Combinatorial Yang-Baxter maps arising from tetrahedron equation

Abstract

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

14 pages. For proceedings of Physics and Mathematics of Nonlinear Phenomena 2015 June 20-17, Gallipoli, Italy

References in corpus (3)

• Tetrahedron equation and quantum R matrices for $q$-oscillator representations of $U_q(A_{2n}^{(2)}),U_q(C_n^{(1)})$ and $U_q(D_{n+1}^{(2)})$
• Classical integrable field theories in discrete 2+1 dimensional space-time
• A polynomial formula for the solution of 3D reflection equation

Cited by in corpus (1)

• Solving the Yang-Baxter, tetrahedron and higher simplex equations using Clifford algebras