Universal K-matrix for quantum symmetric pairs, J. Reine Angew. Math., 2019, 747 (2016)
[10] Bao, H.; Wang, W., A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs, Astérisque, 402 (2018) · Zbl 1411.17001
[11] Chari, V.; Pressley, A., A Guide to Quantum Groups (1994), Cambridge University Press · Zbl 0839.17009
[12] Cherednik, I. V., Factorizing particles on the half-line and root systems, Teor. Mat. Fiz., 61, 35-44 (1984) · Zbl 0575.22021
[13] Ding, J.; Frenkel, I., Isomorphism of two realizations of quantum affine algebra \(U_q( \hat{s l ( n )})\), Commun. Math. Phys., 156, 277-300 (1993) · Zbl 0786.17008
[14] Drinfeld, V. G., Quantum Groups, Proc. ICM-86 Berkeley, vol. 1, 789-820 (1986), Academic Press: Academic Press New York
[15] Doikou, A.; Evangelisti, S.; Feverati, G.; Karaiskos, N., Introduction to quantum integrability, Int. J. Mod. Phys. A, 25, 3307-3351 (2010) · Zbl 1193.81044
[16] Donin, J.; Kulish, P. P.; Mudrov, A. I., On a universal solution to the reflection equation, Lett. Math. Phys., 63, 179-194 (2003) · Zbl 1042.17011
[17] Faddeev, L. D., How algebraic Bethe ansatz works for integrable model, Les-Houches lectures, 59 pp. · Zbl 0934.35170
[18] Faddeev, L. D.; Reshetikhin, N. Y.; Takhtajan, L. A., Quantization of Lie groups and Lie algebras, Leningr. Math. J., 1, 193 (1990), LOMI preprint, Leningrad, 1987 · Zbl 0715.17015
[19] Faddeev, L. D.; Reshetikhin, N. Y.; Takhtajan, L. A., Quantization of Lie groups and Lie algebras, Algebra Anal., 1, 1, 118-206 (1989), (Russian); Faddeev, L. D.; Reshetikhin, N. Y.; Takhtajan, L. A., Quantization of Lie Groups and Lie Algebras, Yang-Baxter Equation in Integrable Systems, Advanced Series in Mathematical Physics, vol. 10, 299-309 (1989), World Scientific: World Scientific Singapore
[20] Fish, C.; Jordan, D., Connected quantized Weyl algebras and quantum cluster algebras, J. Pure Appl. Algebra, 222, 2374-2412 (2018) · Zbl 1417.16030
[21] Freidel, L.; Maillet, J. M., Quadratic algebras and integrable systems, Phys. Lett. B, 262, 278 (1991)
[22] Frenkel, E.; Mukhin, E., The Hopf algebra \(R e p U_q \hat{g l}_\infty \), Sel. Math., 8, 537-635 (2002) · Zbl 1034.17009
[23] Fioravanti, D.; Rossi, M., A braided Yang-Baxter algebra in a theory of two coupled lattice quantum KdV: algebraic properties and ABA representations, J. Phys. A, 35, 3647-3682 (2002) · Zbl 1042.81039
[24] Funk-Neubauer, D., Bidiagonal pairs, the Lie algebra \(s l_2\), and the quantum group \(U_q(s l_2)\), J. Algebra Appl., 12, Article 1250207 pp. (2013); Funk-Neubauer, D., Bidiagonal triples, Linear Algebra Appl., 521, 104-134 (2017) · Zbl 1280.17016
[25] Genest, V. X.; Vinet, L.; Zhedanov, A., The equitable presentation of \(o s p_q(1 | 2)\) and a Q-analog of the Bannai-Ito algebra, Lett. Math. Phys., 105, 1725-1734 (2015) · Zbl 1344.17014
[26] Hlavaty, L., Generalized algebraic framework for open spin chains, J. Phys. A, Math. Gen., 27, 5645 (1994) · Zbl 0884.58111
[27] Hou, B.; Gao, S., Leonard pairs and Leonard triples of Q-Racah type from the quantum algebra \(U_q(s l_2)\), Commun. Algebra, 41, 3762-3774 (2013) · Zbl 1272.05221
[28] Hlavaty, L.; Kundu, A., Quantum integrability of nonultralocal models through Baxterisation of quantised braided algebra, Int. J. Mod. Phys., 11, 2143 (1996) · Zbl 0985.81551
