Representations of quantum affine algebras in their \(R\)-matrix realization. (English) Zbl 1500.17013
In this paper the authors study finite-dimensional irreducible representations, in the R-matrix realization, of the quantum affine algebras in types B, C and D using the isomorphisms between the R-matrix and Drinfeld presentations. In the case of Yangians of types B, C and D, the authors use the Gauss decomposition to establish an equivalence of the descriptions of the representations in the R-matrix and Drinfeld presentations.
Reviewer: Nenad Manojlović (Faro)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
Keywords:
\(R\)-matrix presentation; Drinfeld polynomials; highest weight representation; Gauss decompositionReferences:
| [1] | Arnaudon, Daniel and Molev, Alexander and Ragoucy, Eric, On the {\(R\)}-matrix realization of {Y}angians and their representations, Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 7, 7-8, 1269-1325, (2006) · Zbl 1227.17008 · doi:10.1007/s00023-006-0281-9 |
| [2] | Bazhanov, V. V., Trigonometric solutions of triangle equations and classical {L}ie algebras, Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, 159, 4-6, 321-324, (1985) · doi:10.1016/0370-2693(85)90259-X |
| [3] | Bazhanov, V. V., Integrable quantum systems and classical {L}ie algebras, Communications in Mathematical Physics, 113, 3, 471-503, (1987) · Zbl 0629.58036 · doi:10.1007/BF01221256 |
| [4] | Beck, Jonathan, Braid group action and quantum affine algebras, Communications in Mathematical Physics, 165, 3, 555-568, (1994) · Zbl 0807.17013 · doi:10.1007/BF02099423 |
| [5] | Brundan, Jonathan and Kleshchev, Alexander, Parabolic presentations of the {Y}angian {\(Y({\mathfrak{gl}}_n)\)}, Communications in Mathematical Physics, 254, 1, 191-220, (2005) · Zbl 1128.17012 · doi:10.1007/s00220-004-1249-6 |
| [6] | Chari, Vyjayanthi and Pressley, Andrew, A guide to quantum groups, xvi+651, (1994), Cambridge University Press, Cambridge · Zbl 0839.17009 |
| [7] | Chari, Vyjayanthi and Pressley, Andrew, Quantum affine algebras and their representations, Representations of Groups ({B}anff, {AB}, 1994), CMS Conf. Proc., 16, 59-78, (1995), Amer. Math. Soc., Providence, RI · Zbl 0855.17011 · doi:10.1007/bf00750760 |
| [8] | Ding, Jin Tai and Frenkel, Igor B., Isomorphism of two realizations of quantum affine algebra {\(U_q(\mathfrak{gl}(n))\)}, Communications in Mathematical Physics, 156, 2, 277-300, (1993) · Zbl 0786.17008 · doi:10.1007/BF02098484 |
| [9] | Drinfeld, V. G., Hopf algebras and the quantum {Y}ang–{B}axter equation, 32, 1, 254-258, (1985) · Zbl 0588.17015 |
| [10] | Drinfeld, V. G., Quantum groups, Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ols. 1, 2 ({B}erkeley, {C}alif., 1986), 798-820, (1987), Amer. Math. Soc., Providence, RI · Zbl 0667.16003 |
| [11] | Drinfeld, V. G., A new realization of {Y}angians and of quantum affine algebras, 36, 1, 212-216, (1988) · Zbl 0667.16004 |
| [12] | Frenkel, I. B. and Reshetikhin, N. Yu., Quantum affine algebras and holonomic difference equations, Communications in Mathematical Physics, 146, 1, 1-60, (1992) · Zbl 0760.17006 · doi:10.1007/BF02099206 |
| [13] | Gel’fand, I. M. and Retakh, V. S., A theory of noncommutative determinants and characteristic functions of graphs, Functional Analysis and Its Applications, 26, 4, 231-246, (1992) · Zbl 0799.15003 · doi:10.1007/BF01075044 |
| [14] | Gow, Lucy and Molev, Alexander, Representations of twisted {\(q}-{Y\)}angians, Selecta Mathematica. New Series, 16, 3, 439-499, (2010) · Zbl 1206.81060 · doi:10.1007/s00029-010-0030-2 |
