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 $U_q(A_{n-1}^{(1)})$ and the anti-symmetric tensor representations of $U_{-q^{-1}}(A_{n-1}^{(1)})$, whereas a boundary vector construction interpolates the $q$-oscillator representation of $U_q(D_{n+1}^{(2)})$ and the spin representation of $U_{-q^{-1}}(D_{n+1}^{(2)})$. The intermediate cases are associated with an affinization of quantum super algebras.

28 pages. Minor typo in Prop 2.1 fixed