Colazzo, I., Jespers, E., Kubat, Ł., & Van Antwerpen, A. (2023). Structure algebras of finite set-theoretic solutions of the Yang–Baxter equation. arXiv.
@unpublished{230506023,
archiveprefix = {arXiv},
doi = {2305.06023},
author = {Colazzo, Ilaria and Jespers, Eric and Kubat, Łukasz and Van Antwerpen, Arne},
title = {Structure algebras of finite set-theoretic solutions of the {Y}ang--{B}axter equation},
year = {2023},
keywords = {arXiv},
publisher = {arXiv},
copyright = {arXiv.org perpetual, non-exclusive license}
}
Algebras related to finite bijective or idempotent left non-degenerate solutions (X,r) of the Yang–Baxter equation have been intensively studied. These are the monoid algebras K[M(X,r)] and K[A(X,r)], over a field K, of its structure monoid M(X,r) and left derived structure monoid A(X,r), which have quadratic defining relations.
In this paper we deal with arbitrary finite left non-degenerate solutions (X,r). Via divisibility by generators, i.e., the elements of X, we construct an ideal chain in M(X,r) that has very strong algebraic structural properties on its Rees factors. This allows to obtain characterizations of when the algebras K[M(X,r)] and K[A(X,r)] are left or right Noetherian. Intricate relationships between ring-theoretical and homological properties of these algebras and properties of the solution (X,r) are proven, which extends known results on bijective non-degenerate solutions. Furthermore, we describe the cancellative congruences of A(X,r) and M(X,r) as well as the prime spectrum of K[A(X,r)]. This then leads to an explicit formula for the Gelfand–Kirillov dimension of K[M(X,r)] in terms of the number of orbits in X under actions of certain finite monoids derived from (X,r). It is also shown that the former coincides with the classical Krull dimension of K[M(X,r)] in case the algebra K[M(X,r)] is left or right Noetherian. Finally, we obtain the first structural results for a class of finite degenerate solutions (X,r) of the form r(x,y)=(\lambda_x(y),ρ(y)) by showing that structure algebras of such solution are always right Noetherian.
Colazzo, I., Jespers, E., Kubat, Ł., & Van Antwerpen, A. (2023). Simple solutions of the Yang-Baxter equation. arXiv.
@unpublished{231209687,
archiveprefix = {arXiv},
doi = {2312.09687},
author = {Colazzo, Ilaria and Jespers, Eric and Kubat, Łukasz and Van Antwerpen, Arne},
title = {Simple solutions of the {Y}ang-{B}axter equation},
year = {2023},
keywords = {arXiv},
publisher = {arXiv},
copyright = {arXiv.org perpetual, non-exclusive license}
}
We present a characterization of simple finite non-degenerate bijective set-theoretic solutions of the Yang-Baxter equation in terms of the algebraic structure of the associated permutation skew left braces.
In particular, we prove that they need to have a unique minimal non-zero ideal and modulo this ideal one obtains a trivial skew left brace of cyclic type.
Accepted for publication
Colazzo, I., Ferrara, M., & Trombetti, M. (2022). On derived-indecomposable solutions of the Yang–Baxter equation. In accepted for publication in Publicacions Matemàtiques.
@unpublished{221008598,
doi = {2210.08598},
author = {Colazzo, Ilaria and Ferrara, Maria and Trombetti, Marco},
keywords = {accepted},
title = {On derived-indecomposable solutions of the Yang--Baxter equation},
journal = {accepted for publication in Publicacions Matemàtiques},
year = {2022},
copyright = {arXiv.org perpetual, non-exclusive license}
}
If (X,r) is a finite non-degenerate set-theoretic solution of the Yang–Baxter equation, the additive group of the structure skew brace G(X,r) is an FC-group, i.e. a group whose elements have finitely many conjugates. Moreover, its multiplicative group is virtually abelian, so it is also close to an FC-group itself. If one additionally assumes that the derived solution of (X,r) is indecomposable, then for every element b of G(X,r) there are finitely many elements of the form b∗c and c∗b, with c∈G(X,r). This naturally leads to the study of a brace-theoretic analogue of the class of FC-groups. For this class of skew braces, the fundamental results and their connections with the solutions of the YBE are described: we prove that they have good torsion and radical theories and they behave well with respect to certain nilpotency concepts and finite generation.
