### Abstract

It is shown that square free set theoretic involutive non-degenerate solutions of the Yang-Baxter equation whose associated permutation group (referred to as an involutive Yang-Baxter group) is abelian are retractable in the sense of Etingof, Schedler and Soloviev. This solves a problem of Gateva-Ivanova in the case of abelian IYB groups. It also implies that the corresponding finitely presented abelian-by-finite groups (called the structure groups) are poly Z groups. Secondly, an example of a solution with an abelian involutive Yang-Baxter group which is not a generalized twisted union is constructed. This answers in the negative another problem of Gateva-Ivanova. The constructed solution is of multipermutation level 3. Retractability of solutions is also proved in the case where the natural generators of the IYB group are cyclic permutations. Moreover, it is shown that such solutions are generalized twisted unions. © 2010 Elsevier Inc.

Original language | English |
---|---|

Pages (from-to) | 2472-2484 |

Journal | Advances in Mathematics |

Volume | 224 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Aug 2010 |

### Keywords

- Group of I-type
- Multipermutation solution
- Permutation group
- Set theoretic solution
- Yang-Baxter equation

## Fingerprint Dive into the research topics of 'Retractability of set theoretic solutions of the Yang-Baxter equation'. Together they form a unique fingerprint.

## Cite this

Cedó, F., Jespers, E., & Okniński, J. (2010). Retractability of set theoretic solutions of the Yang-Baxter equation.

*Advances in Mathematics*,*224*(6), 2472-2484. https://doi.org/10.1016/j.aim.2010.02.001