An elementary rebuilding of something and a corresponding algebraic relation

Abstract:
We know that Onsager solved the two-dimensional Ising model with the help of his star-triangle relation. This relation corresponds to what can be called an elementary rebuilding of a planar graph of Ising weights. Onsager's ideas were then developed into the notion of Yang-Baxter equation which describes some elementary rebuildings in mathematical physics as well as in knot theory.

In this report, I review some further equations of this sort. One example is functional tetrahedron, which generalizes Yang-Baxter to higher dimensions. One more example deals with Pachner moves - elementary rebuildings of a manifold triangulation. Here I propose algebraic formulas, originally inspired by Regge calculus, which imitate Pachner moves for three- and four-dimensional manifolds. In three dimensions, some of them can be related to a semiclassical limit of known quantum topological invariants.