Published papers
Colazzo, I., Jespers, E., Kubat, Ł., Van Antwerpen, A., & Verwimp, C. (2023). Finite Idempotent Set-Theoretic Solutions of the Yang–Baxter Equation. International Mathematics Research Notices, rnad183. https://doi.org/10.1093/imrn/rnad183
@article{221207361,
author = {Colazzo, Ilaria and Jespers, Eric and Kubat, Łukasz and Van Antwerpen, Arne and Verwimp, Charlotte},
title = {{Finite Idempotent Set-Theoretic Solutions of the Yang–Baxter Equation}},
journal = {International Mathematics Research Notices},
pages = {rnad183},
year = {2023},
month = aug,
issn = {1073-7928},
doi = {10.1093/imrn/rnad183},
url = {https://doi.org/10.1093/imrn/rnad183},
eprint = {https://academic.oup.com/imrn/advance-article-pdf/doi/10.1093/imrn/rnad183/51112327/rnad183.pdf}
}
It is proven that finite idempotent left non-degenerate set-theoretic solutions \\(X,r)\ of the Yang–Baxter equation on a set \\X\ are determined by a left simple semigroup structure on \\X\ (in particular, a finite union of isomorphic copies of a group) and some maps \\q\ and \\\\varphi _\{x\}\ on \\X\ for \\x\∈X\\. This structure turns out to be a group precisely when the associated Yang–Baxter monoid \\M(X,r)\ is cancellative and all the maps \\\\varphi _\{x\}\ are equal to an automorphism of this group. Equivalently, the Yang–Baxter algebra \\K[M(X,r)]\ is right Noetherian, or in characteristic zero it has to be semiprime. The Yang–Baxter algebra is always a left Noetherian representable algebra of Gelfand–Kirillov dimension one. To prove these results, it is shown that the Yang–Baxter semigroup \\S(X,r)\ has a decomposition in finitely many cancellative semigroups \\S_\{u\}\ indexed by the diagonal, each \\S_\{u\}\ has a group of quotients \\G_\{u\}\ that is finite-by-(infinite cyclic) and the union of these groups carries the structure of a left simple semigroup. The case that \\X\ equals the diagonal is fully described by a single permutation on \\X\\.
Colazzo, I., Jespers, E., Van Antwerpen, A., & Verwimp, C. (2022). Left non-degenerate set-theoretic solutions of the
Yang-Baxter equation and semitrusses. J. Algebra, 610, 409–462. https://doi.org/10.1016/j.jalgebra.2022.07.019
@article{MR4466104,
author = {Colazzo, I. and Jespers, E. and Van Antwerpen, A. and Verwimp, C.},
title = {Left non-degenerate set-theoretic solutions of the
{Y}ang-{B}axter equation and semitrusses},
journal = {J. Algebra},
fjournal = {Journal of Algebra},
volume = {610},
year = {2022},
pages = {409--462},
issn = {0021-8693},
mrclass = {16T25 (16S36 16W22 16Y99 20M25)},
mrnumber = {4466104},
doi = {10.1016/j.jalgebra.2022.07.019},
url = {https://doi.org/10.1016/j.jalgebra.2022.07.019}
}
Catino, F., Colazzo, I., & Stefanelli, P. (2021). Set-theoretic solutions to the Yang-Baxter equation and
generalized semi-braces. Forum Math., 33(3), 757–772. https://doi.org/10.1515/forum-2020-0082
@article{MR4250483,
author = {Catino, Francesco and Colazzo, Ilaria and Stefanelli, Paola},
title = {Set-theoretic solutions to the {Y}ang-{B}axter equation and
generalized semi-braces},
journal = {Forum Math.},
fjournal = {Forum Mathematicum},
volume = {33},
year = {2021},
number = {3},
pages = {757--772},
issn = {0933-7741},
mrclass = {16T25 (16N20 16Y99 81R50)},
mrnumber = {4250483},
mrreviewer = {Run-Qiang Jian},
doi = {10.1515/forum-2020-0082},
url = {https://doi.org/10.1515/forum-2020-0082}
}
Colazzo, I., & Van Antwerpen, A. (2021). The algebraic structure of left semi-trusses. J. Pure Appl. Algebra, 225(2), Paper No. 106467, 15. https://doi.org/10.1016/j.jpaa.2020.106467
@article{MR4125548,
author = {Colazzo, Ilaria and Van Antwerpen, Arne},
title = {The algebraic structure of left semi-trusses},
