General solution of the Yang–Baxter equation with symmetry group 𝑆𝐿(𝑛,ℂ)

Derkachev, S.; Manashov, A.

doi:10.1090/S1061-0022-2010-01106-3

General solution of the Yang–Baxter equation with symmetry group $\mathrm {SL}(n,\mathbb {C})$
HTML articles powered by AMS MathViewer

by S. E. Derkachev and A. N. Manashov
Translated by: B. M. Bekker

St. Petersburg Math. J. 21 (2010), 513-577

DOI: https://doi.org/10.1090/S1061-0022-2010-01106-3

Published electronically: May 20, 2010

PDF | Request permission

Abstract:

The problem of constructing the $\mathrm {R}$-matrix is considered in the case of an integrable spin chain with symmetry group $\mathrm {SL}(n,\mathbb {C})$. A fairly complete study of general $\mathrm {R}$-matrices acting in the tensor product of two continuous series representations of $\mathrm {SL}(n,\mathbb {C})$ is presented. On this basis, $\mathrm {R}$-matrices are constructed that act in the tensor product of Verma modules (which are infinite-dimensional representations of the Lie algebra $\mathrm {sl}(n)$), and also $\mathrm {R}$-matrices acting in the tensor product of finite-dimensional representations of the Lie algebra $\mathrm {sl}(n)$.

