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The alternating central extension of the \(q\)-Onsager algebra. (English) Zbl 1487.17047

Summary: The \(q\)-Onsager algebra \(O_q\) is presented by two generators \(W_0, W_1\) and two relations, called the \(q\)-Dolan/Grady relations. Recently Baseilhac and Koizumi introduced a current algebra \({\mathcal{A}}_q\) for \(O_q\). Soon afterwards, Baseilhac and Shigechi gave a presentation of \({\mathcal{A}}_q\) by generators and relations. We show that these generators give a PBW basis for \({\mathcal{A}}_q\). Using this PBW basis, we show that the algebra \({\mathcal{A}}_q\) is isomorphic to \(O_q \otimes{\mathbb{F}} [z_1, z_2, \ldots]\), where \({\mathbb{F}}\) is the ground field and \(\{ z_n \}_{n=1}^\infty\) are mutually commuting indeterminates. Recall the positive part \(U^+_q\) of the quantized enveloping algebra \(U_q(\widehat{\mathfrak{sl}}_2)\). Our results show that \(O_q\) is related to \({\mathcal{A}}_q\) in the same way that \(U^+_q\) is related to the alternating central extension of \(U^+_q\). For this reason, we propose to call \({\mathcal{A}}_q\) the alternating central extension of \(O_q\).

MSC:

17B65 Infinite-dimensional Lie (super)algebras
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)

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References:

