A unified approach is applied in the consideration of the multi-parametric (colored) Yang-Baxter equations (YBE) and the usual YBE with two-parametric R-matrices, relying on the existence of the arbitrary functions in the general solutions. The colored YBE are considered with the R-matrices defined on two and three dimensional states. We present an exhaustive study and the overall solutions for the YBE with 4 × 4 colored R-matrices. The established classification includes new multi-parametric free fermionic solutions. In the context of the given approach there are obtained the colored solutions to the YBE with 9 × 9 R-matrices having 15 non-zero elements.

We identify three-state Potts paramagnets with gapless edge modes on a triangular lattice protected by (×Z 3 ) 3 ≡ Z 3 × Z 3 × Z 3 symmetry and smaller Z 3 symmetry. We study microscopic models for the gapless edge and discuss the corresponding conformal field theories and their central charges. These are the edge states of s = 1 paramagnets protected by Z 3 ×Z 3 ×Z 3 and Z 3 symmetries, in analogy with the free fermion XX model edge discussed by Levin and Gu for the spin-1/2 Z 2 Ising paramagnet. We show that in spin-1 magnets, there are numerous options to obtain self-dual Hamiltonians and gapless edge modes. These models form the basis for the realization of a variety of symmetry-protected topological phases, opening a window for studies of the unconventional quantum criticalities between them.

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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.

We consider the mutual commutativity of Dunkl operators of the rational Calogero model as zero-curvature condition and calculate the non-local operator, related to these flat connections. This operator has physical meaning of particular scattering matrix of Calogero model and maps the eigenfunctions of Dunkl operator to the wave function of N free particles (plane waves).