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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