Yangians and Baxter’s relations. (English) Zbl 1476.17011
Let \(\mathfrak{g}\) be a finite-dimensional complex simple Lie algebra, and \(\hbar\) be a nonzero complex number. The Yangian \(Y_{\hbar}(\mathfrak{g})\) is a deformation of the universal enveloping algebra of the current \(\mathfrak{g}\otimes_\mathbb{C} \mathbb{C}[t]\). The category \(\mathcal{O}\) of representations of \(Y_{\hbar}(\mathfrak{g})\) is a full subcategory of \(Y_{\hbar}(\mathfrak{g})\)-modules whose objects, viewed as \(\mathfrak{g}\)-modules, belong to the BGG category without integrability assumption.
The author introduces the asymptotic modules in \(\mathcal{O}\) which can be regarded as analogs of Verma modules for Lie algebras, then he derives the three-term Baxter’s TQ relations for these asymptotic modules.
The three-term identities for quantum affine algebras \(U_q(\widehat{\mathfrak{g}})\) with generic \(q\) are also provided in the appendix.
The author introduces the asymptotic modules in \(\mathcal{O}\) which can be regarded as analogs of Verma modules for Lie algebras, then he derives the three-term Baxter’s TQ relations for these asymptotic modules.
The three-term identities for quantum affine algebras \(U_q(\widehat{\mathfrak{g}})\) with generic \(q\) are also provided in the appendix.
Reviewer: Li Luo (Shanghai)
MSC:
| 17B38 | Yang-Baxter equations and Rota-Baxter operators |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
References:
| [1] | Baxter, RJ, Partition function of the eight-vertex lattice model, Ann. Phys., 70, 193-228 (1972) · Zbl 0236.60070 · doi:10.1016/0003-4916(72)90335-1 |
| [2] | Bazhanov, V.; Lukowski, T.; Meneghelli, C.; Staudacher, M., A shortcut to the Q-operator, J. Stat. Mech., 1011, P11002 (2010) · doi:10.1088/1742-5468/2010/11/P11002 |
| [3] | Bazhanov, V.; Frassek, R.; Lukowski, T.; Meneghelli, C.; Staudacher, M., Baxter Q-operators and representations of Yangians, Nucl. Phys. B, 850, 1, 148-174 (2011) · Zbl 1215.81052 · doi:10.1016/j.nuclphysb.2011.04.006 |
| [4] | Bazhanov, VV; Lukyanov, SL; Zamolodchikov, AB, 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 |
| [5] | Bazhanov, VV; Lukyanov, SL; Zamolodchikov, AB, Integrable structure of conformal field theory. III. The Yang-Baxter relation, Commun. Math. Phys., 200, 297-324 (1999) · Zbl 1057.81531 · doi:10.1007/s002200050531 |
| [6] | Braverman, A., Finkelberg, M., Nakajima, H.: Coulomb branches of \(3d\,{\cal{N}} = 4\) quiver gauge theory and slices in the affine Grassmannian, Adv. Theor. Math. Phys. 23: 75-166 (2019), with appendices by A. Braverman, M. Finkelberg, J. Kramnitzer, R. Kodera, H. Nakajima, B. Webster, A. Weekes. arXiv:1604.03625 · Zbl 1479.81044 |
| [7] | Chari, V.; Pressley, A., Yangians and R-matrices, l’Enseign. Math., 36, 267-302 (1990) · Zbl 0726.17013 |
| [8] | Chari, V., Braid group actions and tensor products, Int. Math. Res. Not., 2002, 7, 357-382 (2002) · Zbl 0990.17009 · doi:10.1155/S107379280210612X |
| [9] | Costello, K.; Witten, E.; Yamazaki, M., Gauge theory and integrability, II, ICCM Not., 6, 120-146 (2018) · Zbl 1405.81044 |