References

E. K. Skljanin, L. A. Tahtadžjan, and L. D. Faddeev, Quantum inverse problem method. I, Teoret. Mat. Fiz. 40 (1979), no. 2, 194–220 (Russian, with English summary). MR 549615
L. A. Tahtadžjan and L. D. Faddeev, The quantum method for the inverse problem and the $XYZ$ Heisenberg model, Uspekhi Mat. Nauk 34 (1979), no. 5(209), 13–63, 256 (Russian). MR 562799
P. P. Kulish and E. K. Sklyanin, Quantum spectral transform method. Recent developments, Integrable quantum field theories (Tvärminne, 1981) Lecture Notes in Phys., vol. 151, Springer, Berlin-New York, 1982, pp. 61–119. MR 671263
E. K. Sklyanin, Quantum inverse scattering method. Selected topics, Quantum group and quantum integrable systems, Nankai Lectures Math. Phys., World Sci. Publ., River Edge, NJ, 1992, pp. 63–97. MR 1239668
L. D. Faddeev, How the algebraic Bethe ansatz works for integrable models, Symétries quantiques (Les Houches, 1995) North-Holland, Amsterdam, 1998, pp. 149–219. MR 1616371
A. G. Kuliš and E. K. Skljanin, Solutions of the Yang-Baxter equation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 95 (1980), 129–160, 161 (Russian). Differential geometry, Lie groups and mechanics, III. MR 606023
Michio Jimbo, Introduction to the Yang-Baxter equation, Internat. J. Modern Phys. A 4 (1989), no. 15, 3759–3777. MR 1017340, DOI 10.1142/S0217751X89001503
Michio Jimbo (ed.), Yang-Baxter equation in integrable systems, Advanced Series in Mathematical Physics, vol. 10, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. MR 1061379
V. G. Drinfel′d, Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR 283 (1985), no. 5, 1060–1064 (Russian). MR 802128
V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
Rodney J. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1982. MR 690578
B. Sutherland, A general model for multicomponent quantum systems, Phys. Rev. B 12 (1975), 3795.
P. P. Kulish and N. Yu. Reshetikhin, On $\textrm {GL}_{3}$-invariant solutions of the Yang-Baxter equation and associated quantum systems, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 120 (1982), 92–121 (Russian, with English summary). Questions in quantum field theory and statistical physics, 3. MR 701555
P. P. Kulish and N. Yu. Reshetikhin, Diagonalisation of $\textrm {GL}(N)$ invariant transfer matrices and quantum $N$-wave system (Lee model), J. Phys. A 16 (1983), no. 16, L591–L596. MR 727044
E. K. Sklyanin, The quantum Toda chain, Nonlinear equations in classical and quantum field theory (Meudon/Paris, 1983/1984) Lecture Notes in Phys., vol. 226, Springer, Berlin, 1985, pp. 196–233. MR 802110, DOI 10.1007/3-540-15213-X_{8}0
V. Pasquier and M. Gaudin, The periodic Toda chain and a matrix generalization of the Bessel function recursion relations, J. Phys. A 25 (1992), no. 20, 5243–5252. MR 1192958
L. N. Lipatov, High-energy asymptotics of multicolor $QCD$ and exactly solvable spin models, Pis′ma Zh. Èksper. Teor. Fiz. 59 (1994), no. 9-10, 571–574; English transl., JETP Lett. 59 (1994), 596.
—, Duality symmetry of reggeon interactions in multicolor $QCD$, Nuclear Phys. B 548 (1999), 328.
L. D. Faddeev and G. P. Korchemsky, High-energy $QCD$ as a completely integrable model, Phys. Lett. B 342 (1995), 311.
D. R. Karakhanyan and R. Kirschner, High-energy scattering in gauge theories and integrable spin chains, Proceedings of the 32nd International Symposium Ahrenshoop on the Theory of Elementary Particles (Buckow, 1998), 2000, pp. 139–142. MR 1764063, DOI 10.1002/(SICI)1521-3978(20001)48:1/3<139::AID-PROP139>3.0.CO;2-S
S. É. Derkachov, G. P. Korchemsky, and A. N. Manashov, Noncompact Heisenberg spin magnets from high-energy QCD. I. Baxter $Q$-operator and separation of variables, Nuclear Phys. B 617 (2001), no. 1-3, 375–440. MR 1866770, DOI 10.1016/S0550-3213(01)00457-6