| [15] | Guay, Nicolas and Regelskis, Vidas and Wendlandt, Curtis, Equivalences between three presentations of orthogonal and symplectic {Y}angians, Letters in Mathematical Physics, 109, 2, 327-379, (2019) · Zbl 1472.17051 · doi:10.1007/s11005-018-1108-6 |
| [16] | Hernandez, David, Representations of quantum affinizations and fusion product, Transformation Groups, 10, 2, 163-200, (2005) · Zbl 1102.17009 · doi:10.1007/s00031-005-1005-9 |
| [17] | Jimbo, Michio, Quantum {\(R\)} matrix for the generalized {T}oda system, Communications in Mathematical Physics, 102, 4, 537-547, (1986) · Zbl 0604.58013 · doi:10.1007/BF01221646 |
| [18] | Jimbo, Michio, A {\(q\)}-difference analogue of {\(U({\mathfrak g})\)} and the {Y}ang–{B}axter equation, Letters in Mathematical Physics. A Journal for the Rapid Dissemination of Short Contributions in the Field of Mathematical Physics, 10, 1, 63-69, (1985) · Zbl 0587.17004 · doi:10.1007/BF00704588 |
| [19] | Jing, Naihuan and Liu, Ming and Molev, Alexander, Isomorphism between the {\(R\)}-matrix and {D}rinfeld presentations of {Y}angian in types {\(B\)}, {\(C\)} and {\(D\)}, Communications in Mathematical Physics, 361, 3, 827-872, (2018) · Zbl 1472.17053 · doi:10.1007/s00220-018-3185-x |
| [20] | Jing, Naihuan and Liu, Ming and Molev, Alexander, Isomorphism between the {\(R\)}-matrix and {D}rinfeld presentations of quantum affine algebra: type {\(C\)}, Journal of Mathematical Physics, 61, 3, 031701, 41 pages, (2020) · Zbl 1439.81062 · doi:10.1063/1.5133854 |
| [21] | Jing, Naihuan and Liu, Ming and Molev, Alexander, Isomorphism between the {\(R\)}-matrix and {D}rinfeld presentations of quantum affine algebra: types {\(B\)} and {\(D\)}, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 16, 043, 49 pages, (2020) · Zbl 1495.17024 · doi:10.3842/SIGMA.2020.043 |
| [22] | Khoroshkin, S. and Pakuliak, S. and Tarasov, V., Off-shell {B}ethe vectors and {D}rinfeld currents, Journal of Geometry and Physics, 57, 8, 1713-1732, (2007) · Zbl 1148.17010 · doi:10.1016/j.geomphys.2007.02.005 |
| [23] | Kulish, P. P. and Sklyanin, E. K., Quantum spectral transform method. {R}ecent developments, Integrable Quantum Field Theories ({T}v\"arminne, 1981), Lecture Notes in Phys., 151, 61-119, (1982), Springer, Berlin – New York · Zbl 0734.35071 · doi:10.1007/3-540-11190-5_8 |
| [24] | Molev, Alexander, Yangians and classical {L}ie algebras, Mathematical Surveys and Monographs, 143, xviii+400, (2007), Amer. Math. Soc., Providence, RI · Zbl 1141.17001 · doi:10.1090/surv/143 |
| [25] | Reshetikhin, N. Yu. and Semenov-Tian-Shansky, M. A., Central extensions of quantum current groups, Letters in Mathematical Physics, 19, 2, 133-142, (1990) · Zbl 0692.22011 · doi:10.1007/BF01045884 |
| [26] | Reshetikhin, N. Yu. and Takhtadzhyan, L. A. and Faddeev, L. D., Quantization of {L}ie groups and {L}ie algebras, 1, 1, 193-225, (1990) · Zbl 0715.17015 |
| [27] | Tarasov, V. O., Structure of quantum \(L\)-operators for the \(R\)-matrix of the \(XXZ\)-model, Theoretical and Mathematical Physics, 61, 2, 1065-1072, (1984) · doi:10.1007/BF01029107 |
| [28] | Tarasov, V. O., Irreducible monodromy matrices for the {\(R\)}-matrix of the {\(XXZ\)}-model and local lattice quantum {H}amiltonians, Theoretical and Mathematical Physics, 63, 2, 440-454, (1985) · doi:10.1007/BF01017900 |
| [29] | Wendlandt, Curtis, The {\(R\)}-matrix presentation for the {Y}angian of a simple {L}ie algebra, Communications in Mathematical Physics, 363, 1, 289-332, (2018) · Zbl 1439.17016 · doi:10.1007/s00220-018-3227-4 |
| [30] | Zamolodchikov, Alexander B. and Zamolodchikov, Alexey B., Factorized {\(S\)}-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models, Annals of Physics, 120, 2, 253-291, (1979) · doi:10.1016/0003-4916(79)90391-9 |
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