Geodesic
Khoroshkin–Tolstoy approach to quantum superalgebras
Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 1, pp. 121-149 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The central object of the quantum algebraic approach to the study of quantum integrable models is the universal $R$-matrix, which is an element of the completed tensor product of two copies of a quantum algebra. Various integrability objects are constructed by choosing representations for the factors of this tensor product. There are two approaches to constructing explicit expressions for the universal $R$-matrix. One is based on the use of Lusztig automorphisms, and the other is based on the concepts of normal ordering and $q$-commutator. In the case of a quantum superalgebra, we cannot use the first approach because we do not know an explicit expression for the Lusztig automorphisms. The second approach can be used, although it requires some modifications. In this article, we present the necessary modification of the method and use it to find the $R$-operator for a quantum integrable system related to the quantum superalgebra $\mathrm{U}_q(\mathcal{L}(\mathfrak{sl}_{M | N}))$.
Keywords: quantum integrable systems, quantum superalgebras. 
@article{TMF_2023_215_1_a6,
     author = {A. V. Razumov},
     title = {Khoroshkin{\textendash}Tolstoy approach to quantum superalgebras},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {121--149},
     year = {2023},
     volume = {215},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_215_1_a6/}
}
TY  - JOUR
AU  - A. V. Razumov
TI  - Khoroshkin–Tolstoy approach to quantum superalgebras
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2023
SP  - 121
EP  - 149
VL  - 215
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2023_215_1_a6/
LA  - ru
ID  - TMF_2023_215_1_a6
ER  - 
%0 Journal Article
%A A. V. Razumov
%T Khoroshkin–Tolstoy approach to quantum superalgebras
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2023
%P 121-149
%V 215
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2023_215_1_a6/
%G ru
%F TMF_2023_215_1_a6
A. V. Razumov. Khoroshkin–Tolstoy approach to quantum superalgebras. Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 1, pp. 121-149. http://geodesic.mathdoc.fr/item/TMF_2023_215_1_a6/

[1] H. Boos, F. Göhmann, A. Klümper, Kh. S. Nirov, A. V. Razumov, “Universal ${R}$-matrix and functional relations”, Rev. Math. Phys., 26:6 (2014), 1430005, 66 pp., arXiv: 1205.1631 | DOI | MR

[2] A. V. Razumov, “$\ell$-Vesa i faktorizatsiya transfer-operatorov”, TMF, 208:2 (2021), 310–341, arXiv: 2103.16200 | DOI | DOI

[3] A. V. Razumov, “Quantum groups and functional relations for arbitrary rank”, Nucl. Phys. B, 971 (2021), 115517, 51 pp., arXiv: 2104.12603 | DOI | MR

[4] V. V. Bazhanov, S. L. Lukyanov, A. B. Zamolodchikov, “Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz”, Commun. Math. Phys., 177 (1996), 381–398, arXiv: hep-th/9412229 | DOI

[5] V. V. Bazhanov, S. L. Lukyanov, A. B. Zamolodchikov, “Integrable structure of conformal field theory II. Q-operator and DDV equation”, Commun. Math. Phys., 190:2 (1997), 247–278, arXiv: hep-th/9604044 | DOI | MR

[6] V. V. Bazhanov, S. L. Lukyanov, A. B. Zamolodchikov, “Integrable structure of conformal field theory III. The Yang–Baxter relation”, Commun. Math. Phys., 200:2 (1999), 297–324, arXiv: hep-th/9805008 | DOI | MR

[7] V. G. Drinfeld, “Algebry Khopfa i kvantovoe uravnenie Yanga–Bakstera”, Dokl. AN SSSR, 283:5 (1985), 1060–1064 | MR | Zbl

[8] M. Jimbo, “A$q$-difference analogue of $\mathrm U(\mathfrak g)$ and the Yang–Baxter equation”, Lett. Math. Phys., 10:1 (1985), 63–69 | DOI | MR

[9] V. G. Drinfeld, “Quantum groups”, Proceedings of the International Congress of Mathematicians (Berkeley, CA, August 3–11, 1986), eds. A. E. Gleason, AMS, Providence, RI, 1987, 798–820 | MR

[10] M. Rosso, “An analogue of P.B.W. theorem and the universal $R$-matrix for $U_h \mathrm{sl}(N+1)$”, Commun. Math. Phys., 124:2 (1989), 307–318 | DOI | MR

[11] A. N. Kirillov, N. Reshetikhin, “$q$-Weyl group and a multiplicative formula for universal $R$-matrices”, Commun. Math. Phys., 134:2 (1990), 421–431 | DOI | MR

[12] S. Z. Levendorskii, Ya. S. Soibelman, “Kvantovaya gruppa Veilya i multiplikativnaya formula dlya $R$-matritsy prostoi algebry Li”, Funkts. analiz i ego pril., 25:2 (1991), 73–76 | DOI | MR | Zbl

[13] S. Levendorskii, Ya. Soibelman, V. Stukopin, “The quantum Weyl group and the universal quantum $R$-matrix for affine Lie algebra $A_1^{(1)}$”, Lett. Math. Phys., 27:4 (1993), 253–264 | DOI | MR

