Set-Theoretic Solutions of the Pentagon Equation

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Abstract

A set-theoretic solution of the Pentagon Equation on a non-empty set S is a map s: S2→ S2 such that s23s13s12= s12s23, where s12=s×id, s23=id×s and s13=(τ×id)(id×s)(τ×id) are mappings from S3 to itself and τ: S2→ S2 is the flip map, i.e., τ(x, y) = (y, x). We give a description of all involutive solutions, i.e., s2=id. It is shown that such solutions are determined by a factorization of S as direct product X× A× G and a map σ:A→Sym(X), where X is a non-empty set and A, G are elementary abelian 2-groups. Isomorphic solutions are determined by the cardinalities of A, G and X, i.e., the map σ is irrelevant. In particular, if S is finite of cardinality 2 n(2 m+ 1) for some n, m⩾ 0 then, on S, there are precisely (n+22) non-isomorphic solutions of the Pentagon Equation.

Original languageEnglish
Pages (from-to)1003-1024
Number of pages22
JournalCommunications in Mathematical Physics
Volume380
Issue number2
DOIs
Publication statusPublished - Dec 2020

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