Geodesic
Jordanian deformation of the open XXX spin chain
Teoretičeskaâ i matematičeskaâ fizika, Tome 163 (2010) no. 2, pp. 288-298 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We find the general solution of the reflection equation associated with the Jordanian deformation of the $SL(2)$-invariant Yang $R$-matrix. A special scaling limit of the XXZ model with general boundary conditions leads to the same $K$-matrix. Following the Sklyanin formalism, we derive the Hamiltonian with the boundary terms in explicit form. We also discuss the structure of the spectrum of the deformed XXX model and its dependence on the boundary conditions.
Mots-clés : spin chain
Keywords: boundary condition, quantum group, reflection equation. 
@article{TMF_2010_163_2_a4,
     author = {P. P. Kulish and N. Manoilovich and Z. Nagy},
     title = {Jordanian deformation of the~open {XXX} spin chain},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {288--298},
     year = {2010},
     volume = {163},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2010_163_2_a4/}
}
TY  - JOUR
AU  - P. P. Kulish
AU  - N. Manoilovich
AU  - Z. Nagy
TI  - Jordanian deformation of the open XXX spin chain
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2010
SP  - 288
EP  - 298
VL  - 163
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2010_163_2_a4/
LA  - ru
ID  - TMF_2010_163_2_a4
ER  - 
%0 Journal Article
%A P. P. Kulish
%A N. Manoilovich
%A Z. Nagy
%T Jordanian deformation of the open XXX spin chain
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2010
%P 288-298
%V 163
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2010_163_2_a4/
%G ru
%F TMF_2010_163_2_a4
P. P. Kulish; N. Manoilovich; Z. Nagy. Jordanian deformation of the open XXX spin chain. Teoretičeskaâ i matematičeskaâ fizika, Tome 163 (2010) no. 2, pp. 288-298. http://geodesic.mathdoc.fr/item/TMF_2010_163_2_a4/

[1] L. A. Takhtadzhyan, L. D. Faddeev, UMN, 34:5(209) (1979), 13–63 | DOI | MR

[2] E. K. Sklyanin, L. A. Takhtadzhyan, L. D. Faddeev, TMF, 40:2 (1979), 194–220 | DOI | MR | Zbl

[3] L. D. Faddeev, “How the algebraic Bethe ansatz works for integrable models”, Symétries Quantiques, eds. A. Connes, K. Gawedzki, J. Zinn-Justin, North-Holland, Amsterdam, 1998, 149–219 ; arXiv: hep-th/9605187 | MR | Zbl

[4] P. P. Kulish, E. K. Sklyanin, “Quantum spectral transform method. Recent developments”, Integrable Quantum Field Theories, Lecture Notes in Phys., 151, eds. J. Hietarinta, C. Montonen, Springer, Berlin–New York, 1982, 61–119 | DOI | MR | Zbl

[5] V. G. Drinfeld, “Kvantovye gruppy”, Differentsialnaya geometriya, gruppy Li i mekhanika. 8, Zap. nauchn. sem. LOMI, 155, ed. L. D. Faddeev, Nauka, L., 1986, 18–49 ; V. G. Drinfel'd, “Quantum groups”, Proc. Intern. Congress Math., Vol. 1, 2 (Berkeley, Calif., 1986), AMS, Providence, RI, 1987, 798–820 | MR | Zbl | MR

[6] M. Jimbo, Lett. Math. Phys., 10:1 (1985), 63–69 | DOI | MR | Zbl

[7] V. G. Drinfeld, Algebra i analiz, 1:6 (1989), 114–148 | MR | Zbl

[8] P. P. Kulish, “Twisting of quantum groups and integrable systems”, Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years After NEEDS '79 (Gallipoli, 1999), eds. M. Boiti, L. Martina, F. Pempinelli, B. Prinari, G. Soliani, World Sci., Singapore, 2000, 304–310 | DOI | MR | Zbl

[9] P. P. Kulish, A. A. Stolin, Czech. J. Phys., 47:12 (1997), 1207–1212 | DOI | MR | Zbl

[10] M. Gerstenhaber, A. Giaquinto, S. D. Schack, “Quantum symmetry”, Quantum Groups, Lecture Notes in Math., 1510, ed. P. P. Kulish, Springer, Berlin, 1992, 9–46 | DOI | MR | Zbl

[11] O. V. Ogievetsky, Rend. Circ. Mat. Palermo (2), Suppl. no 37 (1993), 185–199 ; Preprint MPI-Ph/92-99, Munich, 1992 | MR | Zbl

[12] P. P. Kulish, V. D. Lyakhovsky, A. I. Mudrov, J. Math. Phys., 40:9 (1999), 4569–4586 ; arXiv: math/9806014 | DOI | MR | Zbl

[13] S. M. Khoroshkin, A. A. Stolin, V. N. Tolstoy, Comm. Algebra, 26:4 (1998), 1041–1055 | DOI | MR | Zbl

[14] M. Henkel, “Reaction-diffusion processes and their connection with integrable quantum spin chains”, Classical and Quantum Nonlinear Integrable Systems: Theory and Applications, ed. A. Kundu, Institute of Physics, Bristol, 2003, 256–287; arXiv: cond-mat/0303512

[15] E. K. Sklyanin, J. Phys. A, 21:10 (1988), 2375–2389 | DOI | MR | Zbl

[16] V. Chari, A. N. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, Cambridge, 1995 | MR | Zbl

[17] I. V. Cherednik, TMF, 61:1 (1984), 35–44 | DOI | MR | Zbl

[18] P. P. Kulish, E. K. Sklyanin, J. Phys. A, 25:22 (1992), 5963–5975 | DOI | MR | Zbl

[19] L. Freidel, J.-M. Maillet, Phys. Lett. B, 262:2–3 (1991), 278–284 | DOI | MR

[20] P. P. Kulish, R. Sasaki, Progr. Theoret. Phys., 89:3 (1993), 741–761 | DOI | MR

[21] Z. Nagy, J. Avan, A. Doikou, G. Rollet, J. Math. Phys., 46:8 (2005), 083516 | DOI | MR | Zbl

[22] S. Ghoshal, A. B. Zamolodchikov, Internat. J. Modern Phys. A, 9:21 (1994), 3841–3885 | DOI | MR | Zbl

[23] T. Inami, H. Konno, J. Phys. A, 27:24 (1994), L913–L918 | DOI | MR | Zbl

[24] H. J. de Vega, A. González-Ruiz, J. Phys. A, 27:18 (1994), 6129–6137 | DOI | MR | Zbl

[25] A. Doikou, Nucl. Phys. B, 725:3 (2005), 493–530 ; arXiv: math-ph/0409060 | DOI | MR | Zbl

[26] P. P. Kulish, A. I. Mudrov, Lett. Math. Phys., 75:2 (2006), 151–170 | DOI | MR | Zbl

[27] D. Arnaudon, J. Avan, N. Crampé, A. Doikou, L. Frappat, E. Ragoucy, J. Stat. Mech. Theory Exp., 2004:8 (2004), 08005 | DOI | MR | Zbl