This article is cited in 4 papers

Combinatorial Yang–Baxter maps arising from the tetrahedron equation

A. Kuniba

University of Tokyo, Tokyo, Japan

Abstract: We survey the matrix product solutions of the Yang–Baxter equation recently obtained from the tetrahedron equation. They form a family of quantum $\mathscr R$-matrices of generalized quantum groups interpolating the symmetric tensor representations of $U_q(A^{(1)}_{n-1})$ and the antisymmetric tensor representations of $U_{-q^{-1}}(A^{(1)}_{n-1})$. We show that at $q=0$, they all reduce to the Yang–Baxter maps called combinatorial $\mathscr R$-matrices and describe the latter by an explicit algorithm.

Keywords: tetrahedron equation, Yang–Baxter equation, generalized quantum group, Yang–Baxter map.

MSC: 17B37, 16T25, 16T30

DOI: 10.4213/tmf9037

 English version: Theoretical and Mathematical Physics, 2016, 189:1, 1472–1485

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