$\mathcal R$-Matrix and Baxter $\mathcal Q$-Operators for the Noncompact $\mathrm{SL}(N,\mathbb C)$ Invariant Spin Chain

Sergey É Derkachov ; Alexander N. Manashov

Symmetry, integrability and geometry: methods and applications, Tome 2 (2006)

Résumé

The problem of constructing the $SL(N,\mathbb C)$ invariant solutions to the Yang–Baxter equation is considered. The solutions ($\mathcal R$-operators) for arbitrarily principal series representations of $\mathrm{SL}(N,\mathbb C)$ are obtained in an explicit form. We construct the commutative family of the operators $\mathcal Q_k(u)$ which can be identified with the Baxter operators for the noncompact $\mathrm{SL}(N,\mathbb C)$ spin magnet.

Détail

Keywords: Yang–Baxter equation; Baxter operator.

Bibliographie

[1] Theor. Math. Phys., 40:2 (1980), 688–706 | DOI | MR

[2] Faddeev L. D., “How algebraic Bethe ansatz works for integrable model”, Symétries quantiques (Les-Houches Lectures, 1995), North-Holland, Amsterdam, 1998, 149–219 ; <nobr>hep-th/9605187</nobr> | MR | Zbl

[3] Soviet Math. Dokl., 32 (1985), 254–258 | MR

[4] Leningrad Math. J., 1:6 (1990), 1419–1457 | MR

[5] Jimbo M., “A $q$ difference analog of $U(g)$ and the Yang–Baxter equation”, Lett. Math. Phys., 10 (1985), 63–69 | DOI | MR | Zbl

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

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

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

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

[10] Kulish P. P., Reshetikhin N. Y., Sklyanin E. K., “Yang–Baxter equation and representation theory. I”, Lett. Math. Phys., 5 (1981), 393–403 | DOI | MR | Zbl

[11] Kulish P. P., Reshetikhin N. Y., “On $\mathrm GL_3$-invariant solutions of the Yang–Baxter equation and associated quantum systems”, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 120, 1982, 92–121 | MR

[12] Baxter R. J., Exactly solved models in statistical mechanics, Academic Press, London, 1982 | MR | Zbl

[13] Sklyanin E. K., “The quantum Toda chain”, Nonlinear Equations in Classical and Quantum Field Theory (Meudon/Paris, 1983/1984), Lecture Notes in Phys., 226, Springer, Berlin, 1985, 196–233 | MR

[14] Sklyanin E. K., “Quantum inverse scattering method. Selected topics”, Quantum Group and Quantum Integrable Systems (Nankai Lectures in Mathematical Physics), ed. Mo-Lin Ge, World Scientific, Singapore, 1992, 63–97 ; hep-th/9211111 | MR

[15] Derkachov S. E.,, “Factorization of $R$-matrix. I”, Zap. Nauch. Sem. POMI, 335, 2006, 134–163 ; math.QA/0503396 | MR | Zbl

[16] Derkachov S. E., Korchemsky G. P., Manashov A. N., “Noncompact Heisenberg spin magnets from high-energy QCD. I. Baxter $Q$-operator and separation of variables”, Nuclear Phys. B, 617 (2001), 375–440 ; hep-th/0107193 | DOI | MR | Zbl

[17] Derkachov S. E., Korchemsky G. P., Kotanski J., Manashov A. N., “Noncompact Heisenberg spin magnets from high-energy QCD. II. Quantization conditions and energy spectrum”, Nuclear Phys. B, 645 (2002), 237–297 ; hep-th/0204124 | DOI | MR | Zbl

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

[19] Bazhanov V. V., Hibberd A. N., Khoroshkin S. M., “Integrable structure of $W(3)$ conformal field theory, quantum Boussinesq theory and boundary affine Toda theory”, Nuclear Phys. B, 622 (2002), 475–547 ; hep-th/0105177 | DOI | MR | Zbl

[20] Kuznetsov V. B., Nijhoff F. W., Sklyanin E. K., “Separation of variables for the Ruijsenaars system”, Comm. Math. Phys., 189 (1997), 855–877 ; solv-int/9701004 | DOI | MR | Zbl

