Abstract
We introduce a homomorphism from the quantum affine algebras \({U_q(D^{(2)}_{n+1}), U_q (A^{(2)}_{2n})}\), \({U_q(C^{(1)}_{n})}\) to the n-fold tensor product of the q-oscillator algebra \({\mathcal{A}_q}\). Their action commutes with the solutions of the Yang–Baxter equation obtained by reducing the solutions of the tetrahedron equation associated with the modular and the Fock representations of \({\mathcal{A}_q}\). In the former case, the commutativity is enhanced to the modular double of these quantum affine algebras.
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References
Baxter, R.J.: Exactly solved models in statistical mechanics, Dover (2007)
Bazhanov V.V., Baxter R.J.: New solvable lattice models in three dimensions. J. Stat. Phys 69, 453–485 (1992)
Bazhanov, V.V., Mangazeev, V.V., Sergeev, S.M.: Quantum geometry of 3-dimensional lattices. J. Stat. Mech. P07004 (2008)
Bazhanov V.V., Kashaev R.M., Mangazeev V.V., Stroganov Yu.G.: \({({Z}_{N} \times)^{n-1}}\) generalization of the chiral Potts model. Commun. Math. Phys 138, 393–408 (1991)
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)
Date E., Jimbo M., Miki K., Miwa T.: Generalized chiral Potts models and minimal cyclic representations of \({U_q(gl(n,\mathbb{C}))}\). Commun. Math. Phys 137, 133–147 (1991)
Drinfeld, V.G.: Quantum groups. In: Proceedings of the ICM, Vols. 1, 2 (Berkeley, Calif., 1986), pp. 798–820. Am. Math. Soc., Providence (1987)
Faddeev, L.: Modular double of a quantum group. Conférence Moshé Flato 1999, 1, pp. 149–156. Math. Phys. Stud. Kluwer Acad. Publ. (2000)
Frenkel, I.B., Ip, I.C.-H.: Positive representations of split real quantum groups and future perspectives. arXiv:1111.1033
Hayashi T.: Q-analogues of Clifford and Weyl algebras–spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys 127, 129–144 (1990)
Ip, I.C.-H.: Positive representations of split real quantum groups of type B n , C n , F 4, and G 2. arXiv:1205.2940
Jimbo M.: A q-difference analogue of U(g) and the Yang–Baxter equation. Lett. Math. Phys 10, 63–69 (1985)
Kac, V.G.: Infinite dimensional Lie algebras, 3rd edn. Cambridge University Press, Cambridge (1990)
Kapranov M.M., Voevodsky V.A.: 2-Categories and Zamolodchikov tetrahedron equations. Proc. Symp. Pure Math 56, 177–259 (1994)
Kashaev R.V., Volkov A.Yu.: From the tetrahedron equation to universal R-matrices. Am. Math. Soc. Transl. Ser. 2(201), 79–89 (2000)
Kuniba, A., Okado, M.: Tetrahedron and 3D reflection equations from quantized algebra of functions. J. Phys. A Math.Theor. 45, 465206, pp. 27 (2012)
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 in press
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)
Schmüdgen K.: Integrable operator representations of \({\mathbb{R}_q^2}\), \({X_{q,\gamma}}\) and \({\mathrm{SL}_q(2, \mathbb{R})}\). Commun. Math. Phys 159, 217–237 (1994)
Sergeev S.M.: Tetrahedron equations and nilpotent subalgebras of \({\mathcal{U}_q(sl_n)}\). Lett. Math. Phys 83, 231–235 (2008)
Sergeev S.M.: Two-dimensional R-matrices–descendants of three-dimensional R-matrices. Mod. Phys. Lett. A 12, 1393–1410 (1997)
Sergeev S.M., Mangazeev V.V., Stroganov Yu G.: The vertex formulation of the Bazhanov–Baxter model. J. Stat. Phys 82, 31–50 (1994)
Tarasov V.: Cyclic monodromy matrices for sl(n) trigonometric R-matrices. Commun. Math. Phys 158, 459–483 (1993)
Zamolodchikov A.B.: Tetrahedra equations and integrable systems in three-dimensional space, Soviet Phys. JETP 79, 641–664 (1980)
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Kuniba, A., Okado, M. & Sergeev, S. Tetrahedron Equation and Quantum R Matrices for Modular Double of \({{\varvec{{U_q(D^{(2)}_{n+1})}}, \varvec{{U_q (A ^{(2)}_{2n})}}}}\) and \(\varvec{{U_q(C^{(1)}_{n})}}\) . Lett Math Phys 105, 447–461 (2015). https://doi.org/10.1007/s11005-015-0747-0
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DOI: https://doi.org/10.1007/s11005-015-0747-0