Abstract
AbstractThis chapter presents the boundary vector reduction of an n-concatenation of the tetrahedron equation \(RRRR=RRRR\) of the 3D R. It generates infinite families of solutions to the Yang–Baxter equation in matrix product forms. In contrast with the boundary vector reduction starting from \(RLLL = LLLR\) (Chap. 12), they turn out to be the quantum R matrices of the q-oscillator representations of \(U_{q}(A^{(2)}_{2n})\), \(U_{q}(C^{(1)}_{n})\) and \(U_{q}(D^{(2)}_{n+1})\). These algebras have Dynkin diagrams with double arrows. It turns out that the two kinds of boundary vectors correspond to the two directions of the double arrows. For simplicity, we treat the reduction with respect to the third component only.
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