[14] I. Damiani, “La $R$-matrice pour les algèbres quantiques de type affine non tordu”, Ann. Sci. École Norm. Sup., 31:4 (1998), 493–523 | DOI | MR

[15] I. Damiani, “The $R$-matrix for (twisted) affine quantum algebras”, Representations and Quantizations (Shanghai, 1998), China Higher Education Press, Beijing, 2000, 89–144, arXiv: 1111.4085

[16] V. N. Tolstoi, S. M. Khoroshkin, “Universalnaya $R$-matritsa dlya kvantovannykh neskruchennykh affinnykh algebr Li”, Funkts. analiz i ego pril., 26:1 (1992), 85–88 | DOI | MR | Zbl

[17] G. Lusztig, Introduction to Quantum Groups, Progress in Mathematics, 110, Birkhäuser, Boston, MA, 1993 | MR

[18] S. M. Khoroshkin, V. N. Tolstoy, “On Drinfeld's realization of quantum affine algebras”, J. Geom. Phys., 11:1–4 (1993), 445–452 | DOI | MR

[19] H. Yamane, “Quantized enveloping algebras associated with simple Lie superalgebras and their universal $R$-matrices”, Publ. Res. Inst. Math. Sci., 30:1 (1994), 15–87 | DOI | MR

[20] H. Yamane, “On defining relations of affine Lie superalgebras and affine quantized universal enveloping algebras”, Publ. Res. Inst. Math. Sci., 35:3 (1999), 321–390, arXiv: q-alg/9603015 | DOI | MR

[21] M. D. Gould, R. B. Zhang, A. J. Bracken, “Quantum double construction for graded Hopf algebras”, Bull. Austral. Math. Soc., 47:3 (1993), 353–375 | DOI | MR

[22] T. S. Hakobyan, A. G. Sedrakyan, “Universal $R$ matrix of $\mathrm U_q \mathrm {sl}(n, m)$ quantum superalgebras”, J. Math. Phys., 35:5 (1994), 2552–2559 | DOI | MR

[23] S. M. Khoroshkin, V. N. Tolstoy, “Universal $R$-matrix for quantized (super)algebras”, Commun. Math. Phys., 141:3 (1991), 599–617 | DOI | MR

[24] S. M. Khoroshkin, V. N. Tolstoy, “The Cartan–Weyl basis and the universal $R$-matrix for quantum Kac–Moody algebras and superalgebras”, Quantum Symmetries, eds. H.-D. Doebner, V. K. Dobrev, World Sci., Singapore, 1993, 336–351 | MR

[25] S. M. Khoroshkin, V. N. Tolstoy, Twisting of quantum (super)algebras. Connection of Drinfeld's and Cartan–Weyl realizations for quantum affine algebras, arXiv: hep-th/9404036

[26] S. M. Khoroshkin, V. N. Tolstoy, “Twisting of quantum (super-) algebras”, Generalized Symmetries in Physics (Clausthal, 27–29 July, 1993), eds. H.-D. Doebner, V. K. Dobrev, A. G. Ushveridze, World Sci., Singapore, 1994, 42–54 | MR

[27] S. M. Khoroshkin, V. N. Tolstoy, “The uniqueness theorem for the universal $R$-matrix”, Lett. Math. Phys., 24:3 (1992), 231–244 | DOI | MR

[28] Y.-Z. Zhang, M. D. Gould, “Quantum affine algebras and universal $R$-matrix with spectral parameter”, Lett. Math. Phys., 31:2 (1994), 101–110, arXiv: hep-th/9307007 | DOI | MR

[29] A. J. Bracken, M. D. Gould, Y.-Z. Zhang, G. W. Delius, “Infinite families of gauge-equivalent $R$-matrices and gradations of quantized affine algebras”, Internat. J. Modern Phys. B, 8:25–26 (1994), 3679–3691, arXiv: hep-th/9310183 | DOI | MR

[30] A. J. Bracken, M. D. Gould, Y.-Z. Zhang, “Quantised affine algebras and parameter-dependent $R$-matrices”, Bull. Austral. Math. Soc., 51:2 (1995), 177–194 | DOI | MR

[31] H. Boos, F. Göhmann, A. Klümper, Kh. S. Nirov, A. V. Razumov, “Exercises with the universal $R$-matrix”, J. Phys. A: Math. Theor., 43:41 (2010), 415208, 35 pp., arXiv: 1004.5342 | DOI | MR

[32] H. Boos, F. Göhmann, A. Klümper, Kh. S. Nirov, A. V. Razumov, “On the universal $R$-matrix for the Izergin–Korepin model”, J. Phys. A: Math. Theor., 44:35 (2011), 355202, 25 pp., arXiv: 1104.5696 | DOI | MR

[33] V. V. Bazhanov, Z. Tsuboi, “Baxter's $\mathbf{Q}$-operators for supersymmetric spin chains”, Nucl. Phys. B, 805:3 (2008), 451–516, arXiv: 0805.4274 | DOI | MR

