Reflection equation as a tool for studying solutions to the Yang-Baxter equation
| Autor: | Leandro Vendramin, Victoria Lebed |
|---|---|
| Přispěvatelé: | Mathematics |
| Rok vydání: | 2020 |
| Předmět: |
Monoid
Reflection formula Pure mathematics Algebra and Number Theory Yang–Baxter equation 010102 general mathematics Braid group Structure (category theory) Group Theory (math.GR) 01 natural sciences Mathematics - Quantum Algebra 0103 physical sciences Bijection FOS: Mathematics Quantum Algebra (math.QA) 010307 mathematical physics math.GR 0101 mathematics Mathematics - Group Theory Mathematics math.QA |
| DOI: | 10.48550/arxiv.2008.01752 |
| Popis: | Given a right-non-degenerate set-theoretic solution $(X,r)$ to the Yang-Baxter equation, we construct a whole family of YBE solutions $r^{(k)}$ on $X$ indexed by its reflections $k$ (i.e., solutions to the reflection equation for $r$). This family includes the original solution and the classical derived solution. All these solutions induce isomorphic actions of the braid group/monoid on $X^n$. The structure monoids of $r$ and $r^{(k)}$ are related by an explicit bijective $1$-cocycle-like map. We thus turn reflections into a tool for studying YBE solutions, rather than a side object of study. In a different direction, we study the reflection equation for non-degenerate involutive YBE solutions, show it to be equivalent to (any of the) three simpler relations, and deduce from the latter systematic ways of constructing new reflections. Comment: 18 pages, 12 figures. Final version |
| Databáze: | OpenAIRE |
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