[21] Kharchev S., Lebedev D., “Integral representation for the eigenfunctions of quantum periodic Toda chain”, Lett. Math. Phys., 50 (1999), 53–77 ; hep-th/9910265 | DOI | MR | Zbl

[22] Kharchev S., Lebedev D., Semenov-Tian-Shansky M., “Unitary representations of $U_q(\mathfrak{sl}(2,\mathbb R))$, the modular double, and the multiparticle $q$-deformed Toda chains”, Comm. Math. Phys., 225 (2002), 573–609 ; hep-th/0102180 | DOI | MR | Zbl

[23] Derkachov S. E., Korchemsky G. P., Manashov A. N., “Separation of variables for the quantum $\mathrm {SL}(2,\mathbb R)$ spin chain”, JHEP, 07 (2003), Paper 047, 30 pp., ages ; hep-th/0210216 | MR

[24] Derkachov S. E., Korchemsky G. P., Manashov A. N., “Baxter $Q$-operator and separation of variables for the open $\mathrm{SL}(2,\mathbb R)$ spin chain”, JHEP, 10 (2003), Paper 053, 31 pp., ages ; hep-th/0309144 | MR

[25] Kuznetsov V. B., Mangazeev V. V., Sklyanin E. K., “$Q$-operator and factorised separation chain for Jack's symmetric polynomials”, Indag. Mathem., 14 (2003), 451–482 ; math.CA/0306242 | DOI | MR | Zbl

[26] Silantyev A., Transition function for the Toda chain model, nlin.SI/0603017

[27] Iorgov N., “Eigenvectors of open Bazhanov–Stroganov quantum chain”, SIGMA, 2 (2006), Paper 019, 10 pp., ages ; nlin.SI/0602010 | MR | Zbl

[28] Kuznetsov V. B., Sklyanin E. K., “Few remarks on Bäcklund transformations for many-body systems”, J. Phys. A: Math. Gen., 31 (1998), 2241–2251 ; solv-int/9711010 | DOI | MR | Zbl

[29] Sklyanin E. K., Bäcklund transformations and Baxter's $Q$-operator, nlin.SI/0009009

[30] Academie Verlag, Berlin, 1957 | MR | Zbl

[31] Zhelobenko D. P., Shtern A. I., Representations of Lie groups, Nauka, Moscow, 1983 (in Russian) | MR

[32] Knapp A. W., Stein E. M., “Intertwining operators for semisimple groups”, Ann. of Math., 93 (1971), 489–578 | DOI | MR | Zbl

[33] Knapp A. W., Representation theory of semisimple groups, Princeton University Press, Princeton, New Jersey, 1986 | MR | Zbl

[34] Derkachov S. E., Factorization of $R$-matrix and Baxter's $Q$-operator, math.QA/0507252

[35] Perk J. H. H., Au-Yang H., “Yang–Baxter equations”, Encyclopedia of Mathematical Physics, 5, eds. J.-P. Françoise, G. L. Naber and S. T. Tsou, Elsevier, Oxford, 2006, 465–473

[36] Isaev A. P., “Multi-loop Feynman integrals and conformal quantum mechanics”, Nuclear Phys. B, 662 (2003), 461–475 ; hep-th/0303056 | DOI | MR | Zbl

[37] Derkachov S. E., Manashov A. N., “Factorization of the transfer matrices for the quantum $sl(2)$ spin chains and Baxter equation”, J. Phys. A: Math. Gen., 39 (2006), 4147–4159 ; nlin.SI/0512047 | DOI | MR | Zbl

[38] Derkachov S. E., Manashov A. N., “Baxter operators for the quantum $sl(3)$ invariant spin chain”, J. Phys. A: Math. Gen., 39 (2006), 13171–13190 ; nlin.SI/0604018 | DOI | MR | Zbl

[39] Derkachov S. E., “Baxter's $Q$-operator for the homogeneous $XXX$ spin chain”, J. Phys. A: Math. Gen., 32 (1999), 5299–5316 ; solv-int/9902015 | DOI | MR | Zbl