Integrability in $[1+d]$ dimensions: combined local equations and commutativity of the transfer matrices

Abstract

The general concept of integrability in the framework of the Algebraic Bethe Ansatz is discussed and combined non-homogeneous local integrability equations are proposed for general D dimensional vertex statistical models. For 2D the efficiency of the step by step consideration of the Ansatz avoiding the direct analysis of Yang-Baxter equations is demonstrated. Also 3D three-state R-matrices with certain 20-vertex structure are discussed and there are constructed some simple 3D vertex solutions. The simplified or combined versions of the 3D local integrability equations with four-state R-matrices are proposed. A new 3D version of the star-triangle equations ("connected") is constructed.