New solutions to the tetrahedron equation associated with quantized six-vertex models

Abstract

We present a family of new solutions to the tetrahedron equation of the form $RLLL=LLLR$, where $L$ operator may be regarded as a quantized six-vertex model whose Boltzmann weights are specific representations of the $q$-oscillator or $q$-Weyl algebras. When the three $L$'s are associated with the $q$-oscillator algebra, $R$ coincides with the known intertwiner of the quantized coordinate ring $A_q(sl_3)$. On the other hand, $L$'s based on the $q$-Weyl algebra lead to new $R$'s whose elements are either factorized or expressed as a terminating $q$-hypergeometric type series.