From coalgebra to bialgebra for the six-vertex model: the star-triangle relation as a necessary condition for commuting transfer matrices (Q398527)

Summary: Using the most elementary methods and considerations, the solution of the star-triangle condition \(\frac{a^2+b^2-c^2}{2ab}=\frac{(a')^2+(b')^2-(c')^2}{2a'b'}\) is shown to be a necessary condition for the extension of the operator coalgebra of the six-vertex model to a bialgebra. A portion of the bialgebra acts as a spectrum-generating algebra for the algebraic Bethe ansatz, with which higher-dimensional representations of the bialgebra can be constructed. The star-triangle relation is proved to be necessary for the commutativity of the transfer matrices \(T(a,b,c)\) and \(T(a',b',c')\).