Set-theoretical solutions of the pentagon equation on groups

2019/02/12 by Catino, Francesco, Mazzotta, Marzia, Miccoli, Maria Maddalena · 1 citation
#16W35 #20G42 #81R60 #FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1902.04310

Abstract

Let M be a set. A set-theoretical solution of the pentagon equation on M is a map s:M× M\longrightarrow M× M such that s₂₃ s₁₃ s₁₂=s₁₂ s₂₃, where s₁₂=s× id_M, s₂₃=id_M × s and s₁₃=(id_M × τ) s₁₂(id_M × τ), and τ is the flip map, i.e., the permutation on M× M given by τ(x,y)=(y,x), for all x,y∈ M. In this paper we give a complete description of the set-theoretical solutions of the form s(x,y)=(x⋅ y , x∗ y) when either (M,⋅) or (M,∗) is a group; moreover, we raise some questions.

Cited by

• Associative Pentagon Algebras

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