Parametric reflection maps: an algebraic approach
Anastasia Doikou, Marzia Mazzotta, Paola Stefanelli
TL;DR
This work constructs an algebraic framework for the parametric set-theoretic reflection equation by introducing left $p$-shelves and $p$-racks as parametric generalizations of shelves and racks. It establishes reflection criteria for maps associated with these structures, and provides algorithms to derive parametric reflections from ordinary shelves and racks. By leveraging admissible Drinfel'd twists, the authors generate generic parametric Yang–Baxter and reflection maps with simplified constraints, and they extend brace theory to parametric $p$-braces and skew $p$-braces to build rich solution families. The study also develops rack-based Yang–Baxter and reflection algebras in the parametric setting, including explicit $p$-rack reflection operators and their algebraic properties, linking to integrable boundary conditions in quantum and discrete systems.
Abstract
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.