| [10] | Drinfeld, V., A new realization of Yangians and quantum affine algebras, Sov. Math. Dokl., 36, 2, 212-216 (1988) · Zbl 0667.16004 |
| [11] | Feigin, B.; Jimbo, M.; Miwa, T.; Mukhin, E., Finite type modules and Bethe ansatz equations, Ann. Henri Poincaré, 18, 2543-2579 (2017) · Zbl 1407.82019 · doi:10.1007/s00023-017-0577-y |
| [12] | Feigin, B.; Jimbo, M.; Miwa, T.; Mukhin, E., Finite type modules and Bethe ansatz for quantum toroidal \(\mathfrak{gl}_1\), Commun. Math. Phys., 356, 1, 285-327 (2017) · Zbl 1425.17020 |
| [13] | Felder, G., Elliptic Quantum Groups, XIth International Congress of Mathematical Physics, Paris 1994, 211-218 (1995), Cambridge: Int. Press, Cambridge · Zbl 0998.17015 |
| [14] | Felder, G.; Zhang, H., Baxter operators and asymptotic representations, Selecta Math. (N.S.), 23, 4, 2947-2975 (2017) · Zbl 1407.17017 · doi:10.1007/s00029-017-0320-z |
| [15] | Finkelberg, M.; Kamnitzer, J.; Pham, K.; Rybnikov, L.; Weekes, A., Comultiplication for shifted Yangians and quantum open Toda lattice, Adv. Math., 327, 349-389 (2018) · Zbl 1384.81045 · doi:10.1016/j.aim.2017.06.018 |
| [16] | Fourier, G.; Hernandez, D., Schur positivity and Kirillov-Reshetikhin modules, SIGMA, 10, 058 (2014) · Zbl 1372.17018 |
| [17] | 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 |
| [18] | Frenkel, E.; Hernandez, D., Spectra of quantum KdV Hamiltonians, Langlands duality, and affine opers, Commun. Math. Phys., 362, 2, 361-414 (2018) · Zbl 1397.81060 · doi:10.1007/s00220-018-3194-9 |
| [19] | Frenkel, E.; Mukhin, E., Combinatorics of \(q\)-characters of finite-dimensional representations of quantum affine algebras, Commun. Math. Phys., 216, 23-57 (2001) · Zbl 1051.17013 |
| [20] | Frenkel, E.; Reshetikhin, N., The \(q\)-character of representations of quantum affine algebras and deformations of \({\cal{W}} \)-algebras, recent developments in quantum affine algebras and related topics, Contemp. Math., 248, 163-205 (1999) · Zbl 0973.17015 |
| [21] | Gautam, S.; Toledano Laredo, V., Yangians, quantum loop algebras and abelian difference equations, J. Am. Math. Soc., 29, 775-824 (2016) · Zbl 1336.17012 · doi:10.1090/jams/851 |
| [22] | Gautam, S., Toledano Laredo, V.: Elliptic quantum groups and their finite-dimensional representations, Preprint (2017). arXiv:1707.06469 · Zbl 1432.17012 |
| [23] | Guay, N.; Nakajima, H.; Wendlandt, C., Coproduct for Yangians of affine Kac-Moody algebras, Adv. Math., 338, 865-911 (2018) · Zbl 1447.17012 · doi:10.1016/j.aim.2018.09.013 |
| [24] | Guay, N.; Regelskis, V.; Wendlandt, C., Equivalences between three presentations of orthogonal and symplectic Yangians, Lett. Math. Phys., 109, 327-379 (2019) · Zbl 1472.17051 · doi:10.1007/s11005-018-1108-6 |
| [25] | Hernandez, D., The Kirillov-Reshetikhin conjecture and solutions of T-systems, J. Reine Angrew. Math., 596, 63-87 (2006) · Zbl 1160.17010 |
| [26] | Hernandez, D., Representations of quantum affinizations and fusion product, Transform. Groups, 10, 2, 163-200 (2005) · Zbl 1102.17009 · doi:10.1007/s00031-005-1005-9 |
