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Mathematical Proceedings of the Cambridge Philosophical Society Mathematical Proceedings of the Cambridge Philosophical Society

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Cohomologies of n-simplex relations

Published online by Cambridge University Press:  19 May 2016

IGOR G. KOREPANOV ,
GEORGY I. SHARYGIN  and
DMITRY V. TALALAEV
IGOR G. KOREPANOV
Affiliation:
Moscow Technological University, 20 Stromynka Str., Moscow 107996, Russia.
GEORGY I. SHARYGIN
Affiliation:
Institute of Theoretical and Experimental Physics, 25 Bolshaya Cheremushkinskaya Str., Moscow 117218, Russia Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 1 Leninskiye Gory, GSP-1, Moscow 119991, Russia.
DMITRY V. TALALAEV
Affiliation:
Institute of Theoretical and Experimental Physics, 25 Bolshaya Cheremushkinskaya Str., Moscow 117218, Russia Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 1 Leninskiye Gory, GSP-1, Moscow 119991, Russia.
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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 generalise 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 2 , September 2016 , pp. 203 - 222
Copyright
Copyright © Cambridge Philosophical Society 2016 

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