[1] Baseilhac, P., An integrable structure related with tridiagonal algebras, Nucl. Phys. B, 705, 605-619 (2005) · Zbl 1119.81330 · doi:10.1016/j.nuclphysb.2004.11.014
[2] Baseilhac, P., Deformed Dolan-Grady relations in quantum integrable models, Nucl. Phys. B, 709, 491-521 (2005) · Zbl 1160.81392 · doi:10.1016/j.nuclphysb.2004.12.016
[3] Baseilhac, P., The alternating presentation of \(U_q(\widehat{{\mathfrak{g}}{\mathfrak{l}}}_2)\) from Freidel-Maillet algebras, Nucl. Phys. B, 967, 115400 (2021) · Zbl 1490.81095 · doi:10.1016/j.nuclphysb.2021.115400
[4] Baseilhac, P.; Belliard, S., Generalized \(q\)-Onsager algebras and boundary affine Toda field theories, Lett. Math. Phys., 93, 213-228 (2010) · Zbl 1197.81147 · doi:10.1007/s11005-010-0412-6
[5] Baseilhac, P.; Belliard, S., The half-infinite XXZ chain in Onsager’s approach, Nucl. Phys. B, 873, 550-584 (2013) · Zbl 1282.82010 · doi:10.1016/j.nuclphysb.2013.05.003
[6] Baseilhac, P.; Belliard, S., Non-Abelian symmetries of the half-infinite XXZ spin chain, Nucl. Phys. B, 916, 373-385 (2017) · Zbl 1356.82007 · doi:10.1016/j.nuclphysb.2017.01.012
[7] Baseilhac, P., Belliard, S.: An attractive basis for the \(q\)-Onsager algebra. Preprint arXiv:1704.02950 · Zbl 1197.81147
[8] Baseilhac, P.; Koizumi, K., A new (in)finite dimensional algebra for quantum integrable models, Nucl. Phys. B, 720, 325-347 (2005) · Zbl 1194.81122 · doi:10.1016/j.nuclphysb.2005.05.021
[9] Baseilhac, P., Koizumi, K.: A deformed analogue of Onsager’s symmetry in the \(XXZ\) open spin chain. J. Stat. Mech. Theory Exp. 2005, no. 10, P10005 (electronic). arXiv:hep-th/0507053 · Zbl 1456.82223
[10] Baseilhac, P., Koizumi, K.: Exact spectrum of the \(XXZ\) open spin chain from the \(q\)-Onsager algebra representation theory. J. Stat. Mech. Theory Exp. 2007, no. 9, P09006. (electronic). arXiv:hep-th/0703106 · Zbl 1456.82029
[11] Baseilhac, P., Kojima, T.: Correlation functions of the half-infinite XXZ spin chain with a triangular boundary. J. Stat. Mech. (2014) P09004. arXiv:1309.7785 · Zbl 1284.82015
[12] Baseilhac, P.; Kojima, T., Form factors of the half-infinite XXZ spin chain with a triangular boundary, Nucl. Phys. B, 880, 378-413 (2014) · Zbl 1284.82015 · doi:10.1016/j.nuclphysb.2014.01.011
[13] Baseilhac, P.; Kolb, S., Braid group action and root vectors for the \(q\)-Onsager algebra, Transform. Groups, 25, 363-389 (2020) · Zbl 1439.81058 · doi:10.1007/s00031-020-09555-7
[14] Baseilhac, P.; Shigechi, K., A new current algebra and the reflection equation, Lett. Math. Phys., 92, 47-65 (2010) · Zbl 1187.81166 · doi:10.1007/s11005-010-0380-x
[15] Bergman, G., The diamond lemma for ring theory, Adv. Math., 29, 178-218 (1978) · Zbl 0326.16019 · doi:10.1016/0001-8708(78)90010-5
[16] Brualdi, RA, Introductory Combinatorics, 5E (2010), Upper Saddle River: Pearson Prentice Hall, Upper Saddle River
[17] Carter, R., Lie Algebras of Finite and Affine Type. Cambridge Studies in Advanced Mathematics (2005), Cambridge: Cambridge University Press, Cambridge · Zbl 1110.17001 · doi:10.1017/CBO9780511614910
[18] Damiani, I., A basis of type Poincare-Birkoff-Witt for the quantum algebra of \(\widehat{{\mathfrak{s}}{\mathfrak{l}}}_2\), J. Algebra, 161, 291-310 (1993) · Zbl 0803.17003 · doi:10.1006/jabr.1993.1220
[19] Ito, T., TD-pairs and the \(q\)-Onsager algebra, Sugaku Expo., 32, 205-232 (2019) · Zbl 1475.17008 · doi:10.1090/suga/444
[20] Ito, T., Tanabe, K., Terwilliger, P.: Some algebra related to \({P}\)- and \({Q}\)-polynomial association schemes. In: Codes and Association Schemes (Piscataway NJ, 1999), Amer. Math. Soc., Providence, pp. 167-192 (2001). arXiv:math.CO/0406556 · Zbl 0995.05148
[21] Ito, T.; Terwilliger, P., The shape of a tridiagonal pair, J. Pure Appl. Algebra, 188, 145-160 (2004) · Zbl 1037.17013 · doi:10.1016/j.jpaa.2003.10.002
[22] Ito, T.; Terwilliger, P., Tridiagonal pairs of \(q\)-Racah type, J. Algebra, 322, 68-93 (2009) · Zbl 1177.33021 · doi:10.1016/j.jalgebra.2009.04.008
[23] Ito, T.; Terwilliger, P., The augmented tridiagonal algebra, Kyushu J. Math., 64, 81-144 (2010) · Zbl 1236.17022 · doi:10.2206/kyushujm.64.81
[24] Kolb, S., Quantum symmetric Kac-Moody pairs, Adv. Math., 267, 395-469 (2014) · Zbl 1300.17011 · doi:10.1016/j.aim.2014.08.010
[25] Lu, M., Wang, W.: A Drinfeld type presentation of affine \(\iota\) quantum groups I: split ADE type. Preprint arXiv:2009.04542
[26] Lu, M., Ruan, S., Wang, W.: \( \iota\) Hall algebra of the projective line and \(q\)-Onsager algebra. Preprint arXiv:2010.00646
[27] The Sage Developers. Sage Mathematics Software (Version 9.2). The Sage Development Team (2020). http://www.sagemath.org
[28] Terwilliger, P., The subconstituent algebra of an association scheme III, J. Algebraic Combin., 2, 177-210 (1993) · Zbl 0785.05091 · doi:10.1023/A:1022415825656
[29] Terwilliger, P.: Two relations that generalize the \(q\)-Serre relations and the Dolan-Grady relations. In: Physics and Combinatorics 1999 (Nagoya), pp. 377-398. World Scientific Publishing, River Edge (2001). arXiv:math.QA/0307016 · Zbl 1061.16033
[30] Terwilliger, P.: The universal Askey-Wilson algebra. SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011) Paper 069. arXiv:1104.2813 · Zbl 1244.33015
[31] Terwilliger, P., The Lusztig automorphism of the \(q\)-Onsager algebra, J. Algebra, 506, 56-75 (2018) · Zbl 1464.17022 · doi:10.1016/j.jalgebra.2018.03.026
[32] Terwilliger, P., The \(q\)-Onsager algebra and the positive part of \(U_q(\widehat{{{s}}{\mathfrak{l}}}_2)\), Linear Algebra Appl., 521, 19-56 (2017) · Zbl 1386.17018 · doi:10.1016/j.laa.2017.01.027
[33] Terwilliger, P.: The \(q\)-Onsager algebra and the universal Askey-Wilson algebra. SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018) Paper No. 044. arXiv:1801.06083 · Zbl 1390.33038
[34] Terwilliger, P., An action of the free product \({\mathbb{Z}}_2 \star{\mathbb{Z}}_2 \star{\mathbb{Z}}_2\) on the \(q\)-Onsager algebra and its current algebra, Nucl. Phys. B, 936, 306-319 (2018) · Zbl 1400.81123 · doi:10.1016/j.nuclphysb.2018.09.020
[35] Terwilliger, P., The alternating PBW basis for the positive part of \(U_q(\widehat{{\mathfrak{s}}{\mathfrak{l}}}_2)\), J. Math. Phys., 60, 071704 (2019) · Zbl 1421.82012 · doi:10.1063/1.5091801
[36] Terwilliger, P., The alternating central extension for the positive part of \(U_q(\widehat{{\mathfrak{s}}{\mathfrak{l}}}_2)\), Nucl. Phys. B, 947, 114729 (2019) · Zbl 1440.81053 · doi:10.1016/j.nuclphysb.2019.114729
[37] Terwilliger, P., A conjecture concerning the \(q\)-Onsager algebra, Nucl. Phys. B, 966, 115391 (2021) · Zbl 1487.17036 · doi:10.1016/j.nuclphysb.2021.115391
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