| [27] | Hernandez, D.; Jimbo, M., Asymptotic representations and Drinfeld rational fractions, Compos. Math., 148, 5, 1593-1623 (2012) · Zbl 1266.17010 · doi:10.1112/S0010437X12000267 |
| [28] | Hernandez, D.; Leclerc, B., Cluster algebras and category O for representations of Borel subalgebras of quantum affine algebras, Algebra Number Theory, 10, 9, 2015-2052 (2016) · Zbl 1390.17025 · doi:10.2140/ant.2016.10.2015 |
| [29] | Jing, N.; Liu, M.; Molev, A., Isomorphism between the \(R\)-matrix and Drinfeld presentations of Yangians in types \(B, C\) and \(D\), Commun. Math. Phys., 361, 827-872 (2018) · Zbl 1472.17053 |
| [30] | Jing, N., Zhang, H.-L.: Hopf algebraic structures of quantum toroidal algebras, Preprint (2016). arXiv:1604.05416 |
| [31] | Kac, V., Infinite dimensional Lie algebras (1990), Cambridge: Cambridge University Press, Cambridge · Zbl 0716.17022 |
| [32] | Kirillov, A., Reshetikhin, N.: Representation of Yangians and multiplicities of the inclusion of the irreducible components of the tensor product of representations of simple Lie algebras. Zap. Nauch. Sem. LOMI 160, 211-221 (1987). translation in J. Sov. Math. 52, 3156-3164 (1990) · Zbl 0900.16047 |
| [33] | Khoroshkin, S.; Tolstoy, V., Yangian double, Lett. Math. Phys., 36, 373-402 (1996) · Zbl 0860.17017 · doi:10.1007/BF00714404 |
| [34] | Knight, H., Spectra of tensor products of finite dimensional representations of Yangians, J. Algebra, 174, 187-196 (1995) · Zbl 0868.17009 · doi:10.1006/jabr.1995.1123 |
| [35] | Mukhin, E.; Young, C., Affinization of category \({\cal{O}}\) for quantum groups, Trans. Am. Math. Soc., 366, 9, 4815-4847 (2014) · Zbl 1306.17005 |
| [36] | Müller-Schrader, M.: Asymptotic representations of Yangians. Master thesis, Swiss Federal Institute of Technology, Zürich (2018) |
| [37] | Nakajima, H., \(t\)-analog of \(q\)-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Theory, 7, 259-274 (2003) · Zbl 1078.17008 |
| [38] | Tan, Y.; Guay, N., Local Weyl modules and cyclicity of tensor products for Yangians, J. Algebra, 432, 228-251 (2015) · Zbl 1365.17016 · doi:10.1016/j.jalgebra.2015.02.023 |
| [39] | Tan, Y.: Braid group actions and tensor products for Yangians, Preprint (2015). arXiv:1510.01533 |
| [40] | Wendlandt, C., The \(R\)-matrix presentation for Yangian of a simple Lie algebra, Commun. Math. Phys., 363, 289-332 (2018) · Zbl 1439.17016 |
| [41] | Yang, Y., Zhao, G.: Quiver varieties and elliptic quantum groups, Preprint (2017). arXiv:1708.01418 |
| [42] | Zhang, H., Asymptotic representations of quantum affine superalgebras, SIGMA, 13, 066 (2017) · Zbl 1420.17017 |
| [43] | Zhang, H., Length-two representations of quantum affine superalgebras and Baxter operators, Commun. Math. Phys., 358, 815-862 (2018) · Zbl 1409.17009 · doi:10.1007/s00220-017-3022-7 |
| [44] | Zhang, H., Elliptic quantum groups and Baxter relations, Algebra Number Theory, 12, 599-647 (2018) · Zbl 1422.17021 · doi:10.2140/ant.2018.12.599 |
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.
