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Tetrahedron Equation and Quantum R Matrices for q-oscillator Representations of \({U_q(A^{(2)}_{2n})}\), \({U_q(C^{(1)}_{n})}\) and \({U_q(D^{(2)}_{n+1})}\)

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Abstract

The intertwiner of the quantized coordinate ring A q (sl 3) is known to yield a solution to the tetrahedron equation. By evaluating their n-fold composition with special boundary vectors we generate series of solutions to the Yang-Baxter equation. Finding their origin in conventional quantum group theory is a clue to the link between two and three dimensional integrable systems. We identify them with the quantum R matrices associated with the q-oscillator representations of \({U_q(A^{(2)}_{2n})}\), \({U_q(C^{(1)}_n)}\) and \({U_q(D^{(2)}_{n+1})}\).

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References

  1. Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Dover, New York (2007)

    MATH  Google Scholar 

  2. Bazhanov, V.V., Mangazeev V.V., Sergeev, S.M.: Quantum geometry of 3-dimensional lattices. J. Stat. Mech. P07004 (2008)

  3. Bazhanov V.V., Mangazeev V.V., Sergeev S.M.: An integrable 3D lattice model with positive Boltzmann weights. J. Phys. A Math. Theor. 46, 465206 (2013)

    Article  MathSciNet  Google Scholar 

  4. Bazhanov V.V., Sergeev S.M.: Zamolodchikov’s tetrahedron equation and hidden structure of quantum groups. J. Phys. A Math. Theor. 39, 3295–3310 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Drinfeld, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematicians, vols. 1, 2, pp. 798–820 (Berkeley, California, 1986). American Mathematical Society, Providence, RI (1987)

  6. Gasper, G., Rahman,M.: Basic hypergeornerric series. In: Rota, G.C. (ed.) Encyclopedia ofMathematics and its Applications, vol. 35, Cambridge Univ. Press (1990)

  7. Jimbo M.: A q-difference analogue of U(g) and the Yang-Baxter equation. Lett. Math. Phys. 10, 63–69 (1985)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Hayashi T.: Q-analogues of Clifford and Weyl algebras–spinor and oscillator representations of quantum enveloping algebras. Comm. Math. Phys. 127, 129–144 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  9. Kac V.G.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  10. Kapranov M.M., Voevodsky V.A.: 2-Categories and Zamolodchikov tetrahedron equations. Proc. Symposia Pure Math. 56, 177–259 (1994)

    Article  MathSciNet  Google Scholar 

  11. Kashaev, R.M., Volkov, A.Yu.: From the tetrahedron equation to universal R-matrices. L. D. Faddeev’s Seminar on Mathematical Physics, pp. 79–89. Amer. Math. Soc. Transl. Ser. 2, vol. 201. American Mathematical Society, Providence, RI (2000)

  12. Kuniba, A., Okado, M.: Tetrahedron and 3D reflection equations from quantized algebra of functions. J. Phys. A: Math. Theor. 45, 465206 (27pp) (2012)

  13. Kuniba, A., Okado, M.: Tetrahedron equation and quantum R matrices for infinite dimensional modules of \({U_q(A^{(1)}_1)}\) and \({U_q(A^{(2)}_2)}\). J. Phys. A: Math. Theor. 46, 485203 (12pp) (2013)

  14. Kuniba, A., Okado, M., Yamada, Y.: A common structure in PBW bases of the nilpotent subalgebra of \({U_q(\mathfrak{g})}\) and quantized algebra of functions, SIGMA 9, 049, 23 pages (2013)

  15. Kuniba A., Sergeev S.: Tetrahedron equation and quantum R matrices for spin representations of \({B^{(1)}_n, D^{(1)}_n}\) and \({D^{(2)}_{n+1}}\). Commun. Math. Phys. 324, 695–713 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Masuda T., Mimachi K., Nakagami Y., Noumi M., Saburi Y., Ueno K.: Unitary representations of the quantum group SU q (1, 1): II - Matrix elements of unitary representations and the basic hypergoemetric functions. Lett. Math. Phys. 19, 195–204 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Okado M.: Quantum R matrices related to the spin representations of B n and D n . Commun. Math. Phys. 134, 467–486 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Pusz W.: Irreducible unitary representations of quantum Lorentz group. Commun. Math. Phys. 152, 591–626 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. Sergeev S.M.: Two-dimensional R-matrices—descendants of three-dimensional R-matrices. Modern Phys. Lett. A 12, 1393–1410 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Sergeev S.M.: Tetrahedron equations and nilpotent subalgebras of \({\mathcal{U}_q(sl_n)}\). Lett. Math. Phys. 83, 231–235 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Zamolodchikov A.B.: Tetrahedra equations and integrable systems in three-dimensional space. Soviet Phys. JETP 79, 641–664 (1980)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Tokyo, 153-8902, Japan

    Atsuo Kuniba

  2. Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

    Masato Okado

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  1. Atsuo Kuniba
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  2. Masato Okado
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Correspondence to Masato Okado.

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Communicated by N. Reshetikhin

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Kuniba, A., Okado, M. Tetrahedron Equation and Quantum R Matrices for q-oscillator Representations of \({U_q(A^{(2)}_{2n})}\), \({U_q(C^{(1)}_{n})}\) and \({U_q(D^{(2)}_{n+1})}\) . Commun. Math. Phys. 334, 1219–1244 (2015). https://doi.org/10.1007/s00220-014-2147-1

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