journal = {J. Pure Appl. Algebra},
fjournal = {Journal of Pure and Applied Algebra},
volume = {225},
year = {2021},
number = {2},
pages = {Paper No. 106467, 15},
issn = {0022-4049},
mrclass = {16Y99 (16T25 20M17)},
mrnumber = {4125548},
mrreviewer = {Abdelkader Ben Hassine},
doi = {10.1016/j.jpaa.2020.106467},
url = {https://doi.org/10.1016/j.jpaa.2020.106467}
}
Catino, F., Colazzo, I., & Stefanelli, P. (2020). The matched product of the solutions to the Yang-Baxter
equation of finite order. Mediterr. J. Math., 17(2), Paper No. 58, 22. https://doi.org/10.1007/s00009-020-1483-y
@article{MR4067191,
author = {Catino, Francesco and Colazzo, Ilaria and Stefanelli, Paola},
title = {The matched product of the solutions to the {Y}ang-{B}axter
equation of finite order},
journal = {Mediterr. J. Math.},
fjournal = {Mediterranean Journal of Mathematics},
volume = {17},
year = {2020},
number = {2},
pages = {Paper No. 58, 22},
issn = {1660-5446},
mrclass = {16T25 (16N20 16Y99 81R50)},
mrnumber = {4067191},
mrreviewer = {\L ukasz Kubat},
doi = {10.1007/s00009-020-1483-y},
url = {https://doi.org/10.1007/s00009-020-1483-y}
}
Colazzo, I., Jespers, E., & Kubat, Ł. (2020). Set-theoretic solutions of the pentagon equation. Comm. Math. Phys., 380(2), 1003–1024. https://doi.org/10.1007/s00220-020-03862-6
@article{MR4170296,
author = {Colazzo, Ilaria and Jespers, Eric and Kubat, \L ukasz},
title = {Set-theoretic solutions of the pentagon equation},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {380},
year = {2020},
number = {2},
pages = {1003--1024},
issn = {0010-3616},
mrclass = {20K10 (16T25)},
mrnumber = {4170296},
mrreviewer = {Jo\~{a}o Matheus Jury Giraldi},
doi = {10.1007/s00220-020-03862-6},
url = {https://doi.org/10.1007/s00220-020-03862-6}
}
Catino, F., Colazzo, I., & Stefanelli, P. (2020). The matched product of set-theoretical solutions of the
Yang-Baxter equation. J. Pure Appl. Algebra, 224(3), 1173–1194. https://doi.org/10.1016/j.jpaa.2019.07.012
@article{MR4009573,
author = {Catino, Francesco and Colazzo, Ilaria and Stefanelli, Paola},
title = {The matched product of set-theoretical solutions of the
{Y}ang-{B}axter equation},
journal = {J. Pure Appl. Algebra},
fjournal = {Journal of Pure and Applied Algebra},
volume = {224},
year = {2020},
number = {3},
pages = {1173--1194},
issn = {0022-4049},
mrclass = {16T25 (16N20 16Y99 81R50)},
mrnumber = {4009573},
mrreviewer = {Jo\~{a}o Matheus Jury Giraldi},
doi = {10.1016/j.jpaa.2019.07.012},
url = {https://doi.org/10.1016/j.jpaa.2019.07.012}
}
Catino, F., Colazzo, I., & Stefanelli, P. (2019). Skew left braces with non-trivial annihilator. J. Algebra Appl., 18(2), 1950033, 23. https://doi.org/10.1142/S0219498819500336
@article{MR3917122,
author = {Catino, Francesco and Colazzo, Ilaria and Stefanelli, Paola},
title = {Skew left braces with non-trivial annihilator},
journal = {J. Algebra Appl.},
fjournal = {Journal of Algebra and its Applications},
volume = {18},
year = {2019},
number = {2},
pages = {1950033, 23},
issn = {0219-4988},
mrclass = {16T25 (16N20 16Y99 20B15 20G99 81R50)},
mrnumber = {3917122},
mrreviewer = {Loic Foissy},
doi = {10.1142/S0219498819500336},
url = {https://doi.org/10.1142/S0219498819500336}
}
Catino, F., Colazzo, I., & Stefanelli, P. (2017). Semi-braces and the Yang-Baxter equation. J. Algebra, 483, 163–187. https://doi.org/10.1016/j.jalgebra.2017.03.035
@article{MR3649817,
author = {Catino, Francesco and Colazzo, Ilaria and Stefanelli, Paola},
title = {Semi-braces and the {Y}ang-{B}axter equation},
journal = {J. Algebra},
fjournal = {Journal of Algebra},
volume = {483},
year = {2017},
pages = {163--187},
issn = {0021-8693},
mrclass = {16T25 (16N20 16Y99 81R50)},
mrnumber = {3649817},