Marc Kirch and Alexander N. Manashov, Noncompact $\textrm {SL}(2,\Bbb R)$ spin chain, J. High Energy Phys. 6 (2004), 035, 31. MR 2084421, DOI 10.1088/1126-6708/2004/06/035
Andrei G. Bytsko and Jörg Teschner, Quantization of models with non-compact quantum group symmetry: modular $XXZ$ magnet and lattice sinh-Gordon model, J. Phys. A 39 (2006), no. 41, 12927–12981. MR 2277462, DOI 10.1088/0305-4470/39/41/S11
S. É. Derkachov, Baxter’s $Q$-operator for the homogeneous $XXX$ spin chain, J. Phys. A 32 (1999), no. 28, 5299–5316. MR 1720341, DOI 10.1088/0305-4470/32/28/309
V. B. Kuznetsov, M. Salerno, and E. K. Sklyanin, Quantum Bäcklund transformation for the integrable DST model, J. Phys. A 33 (2000), no. 1, 171–189. MR 1748470, DOI 10.1088/0305-4470/33/1/311
G. P. Pronko, On Baxter’s $Q$-operator for the $XXX$ spin chain, Comm. Math. Phys. 212 (2000), no. 3, 687–701. MR 1779164, DOI 10.1007/s002200000235
A. E. Kovalsky and G. P. Pronko, Baxter $Q$-operators for integrable $DST$ chain, nlin.SI/0203030.
—, Baxter’s $Q$-operators for the simplest $q$-deformed model, nlin.SI/0307040.
A. Yu. Volkov, Quantum lattice KdV equation, Lett. Math. Phys. 39 (1997), no. 4, 313–329. MR 1449577, DOI 10.1023/A:1007390915590
Alexander Antonov and Boric Feigin, Quantum group representations and the Baxter equation, Phys. Lett. B 392 (1997), no. 1-2, 115–122. MR 1435226, DOI 10.1016/S0370-2693(96)01526-2
A. A. Belavin, A. V. Odesskiĭ, and R. A. Usmanov, New relations in the algebra of the Baxter $Q$-operators, Teoret. Mat. Fiz. 130 (2002), no. 3, 383–413 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 130 (2002), no. 3, 323–350. MR 1920471, DOI 10.1023/A:1014758721234
Marco Rossi and Robert Weston, A generalized $Q$-operator for $U_q(\widehat {\mathfrak {sl}_2})$ vertex models, J. Phys. A 35 (2002), no. 47, 10015–10032. MR 1957841, DOI 10.1088/0305-4470/35/47/304
A. Zabrodin, Commuting difference operators with elliptic coefficients from Baxter’s vacuum vectors, J. Phys. A 33 (2000), no. 20, 3825–3850. MR 1767042, DOI 10.1088/0305-4470/33/20/308
Christian Korff, Solving Baxter’s $TQ$-equation via representation theory, Noncommutative geometry and representation theory in mathematical physics, Contemp. Math., vol. 391, Amer. Math. Soc., Providence, RI, 2005, pp. 199–211. MR 2184024, DOI 10.1090/conm/391/07330
Vladimir V. Bazhanov, Sergei L. Lukyanov, and Alexander B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Comm. Math. Phys. 177 (1996), no. 2, 381–398. MR 1384140
Vladimir V. Bazhanov, Anthony N. Hibberd, and Sergey M. Khoroshkin, Integrable structure of $\scr W_3$ conformal field theory, quantum Boussinesq theory and boundary affine Toda theory, Nuclear Phys. B 622 (2002), no. 3, 475–547. MR 1880098, DOI 10.1016/S0550-3213(01)00595-8
Vladimir V. Bazhanov and Zengo Tsuboi, Baxter’s Q-operators for supersymmetric spin chains, Nuclear Phys. B 805 (2008), no. 3, 451–516. MR 2449804, DOI 10.1016/j.nuclphysb.2008.06.025
S. M. Khoroshkin and V. N. Tolstoy, Universal $R$-matrix for quantized (super)algebras, Comm. Math. Phys. 141 (1991), no. 3, 599–617. MR 1134942
S. M. Khoroshkin and V. N. Tolstoy, The uniqueness theorem for the universal $R$-matrix, Lett. Math. Phys. 24 (1992), no. 3, 231–244. MR 1166753, DOI 10.1007/BF00402899
S. M. Khoroshkin, A. A. Stolin, and V. N. Tolstoy, Generalized Gauss decomposition of trigonometric $R$-matrices, Modern Phys. Lett. A 10 (1995), no. 19, 1375–1392. MR 1341337, DOI 10.1142/S0217732395001496
Takeo Kojima, Baxter’s $Q$-operator for the $W$-algebra $W_N$, J. Phys. A 41 (2008), no. 35, 355206, 16. MR 2426018, DOI 10.1088/1751-8113/41/35/355206
S. E. Derkachov, Factorization of the $R$-matrix and Baxter’s $Q$-operator, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 347 (2007), no. Vopr. Kvant. Teor. Polya i Stat. Fiz. 20, 144–166, 241–242 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 151 (2008), no. 2, 2880–2893. MR 2458889, DOI 10.1007/s10958-008-9010-x
