My research

My research lies at the intersection of non-commutative algebra, group theory, and mathematical physics. I focus on algebraic structures, such as skew braces and their generalisations, which play a key role in equations like the Yang-Baxter and Pentagon equations. In addition, I use computer algebra systems, particularly GAP, to explore and analyse these structures, providing computational insights that complement the theoretical aspects of my work.

Keywords:

Set–theoretical solutions of the Yang–Baxter equation, Skew braces, trusses and generalisations, Set–theoretical solutions of the pentagon equation, Regular subgroups of the holomorph, Hopf–Galois extensions

Preprints

1. Colazzo, I., & Antwerpen, A. V. (2024). On the cabling of non-involutive set-theoretic solutions of the Yang–Baxter equation. arXiv.
   ARXIV BIB ABSTRACT

   @unpublished{241023821,
     title = {On the cabling of non-involutive set-theoretic solutions of the Yang--Baxter equation},
     author = {Colazzo, Ilaria and Antwerpen, Arne Van},
     year = {2024},
     archiveprefix = {arXiv},
     publisher = {arXiv},
     copyright = {arXiv.org perpetual, non-exclusive license},
     keywords = {arXiv},
     doi = {2410.23821}
   }
   

   In this paper, we propose an extension of the cabling methods to bijective non-degenerate solutions of the Yang–Baxter equation, with applications to indecomposable and simple solutions. We address two main challenges in extending this technique to the non-involutive case. First, we establish that the indecomposability of a solution can be assessed through its injectivization or the associated biquandle. Second, we show that the canonical embedding into the structure monoid resolves issues related to the pullback of subsolutions. Our results extend the theorems of Lebed, Ramirez and Vendramin to non-involutive solutions also providing numerical criteria for indecomposability.

2. Colazzo, I., Okniński, J., & Van Antwerpen, A. (2024). Bijective solutions to the Pentagon Equation. arXiv.
   ARXIV BIB ABSTRACT

   @unpublished{240520406,
     archiveprefix = {arXiv},
     doi = {2405.20406},
     author = {Colazzo, Ilaria and Okniński, Jan and Van Antwerpen, Arne},
     title = {Bijective solutions to the Pentagon Equation},
     year = {2024},
     keywords = {arXiv},
     publisher = {arXiv},
     copyright = {arXiv.org perpetual, non-exclusive license}
   }
   

   A complete classification of all finite bijective set-theoretic solutions (S,s) to the Pentagon Equation is obtained. First, it is shown that every such a solution determines a semigroup structure on the set S that is the direct product E \times G of a semigroup of left zeros E and a group G. Next, we prove that this leads to a decomposition of the set S as a Cartesian product X\times A\times G, for some sets X, A and to a discovery of a hidden group structure on A. Then an unexpected structure of a matched product of groups A, G is found such that the solution (S,s) can be explicitly described as a lift of a solution determined on the set A\times G by this matched product of groups. Conversely, every matched product of groups leads to a family of solutions arising in this way. Moreover, a simple criterion for the isomorphism of two solutions is obtained. The results provide a far reaching extension of the results of Colazzo, Jespers and Kubat, dealing with the special case of the so called involutive solutions. Connections to the solutions to the Yang–Baxter equation and to the theory of skew braces are derived.

3. Colazzo, I., Jespers, E., Kubat, Ł., & Van Antwerpen, A. (2024). Simple solutions of the Yang-Baxter equation. arXiv.
   ARXIV BIB ABSTRACT

   @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 = {2024},
     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. 

4. Colazzo, I., Jespers, E., Kubat, Ł., & Van Antwerpen, A. (2023). Structure algebras of finite set-theoretic solutions of the Yang–Baxter equation. arXiv.
   ARXIV BIB ABSTRACT

   @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.

Upcoming talks

1. Colazzo, I. (2025-07-11). Matched pairs of groups and the combinatorial solutions to the pentagon equation. 15th Ukraine Algebra Conference; Lviv (Ukraine) - 08-11/07/2025.