Parametric reflection maps: an algebraic approach

TL;DR

This paper introduces a new algebraic framework for parametric reflection maps using generalized shelves and racks, leading to simplified solutions and novel algebraic structures like p-braces for the parametric Yang-Baxter equation.

Contribution

It develops the theory of parametric (p)-shelves and racks, introduces p-braces, and constructs algebraic structures for the parametric reflection equation, advancing algebraic understanding in this area.

Findings

Derived generic invertible solutions via parametric twist.

Introduced (skew) p-braces generalizing known braces.

Defined p-rack Yang-Baxter and reflection operators.

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.