[29] Heckenberger, I.; Kolb, S., Right coideal subalgebras of the Borel part of a quantized enveloping algebra, Int. Math. Res. Not., 2, 419-451 (2011) · Zbl 1277.17011
[30] Ito, T.; Terwilliger, P.; Weng, C., The quantum algebra \(U_q(s l_2)\) and its equitable presentation, J. Algebra, 298, 284-301 (2006) · Zbl 1090.17004
[31] Ito, T.; Rosengren, H.; Terwilliger, P., Evaluation modules for the Q-tetrahedron algebra, Linear Algebra Appl., 451, 107-168 (2014) · Zbl 1294.17014
[32] Ito, T.; Terwilliger, P., Tridiagonal pairs and the quantum affine algebra \(U_q(L(s l_2))\), Ramanujan J., 13, 39-62 (2007) · Zbl 1128.16028
[33] Jantzen, J. C., Lectures on Quantum Groups, Graduate Studies in Mathematics, vol. 6 (1996), Amer. Math. Soc.: Amer. Math. Soc. Providence RI · Zbl 0842.17012
[34] Jimbo, M., A Q-difference analog of \(U( \hat{g})\) and the Yang-Baxter equation, Lett. Math. Phys., 10, 63-69 (1985); Jimbo, M., A q-analog of \(U(g l(N + 1))\), Hecke algebras, and the Yang-Baxter equation, Lett. Math. Phys., 11, 247-252 (1986) · Zbl 0602.17005
[35] Jing, N.; Liu, M.; Molev, A., Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: type C, J. Math. Phys., 61, Article 031701 pp. (2020) · Zbl 1439.81062
[36] Jing, N.; Liu, M.; Molev, A., Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: types B and D, SIGMA, 16, Article 043 pp. (2020) · Zbl 07209356
[37] Kolb, S., Quantum symmetric pairs and the reflection equation, Algebr. Represent. Theory, 11, 519-544 (2008) · Zbl 1190.17007
[38] Kolb, S., Quantum symmetric Kac-Moody pairs, Adv. Math., 267, 395-469 (2014) · Zbl 1300.17011
[39] Kolb, S., Braided module categories via quantum symmetric pairs, Proc. Lond. Math. Soc., 121, 1, 1-31 (2020) · Zbl 07199116
[40] Kolb, S.; Stokman, J. V., Reflection equation algebras, coideal subalgebras, and their centres, Sel. Math., 15 (2009) · Zbl 1236.17023
[41] Kulish, P. P.; Reshetkhin, N. Yu., Quantum linear problem for the sine-Gordon equation and higher representations, Questions of Quantum Field Theory and Statistical Physics. 2. Questions of Quantum Field Theory and Statistical Physics. 2, J. Sov. Math., 23, 4 (1983)
[42] Kulish, P. P.; Sklyanin, E., Algebraic structures related to reflection equations, J. Phys. A, 25, 5963-5975 (1992) · Zbl 0774.17019
[43] Kundu, A., Exact Bethe ansatz solution of nonultralocal quantum mKdV model, Mod. Phys. Lett. A, 10, 2955 (1995) · Zbl 1022.81744
[44] Letzter, G., Symmetric pairs for quantized enveloping algebras, J. Algebra, 220, 729-767 (1999) · Zbl 0956.17007
[45] Liashyk, A.; Pakuliak, S. Z., On the R-matrix realization of quantum loop algebras
[46] Lentner, S.; Vocke, K., Constructing new Borel subalgebras of quantum groups with a non-degeneracy property
[47] Lentner, S.; Vocke, K., On Borel subalgebras of quantum groups · Zbl 1477.16037
[48] Liu, J.; Xu, Q.; Jiang, W., The equitable presentation for the quantum group \(\nu_q(s l_2)\), J. Math. Phys., 60, Article 091703 pp. (2019) · Zbl 1422.81133
[49] Lukyanenko, I.; Isaac, P.; Links, J., On the boundaries of quantum integrability for the spin-1/2 Richardson-Gaudin system, Nucl. Phys. B, 886, 364-398 (2014) · Zbl 1325.82007
[50] Mazzocco, M.; Rubtsov, V., Confluence on the Painlevé monodromy manifolds, their Poisson structure and quantisation
[51] Molev, A. I.; Ragoucy, E.; Sorba, P., Coideal subalgebras in quantum affine algebras, Rev. Math. Phys., 15, 789-822 (2003) · Zbl 1129.17302
