Letters in Mathematical Physics

, Volume 105, Issue 1, pp 45–61 | Cite as

Tetrahedron Equation, Weyl Group, and Quantum Dilogarithm

  • Andrei Bytsko
  • Alexander Volkov


We derive a family of solutions to the tetrahedron equation using the RTT presentation of a two parametric quantized algebra of regular functions on an upper triangular subgroup of GL(n). The key ingredients of the construction are the longest element of the Weyl group, the quantum dilogarithm function, and central elements of the quantized division algebra of rational functions on the subgroup in question.

Mathematics Subject Classification

20G42 16K20 16T25 


tetrahedron equation quantum dilogarithm quantum group 


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  1. 1.
    Bytsko, A., Volkov, A.: Tetrahedron equation and cyclic quantum dilogarithm identities. Int. Math. Res. Notices (to appear). arXiv:1304.1641
  2. 2.
    Cliff G.: The division ring of quotients of the coordinate ring of the quantum general linear group. J. London Math. Soc. (2) 51(3), 503–513 (1995)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Drinfeld V.G.: Quantum groups. J. Sov. Math. 41(2), 898–915 (1988)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Faddeev L.D.: Discrete Heisenberg–Weyl group and modular group. Lett. Math. Phys. 34(3), 249–254 (1995)ADSCrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Faddeev L.D., Kashaev R.M.: Quantum dilogarithm. Mod. Phys. Lett. A 9(5), 427–434 (1994)ADSCrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Faddeev L.D., Reshetikhin N.Yu, Takhtadzhyan L.A.: Quantization of Lie groups and Lie algebras. Leningrad Math. J 1(1), 193–225 (1990)MathSciNetMATHGoogle Scholar
  7. 7.
    Faddeev L.D., Volkov A.Yu.: Abelian current algebra and the Virasoro algebra on the lattice. Phys. Lett. B 315(3–4), 311–318 (1993)ADSCrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Kapranov, M., Voevodsky, V.: 2–Categories and Zamolodchikov tetrahedra equations. In: Algebraic groups and their generalizations: quantum and infinite–dimensional methods, Proc. Sympos. Pure Math. Part 2, vol. 56, pp. 177–259. AMS, Providence, RI (1994)Google Scholar
  9. 9.
    Kashaev, RM., Volkov, A.Yu.: From the tetrahedron equation to universal R–matrices. In: L.D. Faddeev’s Seminar on Mathematical Physics, AMS Transl. Ser. 2, vol. 201, pp. 79–89. AMS, Providence, RI (2000)Google Scholar
  10. 10.
    Kazhdan, D., Soibelman, Ya.: Representations of the quantized function algebras, 2–categories and Zamolodchikov tetrahedra equation. In: The Gel’fand Mathematical Seminars, 1990–1992, pp. 163–171. Birkhäuser (1993)Google Scholar
  11. 11.
    Kuniba, A., Okado, M.: Tetrahedron and 3D reflection equations from quantized algebra of functions. J. Phys. A 45(46), 465206 (2012)Google Scholar
  12. 12.
    Mosin, V.G., Panov, A.N.: Quotient skew fields and central elements of multiparametric quantizations. Sbornik. Math. 187(6), 835–855 (1996)Google Scholar
  13. 13.
    Panov A.N.: Skew fields of twisted rational functions and the skew field of rational functions on GL q(n, K). St. Petersburg Math. J. 7(1), 129–143 (1996)MathSciNetGoogle Scholar
  14. 14.
    Reshetikhin N.: Multiparameter quantum groups and twisted quasitriangular Hopf algebras. Lett. Math. Phys. 20(4), 331–335 (1990)ADSCrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Schützenberger, M.P.: Une interprétation de certaines solutions de l’équation fonctionnelle: F(x + y) = F(x)F(y). C. R. Acad. Sci. Paris. 236, 352–353 (1953)Google Scholar
  16. 16.
    Sergeev S.M.: Two–dimensional R–matrices—descendants of three–dimensional R–matrices. Mod. Phys. Lett. A 12(19), 1393–1410 (1997)ADSCrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Soibelman Ya.S., Vaksman L.L.: An algebra of functions on the quantum group SU(2). Funct. Anal. Appl 22(3), 170–181 (1989)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Soibelman Ya.S., Vaksman L.L.: Algebra of functions on the quantum group SU(n + 1), and odd–dimensional quantum spheres. Leningrad Math. J 2(5), 1023–1042 (1991)MathSciNetGoogle Scholar
  19. 19.
    Zamolodchikov A.B.: Tetrahedron equations and the relativistic S–matrix of straight–strings in 2+1–dimensions. Commun. Math. Phys. 79(4), 489–505 (1981)ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Section of MathematicsUniversity of GenevaGeneva 4Switzerland
  2. 2.Steklov Mathematical InstituteRussian Academy of SciencesSt. PetersburgRussia

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