[34] G. Boos, F. Geman, A. Klyumper, Kh. S. Nirov, A. V. Razumov, “Universalnye ob'ekty integriruemosti”, TMF, 174:1 (2013), 25–45, arXiv: 1205.4399 | DOI | DOI | MR | Zbl

[35] A. V. Razumov, “Monodromy operators for higher rank”, J. Phys. A: Math. Theor., 46:38 (2013), 385201, 24 pp., arXiv: 1211.3590 | DOI | MR

[36] V. V. Bazhanov, A. N. Hibberd, S. M. Khoroshkin, “Integrable structure of $\mathcal W_3$ conformal field theory, quantum Boussinesq theory and boundary affine Toda theory”, Nucl. Phys. B, 622:3 (2002), 475–574, arXiv: hep-th/0105177 | DOI | MR

[37] T. Kojima, “Baxter's $Q$-operator for the $W$-algebra $W_N$”, J. Phys. A: Math. Theor., 41:35 (2008), 355206, 16 pp., arXiv: 0803.3505 | DOI | MR

[38] H. Boos, F. Göhmann, A. Klümper, Kh. S. Nirov, A. V. Razumov, “Quantum groups and functional relations for higher rank”, J. Phys. A: Math. Theor., 47:27 (2014), 275201, 47 pp., arXiv: 1312.2484 | DOI | MR

[39] Kh. S. Nirov, A. V. Razumov, “Quantum groups and functional relations for lower rank”, J. Geom. Phys., 112 (2017), 1–28, arXiv: 1412.7342 | DOI

[40] H. Boos, M. Jimbo, T. Miwa, F. Smirnov, Y. Takeyama, “Hidden Grassmann structure in the XXZ model”, Commun. Math. Phys., 272:1 (2007), 263–281, arXiv: hep-th/0606280 | DOI | MR

[41] H. Boos, M. Jimbo, T. Miwa, F. Smirnov, Y. Takeyama, “Hidden Grassmann structure in the XXZ model II: Creation operators”, Commun. Math. Phys., 286:3 (2009), 875–932, arXiv: 0801.1176 | DOI | MR

[42] H. Boos, M. Jimbo, T. Miwa, F. Smirnov, “Hidden Grassmann structure in the XXZ model IV: CFT limit”, Commun. Math. Phys., 299:3 (2010), 825–866, arXiv: 0911.3731 | DOI | MR

[43] A. Klümper, Kh. S. Nirov, A. V. Razumov, “Reduced qKZ equation: general case”, J. Phys. A: Math. Theor., 53:1 (2020), 015202, 35 pp., arXiv: 1905.06014 | DOI | MR | Zbl

[44] A. V. Razumov, “Reduced qKZ equation and genuine qKZ equation”, J. Phys. A: Math. Theor., 53:40 (2020), 405204, 32 pp., arXiv: 2004.02624 | DOI | MR

[45] I. C.-H. Ip, A. M. Zeitlin, “$Q$-operator and fusion relations for $U_q(C^{(2)}(2))$”, Lett. Math. Phys., 104:8 (2014), 1019–1043, arXiv: 1312.4063 | DOI | MR

[46] J. H. H. Perk, C. L. Schultz, “New families of commuting transfer matrices in $q$-state vertex models”, Phys. Lett. A, 84:8 (1981), 407–410 | DOI | MR

[47] V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, Cambridge, 1994 | MR

[48] V. G. Kac, “Lie superalgebras”, Adv. Math., 26:1 (1977), 8–96 | DOI | MR

[49] M. Jimbo, “A $q$-analogue of $\mathrm U(\mathfrak{gl}(N + 1))$, Hecke algebra, the Yang–Baxter equation”, Lett. Math. Phys., 11:3 (1986), 247–252 | DOI | MR

[50] J. van de Leur, Contragredient Lie superalgebras of finite growth, PhD thesis, Utrecht University, Utrecht, 1986

[51] J. W. van de Leur, “A classification of contragredient Lie superalgebra of finite growth”, Commun. Algebra, 17:8 (1989), 1815–1841 | DOI | MR

[52] C. Meneghelli, J. Teschner, Integrable light-cone lattice discretizations from the universal $R$-matrix, arXiv: 1504.04572

[53] Kh. S. Nirov, A. V. Razumov, “Vertex models and spin chains in formulas and pictures”, SIGMA, 15 (2019), 068, 67 pp., arXiv: 1811.09401 [math-ph] | DOI | MR

[54] H. Zhang, “Representations of quantum affine superalgebras”, Math. Z., 278:3–4 (2014), 663–703, arXiv: 1309.5250 | DOI | MR

[55] T. Tanisaki, “Killing forms, Harish–Chandra homomorphisms and universal $R$-matrices for quantum algebras”, Infinite Analysis (Kyoto University, Kyoto, June 1 –August 31, 1991), Advanced Series in Mathematical Physics, 16, eds. A. Tsuchiya, T. Eguchi, M. Jimbo, World Sci., Singapore, 1992, 941–962 | DOI | MR

[56] R. A. Usmani, “Inversion of a tridiagonal Jacobi matrix”, Linear Algebra Appl., 212/213 (1994), 413–414 | DOI | MR