Cohomologies of $n$-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.

25 pages, 4 figures; v3: added the proof of nontriviality in Example 2, formulas in this example corrected

References in corpus (1)

• An integrable 3D lattice model with positive Boltzmann weights

Cited by in corpus (7)

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• Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Nonconstant hexagon relations and their cohomology
• Grassmannian-parameterized solutions to direct-sum polygon and simplex equations
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Set-theoretical solutions to the Zamolodchikov tetrahedron equation on associative rings and Liouville integrability
• Odd-gon relations and their cohomology