[52] Nijhoff, F.; Capel, H., Integrable quantum mappings and non-ultralocal Yang-Baxter structures, Phys. Lett. A, 163, 49-56 (1992)
[53] Nomura, K., Linear transformations that are tridiagonal with respect to the three decompositions for an LR triple, Linear Algebra Appl., 486, 173-203 (2015); Nomura, K., An LR pair that can be extended to an LR triple, Linear Algebra Appl., 493, 336-357 (2016) · Zbl 1358.15009
[54] Parmentier, S., On coproducts of quasi-triangular Hopf algebras, Algebra Anal., 6, 4, 204-222 (1994) · Zbl 0839.17008
[55] Regelskis, V.; Vlaar, B., Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type
[56] Reshetikhin, N. Yu., Multiparameter quantum groups and twisted quasitriangular Hopf algebras, Lett. Math. Phys., 20, 331-335 (1990) · Zbl 0719.17006
[57] Reshetikhin, N. Yu.; Semenov Tian-Shansky, M., Central extensions of quantum current roups, Lett. Math. Phys., 19, 133-142 (1990) · Zbl 0692.22011
[58] Sang, M.; Gao, S.; Hou, B., Leonard pairs and quantum algebra \(U_q(s l_2)\), Linear Algebra Appl., 510, 346-360 (2016) · Zbl 1403.17021
[59] Sklyanin, E. K., An algebra generated by quadratic relations, Usp. Mat. Nauk, 40, 2, 214 (1985)
[60] Sklyanin, E. K., Boundary conditions for integrable quantum systems, J. Phys. A, 21, 2375-2389 (1988) · Zbl 0685.58058
[61] Slavnov, N., Algebraic Bethe ansatz · Zbl 1483.82003
[62] Terwilliger, P., Two relations that generalize the q-Serre relations and the Dolan-Grady relations, (Kirillov, A. N.; Tsuchiya, A.; Umemura, H., Proceedings of the Nagoya 1999 International workshop on physics and combinatorics (1999)), 377-398 · Zbl 1061.16033
[63] Terwilliger, P., The equitable presentation for the quantum group \(U_q(g)\) associated with a symmetrizable Kac-Moody algebra g, J. Algebra, 298, 302-319 (2006) · Zbl 1106.17021
[64] Terwilliger, P., Finite-dimensional irreducible \(U_q(s l_2)\)-modules from the equitable point of view, Linear Algebra Appl., 439, 358-400 (2013) · Zbl 1354.17012
[65] Terwilliger, P., The universal Askey-Wilson algebra and the equitable presentation of \(U_q(s l_2)\), SIGMA, 7, Article 099 pp. (2011) · Zbl 1244.33016
[66] Terwilliger, P., Billiard arrays and finite-dimensional irreducible \(U_q(s l_2)\)-modules, Linear Algebra Appl., 461, 211-270 (2014) · Zbl 1352.17021
[67] Terwilliger, P., Lowering-raising triples and \(U_q(s l_2)\), Linear Algebra Appl., 486, 1-172 (2015) · Zbl 1358.17021
[68] Terwilliger, P., The Lusztig automorphism of \(U_q(s l_2)\) from the equitable point of view, J. Algebra Appl., 16, Article 1750235 pp. (2017) · Zbl 1386.17019
[69] Terwilliger, P., Using Catalan words and a Q-shuffle algebra to describe a PBW basis for the positive part of \(U_q( \hat{s l_2})\), J. Algebra, 525, 359-373 (2019) · Zbl 1462.17021
[70] Terwilliger, P., Tridiagonal pairs of Q-Racah type and the Q-tetrahedron algebra, J. Pure Appl. Algebra, 225, Article 106632 pp. (2021) · Zbl 1481.17025
[71] Terwilliger, P., The alternating central extension of the Q-Onsager algebra · Zbl 07424944
[72] Vocke, K., On right coideal subalgebras of quantum groups
[73] Yang, Y., Upper triangular matrices and billiard arrays, Linear Algebra Appl., 493, 508-536 (2016); Yang, Y., Some Q-exponential formulas for finite-dimensional \(\square_q\)-modules, Algebr. Represent. Theory, 23, 467-482 (2020) · Zbl 07210914
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