PreprintarXiv licencearXiv
Dual Reflection Groups from Solutions to the Yang--Baxter Equation
Abstract
Suppose that r: X2 → X2 is a nondegenerate, involutive solution to the set-theoretic Yang--Baxter equation. To this data, we can associate an Artin--Schelter regular algebra A and an infinite family of groups Gk, where k geqslant 1. For each k, we show that Gk is a dual reflection group for A, in the sense the component of A graded by 1Gk is again Artin--Schelter regular. We also show that, using the additional structure afforded by the underlying solution to the Yang--Baxter equation, one can determine the centre, Koszul complex, and associated twisted superpotential for A.
Related
- Gaussian solutions to the Yang--Baxter equation and their twists
- Centralizers of maps as automorphism groups of Yang--Baxter solutions
- Dual reflection groups for three-dimensional Artin--Schelter regular algebras
- NS-Leibniz bialgebras, the Yang--Baxter equation and relative Rota--Baxter operators
- Rota---Baxter operators on extraspecial groups
BibTeX
@misc{crawford2026dual,
title = {{Dual Reflection Groups from Solutions to the Yang--Baxter Equation}},
author = {Simon Crawford},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26042},
eprint = {2610.26042},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26042},
}
RIS
TY - UNPB
TI - Dual Reflection Groups from Solutions to the Yang--Baxter Equation
AU - Simon Crawford
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26042
UR - https://arxiv.org/abs/2610.26042
AB - Suppose that r: X^2 → X^2 is a nondegenerate, involutive solution to the set-theoretic Yang--Baxter equation. To this data, we can associate an Artin--Schelter regular algebra A and an infinite family of groups G_k, where k geqslant 1. For each k, we show that G_k is a dual reflection group for A, in the sense the component of A graded by 1_G_k is again Artin--Schelter regular. We also show that, using the additional structure afforded by the underlying solution to the Yang--Baxter equation, one can determine the centre, Koszul complex, and associated twisted superpotential for A.
ER -