Parametric reflection maps: an algebraic approach

Anastasia Doikou

doi:10.1080/00927872.2026.2621983

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Parametric reflection maps: an algebraic approach

2026-02-01

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Anastasia Doikou

Marzia Mazzotta

Paola Stefanelli *

DOI：10.1080/00927872.2026.2621983

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Abstract

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We study solutions of the parametric set-theoretic reflection equation from an algebraic perspective by employing recently derived generalizations of the familiar shelves and racks, called parametric (p)-shelves and racks. Generic invertible solutions of the set-theoretic reflection equation are also obtained by a suitable parametric twist. The twist leads to considerably simplified constraints compared to the ones obtained from general set-theoretic reflections. In this context, novel algebraic structures of (skew) p-braces that generalize the known (skew) braces and are suitable for the parametric Yang-Baxter equation are introduced. The p-rack Yang-Baxter and reflection operators as well as the associated algebraic structures are defined setting up the frame for formulating the p-rack reflection algebra.

Keywords:

Brace

rack

reflection equation

skew brace

Yang-Baxter equation