Cohomologies ofn-simplex relations

Abstract

A theory of (co)homologies related to set-theoretic n-simplex relations is constructed in analogy with the known quandle and Yang-Baxter (co)homologies, with emphasis made on the tetrahedron case. In particular, this permits us to generalize Hietarinta's idea of "permutation-type" solutions to the quantum (or "tensor") n-simplex equations. Explicit examples of solutions to the tetrahedron equation involving nontrivial cocycles are presented.

Simplex and Polygon Equations

“…The Yang-Baxter equation is a member of a family, called simplex equations [18] (also see, e.g., [39,68,74,75,77]). The N -simplex equation is an equation imposed on a mapR : V ⊗N → V ⊗N , respectivelyR : U N → U N for the set-theoretical version.…”

Zamolodchikov Tetrahedral Equation and Higher Hamiltonians of 2d Quantum Integrable Systems

“…It turns out that the condition of equivalence of the coloring obtained by these two ways is equivalent to the tetrahedral equation on Φ. The problem is described in more details in [9].…”

“…The N -cube 2-faces coloring problem allows us to construct a complex analogous to those calculating the Yang-Baxter cohomology for the case of set-theoretic tetrahedral equations in [9]. The 3-cocycles of the complex play a special role in this subject, they are determined by the condition ϕ(a 1 , a 2 , a 3 )ϕ(a 1 , a 4 , a 5 )ϕ(a 2 , a 4 , a 6 )ϕ(a 3 , a 5 , a 6 ) = ϕ(a 3 , a 5 , a 6 )ϕ(a 2 , a 4 , a 6 )ϕ(a 1 , a 4 , a 5 )ϕ(a 1 , a 2 , a 3 ) in the notation of Fig.…”

“…To construct such matrix solution from the electric solution for the FTE one should realize the finite-dimensional reduction of the latter. In [9,Lemma 6] it is demonstrated that the electric solution reduces correctly to the set X obtained as follows: consider the residue ring Z/p k Z, where p is either a prime number of the form p = 4l + 1, or p = 2, and k is an integer ≥ 2 (the Legendre symbol − 1 p equals 1). We fix one square root of −1 and call it ε ∈ Z/pZ.…”

Towards integrable structure in 3d Ising model

“…The main part of the work involves the combinatorics of the n-simplicial complex [7]. We first establish a recursion procedure on the spaces of solutions for the n-simplex equation.…”