S. É. Derkachov and A. N. Manashov, Factorization of the transfer matrices for the quantum $\textrm {sl}(2)$ spin chains and Baxter equation, J. Phys. A 39 (2006), no. 16, 4147–4159. MR 2220363, DOI 10.1088/0305-4470/39/16/003
S. É. Derkachov and A. N. Manashov, Baxter operators for the noncompact quantum $\textrm {sl}(3)$ invariant spin chain, J. Phys. A 39 (2006), no. 42, 13171–13190. MR 2266052, DOI 10.1088/0305-4470/39/42/001
A. V. Belitsky, S. É. Derkachov, G. P. Korchemsky, and A. N. Manashov, The Baxter $Q$-operator for the graded $\textrm {SL}(2|1)$ spin chain, J. Stat. Mech. Theory Exp. 1 (2007), P01005, 63. MR 2284009, DOI 10.1088/1742-5468/2007/01/p01005
Sergey É. Derkachov and Alexander N. Manashov, $\scr R$-matrix and Baxter $\scr Q$-operators for the noncompact $\textrm {SL}(N,{\Bbb C})$ invariant spin chain, SIGMA Symmetry Integrability Geom. Methods Appl. 2 (2006), Paper 084, 20. MR 2264900, DOI 10.3842/SIGMA.2006.084
S. E. Derkachov, Factorization of the $R$-matrix. I, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 335 (2006), no. Vopr. Kvant. Teor. Polya i Stat. Fiz. 19, 134–163; English transl., J. Math. Sci. (N.Y.) 143 (2007), no. 1, 2773–2790. MR 2269755, DOI 10.1007/s10958-007-0164-8
I. M. Gel′fand and G. E. Šilov, Obobshchennye funksii i deĭ stviya iad nimi, Obobščennye funkcii, Vypusk 1. [Generalized functions, part 1], Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1958 (Russian). MR 0097715
I. M. Gel′fand and M. A. Naĭmark, Unitarnye predstavleniya klassičeskih grupp, Izdat. Nauk SSSR, Moscow-Leningrad, 1950 (Russian). Trudy Mat. Inst. Steklov. no. 36,. MR 0046370
I. M. Gel′fand, M. I. Graev, and N. Ja. Vilenkin, Obobshchennye funktsii, Vyp. 5. Integral′naya geometriya i svyazannye s neĭ voprosy teorii predstavleniĭ, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1962 (Russian). MR 0160110
D. P. Želobenko, Classical groups. Spectral analysis of finite-dimensional representations, Uspehi Mat. Nauk 17 (1962), no. 1 (103), 27–120 (Russian). MR 0136664
D. P. Zhelobenko, Kompaktnye gruppy Li i ikh predstavleniya, Izdat. “Nauka”, Moscow, 1970 (Russian). MR 0473097
M. A. Naĭmark, Teoriya predstavleniĭ grupp, Izdat. “Nauka”, Moscow, 1976 (Russian). MR 0578530
H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 14, Springer-Verlag, New York-Heidelberg, 1972. MR 0349820
Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR 855239, DOI 10.1515/9781400883974
P. P. Kulish, N. Yu. Reshetikhin, and E. K. Sklyanin, Yang-Baxter equations and representation theory. I, Lett. Math. Phys. 5 (1981), no. 5, 393–403. MR 649704, DOI 10.1007/BF02285311
A. N. Vasil′ev, Quantum field renormalization group in the critical behavior theory and in stochastic dynamics, S.-Peterburg. Inst. Yader. Fiz., St. Petersburg, 1998. (Rissian)
Alexandre Yu. Volkov, Noncommutative hypergeometry, Comm. Math. Phys. 258 (2005), no. 2, 257–273. MR 2171695, DOI 10.1007/s00220-005-1342-5
A. P. Isaev, Multi-loop Feynman integrals and conformal quantum mechanics, Nuclear Phys. B 662 (2003), no. 3, 461–475. MR 1985276, DOI 10.1016/S0550-3213(03)00393-6
E. K. Sklyanin, Classical limits of $\textrm {SU}(2)$-invariant solutions of the Yang-Baxter equation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 146 (1985), 119–136, 203, 206–207 (Russian, with English summary). Translated in J. Soviet Math. 40 (1988), no. 1, 93–107; Differential geometry, Lie groups and mechanics. VII. MR 836552
George Gasper, Elementary derivations of summation and transformation formulas for $q$-series, Special functions, $q$-series and related topics (Toronto, ON, 1995) Fields Inst. Commun., vol. 14, Amer. Math. Soc., Providence, RI, 1997, pp. 55–70. MR 1448679
George Gasper and Mizan Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR 1052153