Tetrahedron equation and generalized quantum groups

We construct -families of solutions of the Yang–Baxter equation from n-products of three-dimensional R and L operators satisfying the tetrahedron equation. They are identified with the quantum R matrices for the Hopf algebras known as generalized quantum groups. Depending on the number of Rʼs and Lʼs involved in the product, the trace construction interpolates the symmetric tensor representations of and the antisymmetric tensor representations of , whereas a boundary vector construction interpolates the q-oscillator representation of and the spin representation of . The intermediate cases are associated with an affinization of quantum superalgebras.