South Ural State University,

76 Lenina Ave.,

454080 Chelyabinsk,

RUSSIA

E-mail: kig@susu.ac.ru

**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.