### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 067, 23 pages      arXiv:1803.01586      https://doi.org/10.3842/SIGMA.2018.067

### Tetrahedron Equation and Quantum $R$ Matrices for $q$-Oscillator Representations Mixing Particles and Holes

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

Received March 15, 2018, in final form June 23, 2018; Published online July 04, 2018

Abstract
We construct $2^n+1$ solutions to the Yang-Baxter equation associated with the quantum affine algebras $U_q\big(A^{(1)}_{n-1}\big)$, $U_q\big(A^{(2)}_{2n}\big)$, $U_q\big(C^{(1)}_n\big)$ and $U_q\big(D^{(2)}_{n+1}\big)$. They act on the Fock spaces of arbitrary mixture of particles and holes in general. Our method is based on new reductions of the tetrahedron equation and an embedding of the quantum affine algebras into $n$ copies of the $q$-oscillator algebra which admits an automorphism interchanging particles and holes.

Key words: tetrahedron equation; Yang-Baxter equation; quantum groups; $q$-oscillator representations.

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References

1. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982.
2. Bazhanov V.V., Mangazeev V.V., Sergeev S.M., Quantum geometry of 3-dimensional lattices, J. Stat. Mech. Theory Exp. 2008 (2008), P07004, 27 pages, arXiv:0801.0129.
3. Bazhanov V.V., Sergeev S.M., Zamolodchikov's tetrahedron equation and hidden structure of quantum groups, J. Phys. A: Math. Gen. 39 (2006), 3295-3310, hep-th/0509181.
4. Drinfel'd V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
5. Hayashi T., $q$-analogues of Clifford and Weyl algebras - spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys. 127 (1990), 129-144.
6. Jimbo M., A $q$-difference analogue of $U({\mathfrak g})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69.
7. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
8. Kapranov M.M., Voevodsky V.A., $2$-categories and Zamolodchikov tetrahedra equations, in Algebraic Groups and their Generalizations: Quantum and Infinite-Dimensional Methods (University Park, PA, 1991), Proc. Sympos. Pure Math., Vol. 56, Amer. Math. Soc., Providence, RI, 1994, 177-259.
9. Kashaev R.M., Volkov A.Yu., From the tetrahedron equation to universal $R$-matrices, in L.D. Faddeev's Seminar on Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 201, Amer. Math. Soc., Providence, RI, 2000, 79-89, math.QA/9812017.
10. Kuniba A., Combinatorial Yang-Baxter maps arising from tetrahedron equation, Theoret. and Math. Phys. 189 (2016), 1472-1485, arXiv:1509.02245.
11. Kuniba A., Okado M., Tetrahedron and 3D reflection equations from quantized algebra of functions, J. Phys. A: Math. Theor. 45 (2012), 465206, 27 pages, arXiv:1208.1586.
12. Kuniba A., Okado M., Tetrahedron equation and quantum $R$ matrices for $q$-oscillator representations of $U_q\big(A^{(2)}_{2n}\big)$, $U_q\big(C^{(1)}_n\big)$ and $U_q\big(D^{(2)}_{n+1}\big)$, Comm. Math. Phys. 334 (2015), 1219-1244, arXiv:1311.4258.
13. Kuniba A., Okado M., Sergeev S., Tetrahedron equation and generalized quantum groups, J. Phys. A: Math. Theor. 48 (2015), 304001, 38 pages, arXiv:1503.08536.
14. Kuniba A., Okado M., Sergeev S., Tetrahedron equation and quantum $R$ matrices for modular double of $U_q\big(D^{(2)}_{n+1}\big)$, $U_q \big(A^{(2)}_{2n}\big)$ and $U_q\big(C^{(1)}_n\big)$, Lett. Math. Phys. 105 (2015), 447-461, arXiv:1409.1986.
15. Kuniba A., Okado M., Yamada Y., Box-ball system with reflecting end, J. Nonlinear Math. Phys. 12 (2005), 475-507, nlin.SI/0411044.
16. Kuniba A., Pasquier V., Matrix product solutions to the reflection equation from three dimensional integrability, J. Phys. A: Math. Theor. 51 (2018), 255204, 26 pages, arXiv:1802.09164.
17. Kuniba A., Sergeev S., Tetrahedron equation and quantum $R$ matrices for spin representations of $B^{(1)}_n$, $D^{(1)}_n$ and $D^{(2)}_{n+1}$, Comm. Math. Phys. 324 (2013), 695-713, arXiv:1203.6436.
18. Sergeev S.M., Two-dimensional $R$-matrices - descendants of three-dimensional $R$-matrices, Modern Phys. Lett. A 12 (1997), 1393-1410.
19. Sergeev S.M., Supertetrahedra and superalgebras, J. Math. Phys. 50 (2009), 083519, 21 pages.
20. Zamolodchikov A.B., Tetrahedra equations and integrable systems in three-dimensional space, Soviet Phys. JETP 52 (1980), 325-336.