mrreviewer = {Leandro Vendramin},
doi = {10.1016/j.jalgebra.2017.03.035},
url = {https://doi.org/10.1016/j.jalgebra.2017.03.035}
}
Catino, F., Colazzo, I., & Stefanelli, P. (2016). Regular subgroups of the affine group and asymmetric product
of radical braces. J. Algebra, 455, 164–182. https://doi.org/10.1016/j.jalgebra.2016.01.038
@article{MR3478858,
author = {Catino, Francesco and Colazzo, Ilaria and Stefanelli, Paola},
title = {Regular subgroups of the affine group and asymmetric product
of radical braces},
journal = {J. Algebra},
fjournal = {Journal of Algebra},
volume = {455},
year = {2016},
pages = {164--182},
issn = {0021-8693},
mrclass = {16Y99 (20B15)},
mrnumber = {3478858},
mrreviewer = {Leandro Vendramin},
doi = {10.1016/j.jalgebra.2016.01.038},
url = {https://doi.org/10.1016/j.jalgebra.2016.01.038}
}
Catino, F., Colazzo, I., & Stefanelli, P. (2015). On regular subgroups of the affine group. Bull. Aust. Math. Soc., 91(1), 76–85. https://doi.org/10.1017/S000497271400077X
@article{MR3294261,
author = {Catino, Francesco and Colazzo, Ilaria and Stefanelli, Paola},
title = {On regular subgroups of the affine group},
journal = {Bull. Aust. Math. Soc.},
fjournal = {Bulletin of the Australian Mathematical Society},
volume = {91},
year = {2015},
number = {1},
pages = {76--85},
issn = {0004-9727},
mrclass = {20N99 (16Y30)},
mrnumber = {3294261},
mrreviewer = {Elena Zizioli},
doi = {10.1017/S000497271400077X},
url = {https://doi.org/10.1017/S000497271400077X}
}
Other publications
Brzeziński, T., Colazzo, I., Doikou, A., & Vendramin, L. (2023). Mini-workshop: Skew Braces and the Yang-Baxter Equation. In Oberwolfach Reports (No.1; Vol. 20, Number 1, pp. 536–567).
@report{oberwolfach,
title = {Mini-workshop: Skew Braces and the Yang-Baxter Equation},
author = {Brzeziński, Tomasz and Colazzo, Ilaria and Doikou, Anastasia and Vendramin, Leandro},
journal = {Oberwolfach Reports},
volume = {20},
number = {1},
pages = {536--567},
year = {2023},
doi = {10.4171/OWR/2023/9}
}
The workshop was focused on the interplay between set-theoretic solutions to the Yang–Baxter equation and several algebraic structures used to construct and understand new solutions. In this vein, the YBE and properties of these algebraic structures are used as a source of inspiration to study other mathematical problems not directly related to the YBE.
Colazzo, I. (2022). Braces: between regular subgroups and solutions of the Yang-Baxter equation. In Algebra for cryptography. With a preface by Massimiliano Sala (pp. 35–38). Rome: Aracne Editrice.
@report{extended_abstract,
author = {Colazzo, Ilaria},
title = {Braces: between regular subgroups and solutions of the {Yang}-{Baxter} equation},
booktitle = {Algebra for cryptography. With a preface by Massimiliano Sala},
isbn = {979-12-5994-328-6; 979-12-5994-622-5},
pages = {35--38},
year = {2022},
publisher = {Rome: Aracne Editrice},
language = {English},
keywords = {16T25},
zbmath = {7623857}
}
Colazzo, I. (2017). Left semi-braces and the Yang-Baxter equation. Università del Salento; Ph.D. Thesis.
@thesis{Colazzo_thesis,
author = {Colazzo, I.},
title = {Left semi-braces and the Yang-Baxter equation},
year = {2017},
school = {Università del Salento},
note = {Ph.D. Thesis},
file = {thesis-colazzo.pdf}
}
This thesis focuses on the new algebraic structure: semi-brace. We study basic properties of this structure and we show that semi-braces are a generalization of braces. Moreover we introduce new constructions of semi-braces, the asymmetric product and the matched product, in order to obtain several examples of semi- braces. Finally, we prove that we may construct left non-degenerate solutions of the Yang-Baxter equation through left semi-braces.