Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



On higher spin $ U_q(\operatorname{sl}_2)$-invariant $ R$-matrices

Author: A. G. Bytsko
Translated by: the author
Original publication: Algebra i Analiz, tom 17 (2005), nomer 3.
Journal: St. Petersburg Math. J. 17 (2006), 393-408
MSC (2000): Primary 81R50
Published electronically: March 9, 2006
MathSciNet review: 2167842
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The spectral decomposition of regular $ U_q(\operatorname{sl}_2)$-invariant solutions of the Yang-Baxter equation is studied. An algorithm for the search of all possible spin $ s$ solutions is developed, also allowing reconstruction of the $ R$-matrix by a given nearest neighbor spin chain Hamiltonian. The algorithm is based on reduction of the Yang-Baxter equation to certain subspaces. As an application, a complete list of nonequivalent regular $ U_q(\operatorname{sl}_2)$-invariant $ R$-matrices is obtained for generic $ q$ and spins $ s\leq\frac{3}{2}$. Some further results about spectral decompositions for higher spins are also proved. In particular, it is shown that certain types of regular $ \operatorname{sl}_2$-invariant $ R$-matrices have no $ U_q(\operatorname{sl}_2)$-invariant counterparts.

References [Enhancements On Off] (What's this?)

  • 1. P. P. Kuliš and N. Ju. Rešetihin, Quantum linear problem for the sine-Gordon equation and higher representations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 101 (1981), 101–110, 207 (Russian). Questions in quantum field theory and statistical physics, 2. MR 623928
  • 2. E. K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation, Funktsional. Anal. i Prilozhen. 16 (1982), no. 4, 27–34, 96 (Russian). MR 684124
  • 3. -, On an algebra generated by quadratic relations, Uspekhi Mat. Nauk 40 (1985), no. 2, 214. (Russian)
  • 4. Michio Jimbo, A 𝑞-difference analogue of 𝑈(𝔤) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63–69. MR 797001, 10.1007/BF00704588
  • 5. 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
  • 6. Marc Rosso, Finite-dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra, Comm. Math. Phys. 117 (1988), no. 4, 581–593. MR 953821
  • 7. J. B. McGuire, Study of exactly soluble one-dimensional 𝑁-body problems, J. Mathematical Phys. 5 (1964), 622–636. MR 0161667
  • 8. C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312–1315. MR 0261870
  • 9. 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, 10.1007/BF02285311
  • 10. Alexander B. Zamolodchikov and Alexey B. Zamolodchikov, Factorized 𝑆-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models, Ann. Physics 120 (1979), no. 2, 253–291. MR 546461, 10.1016/0003-4916(79)90391-9
  • 11. R. J. Baxter, The inversion relation method for some two-dimensional exactly solved models in lattice statistics, J. Statist. Phys. 28 (1982), no. 1, 1–41. MR 664123, 10.1007/BF01011621
  • 12. Tom Kennedy, Solutions of the Yang-Baxter equation for isotropic quantum spin chains, J. Phys. A 25 (1992), no. 10, 2809–2817. MR 1166928
  • 13. Michio Jimbo, Quantum 𝑅 matrix for the generalized Toda system, Comm. Math. Phys. 102 (1986), no. 4, 537–547. MR 824090
  • 14. H. N. V. Temperley and E. H. Lieb, Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1549, 251–280. MR 0498284
  • 15. A. N. Kirillov and N. Yu. Reshetikhin, Representations of the algebra 𝑈_{𝑞}(𝑠𝑙(2)),𝑞-orthogonal polynomials and invariants of links, Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys., vol. 7, World Sci. Publ., Teaneck, NJ, 1989, pp. 285–339. MR 1026957
  • 16. Masao Nomura, Relations for Clebsch-Gordan and Racah coefficients in 𝑠𝑢_{𝑞}(2) and Yang-Baxter equations, J. Math. Phys. 30 (1989), no. 10, 2397–2405. MR 1016311, 10.1063/1.528612
  • 17. P. P. Kulish and E. K. Sklyanin, Quantum spectral transform method. Recent developments, Lecture Notes in Phys., vol. 151, Springer, Berlin-New York, 1982, pp. 61–119. MR 671263
  • 18. 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
  • 19. V. F. R. Jones, On a certain value of the Kauffman polynomial, Comm. Math. Phys. 125 (1989), no. 3, 459–467. MR 1022523
  • 20. Zhong Qi Ma, Yang-Baxter equation and quantum enveloping algebras, Advanced Series on Theoretical Physical Science, vol. 1, World Scientific Publishing Co., Inc., River Edge, NJ; Institute of Theoretical Physics, Beijing, 1993. MR 1337274

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 81R50

Retrieve articles in all journals with MSC (2000): 81R50

Additional Information

A. G. Bytsko
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

Keywords: Quantum Lie algebra, Hopf algebra, spin, Yang--Baxter equation
Published electronically: March 9, 2006
Additional Notes: Supported by the INTAS (grant YS-03-55-962) and by RFBR (grant nos. 02-01-00085 and 03-01-00593)
Dedicated: Dedicated to Professor L. D. Faddeev on the occasion of his 70th birthday
Article copyright: © Copyright 2006 American Mathematical Society