Tetrahedron equation and generalized quantum groups

Abstract

We construct 2n-families of solutions of the YangBaxter equation from nproducts 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 1's and L's involved in the product, the trace construction interpolates the symmetric tensor representations of Uq (An-1(1)) and the antisymmetric tensor representations of U-q-1 (An-1(1)), whereas a boundary vector construction interpolates the q-oscillator representation of Uq (Dn 1) (2) + and the spin representation of Uq (Dn+1(2)) . The intermediate cases are associated with an affinization of quantum superalgebras.