Tetrahedron equation and generalized quantum groups

Abstract

We construct $2^n$-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 $R$'s and $L$'s involved in the product, the trace construction interpolates the symmetric tensor representations of $U_q(A^{(1)}_{n-1})$ and the anti-symmetric tensor representations of $U_{-q^{-1}}(A^{(1)}_{n-1})$, whereas a boundary vector construction interpolates the $q$-oscillator representation of $U_q(D^{(2)}_{n+1})$ and the spin representation of $U_{-q^{-1}}(D^{(2)}_{n+1})$. The intermediate cases are associated with an affinization of quantum super algebras.