L. Vendramin’s research while affiliated with Vrije Universiteit Brussel and other places

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Publications (92)


Fig. 1. Comparing the runtimes of the implementation of AMV22 and our approaches building on SAT Modulo Symmetries.
Incremental SAT-Based Enumeration of Solutions to the Yang-Baxter Equation
  • Preprint
  • File available

January 2025

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5 Reads

Daimy Van Caudenberg

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Leandro Vendramin

We tackle the problem of enumerating set-theoretic solutions to the Yang-Baxter equation. This equation originates from statistical and quantum mechanics, but also has applications in knot theory, cryptography, quantum computation and group theory. Non-degenerate, involutive solutions have been enumerated for sets up to size 10 using constraint programming with partial static symmetry breaking; for general non-involutive solutions, a similar approach was used to enumerate solutions for sets up to size 8. In this paper, we use and extend the SAT Modulo Symmetries framework (SMS), to expand the boundaries for which solutions are known. The SMS framework relies on a minimality check; we present two solutions to this, one that stays close to the original one designed for enumerating graphs and a new incremental, SAT-based approach. With our new method, we can reproduce previously known results much faster and also report on results for sizes that have remained out of reach so far. This is an extended version of a paper to appear in the proceedings of the 31st International Conference on Tools and Algorithms for the Construction and Analysis of Systems.

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Pointed Hopf algebras of odd dimension and Nichols algebras over solvable groups

November 2024

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8 Reads

We classify finite-dimensional Nichols algebras of Yetter-Drinfeld modules with indecomposable support over finite solvable groups in characteristic 0, using a variety of methods including reduction to positive characteristic. As a consequence, all Nichols algebras over groups of odd order are of diagonal type, which allows us to describe all pointed Hopf algebras of odd dimension.


Mini-Workshop: Bridging Number Theory and Nichols Algebras via Deformations

September 2024

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13 Reads

Oberwolfach Reports

Nichols algebras are graded Hopf algebra objects in braided tensor categories. They appeared first in a paper by Nichols in 1978 in the search for new examples of Hopf algebras. Rediscovered later several times, they also provide a conceptual explanation of the construction of quantum groups. The aim of the workshop is to review recent developments in the field, initiate collaborations, and discuss new approaches to open problems.


Hopf formulae for homology of skew braces

September 2024

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18 Reads

The variety of skew braces contains several interesting subcategories as subvarieties, as for instance the varieties of radical rings, of groups and of abelian groups. In this article the methods of non-abelian homological algebra are applied to establish some new Hopf formulae for homology of skew braces, where the coefficient functors are the reflectors from the variety of skew braces to each of the three above-mentioned subvarieties. The corresponding central extensions of skew braces are characterized in purely algebraic terms, leading to some new results, such as an explicit Stallings-Stammbach exact sequence associated with any exact sequence of skew braces, and a new result concerning central series.





On computing finite index subgroups of PSL 2 (ℤ)

March 2024

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9 Reads

Journal of Algebra and Its Applications

We present a method to compute finite index subgroups of PSL 2 (ℤ). Our strategy follows Kulkarni’s ideas, the main contribution being a recursive method to compute bivalent trees as well as their automorphism groups. As a concrete application, we compute all subgroups of index up to [Formula: see text]. We then use this database to produce tables with several arithmetical properties.


The prime spectrum of an 𝐿-algebra

February 2024

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36 Reads

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1 Citation

Proceedings of the American Mathematical Society

We prove that the lattice of ideals of an arbitrary L L -algebra is distributive. As a consequence, a spectral theory applies with no restriction. We also study the spectrum (i.e. the set of prime ideals) of L L -algebras and characterize prime ideals in topological terms.


Citations (53)


... Databases of solutions up to size 10 have inspired a vast body of research in math research. For instance, a recent survey [31] contains several mathematical conjectures that were inspired by mining the database of solutions up to size 10 for interesting properties (see for instance Problems 57 and 61). The list of solutions for size 11 will be useful for similar purposes and provides a substantial number of decomposable or multipermutation solutions (both of which are very important in the combinatorial theory of the Yang-Baxter equation). ...

Reference:

Incremental SAT-Based Enumeration of Solutions to the Yang-Baxter Equation
Skew Braces: A Brief Survey
  • Citing Chapter
  • May 2024

... Smoktunowicz studied in [13] the left braces that are both left nilpotent and right nilpotent. They turn out to coincide with the centrally nilpotent left braces, a notion introduced in [2] for skew left braces (see also [9]). ...

Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation
  • Citing Article
  • September 2022

Communications in Contemporary Mathematics

... In the first part of the paper, following [15,Question 3.4] and [16,Question 8.2.11], we discuss a possible converse to Kinnear's result on semidirect product of semiprime skew left brace. In this context, we show that if a skew left braces semidirect product B 1 ⋊ B 2 is semiprime, then so B 1 is, provided that B 1 is Artinian [14] (this is the case if B 1 is finite); on ther other hand, we show that an analog result for B 2 does not follows (even in the finite case), therefore we can not provide a characterization similar to the one of solvable skew left braces. Moreover, we show a "strongly" version of Kinnear's result, i.e. we prove that the semidirect product of strongly semiprime skew left braces is strongly semiprime. ...

Radical and weight of skew braces and their applications to structure groups of solutions of the Yang–Baxter equation
  • Citing Article
  • July 2021

Advances in Mathematics

... In the present study, we focus on solutions of the parametric, set-theoretic Yang-Baxter and reflection equations. The set-theoretic Yang-Baxter equation first introduced by Drinfel'd [19], whereas a plethora of studies followed on the investigation of solutions of the equation and associated algebraic structures, see for instance [4,8,10,15,16], [25]- [27], [21,22,28,29,30,31,34,35,37,38,39], [42]- [44], [47,49]. The parametric set-theoretical reflection equation together with the first examples of solutions appeared for the first time in [6], while a more systematic study and a classification inspired by maps appearing in integrable discrete systems [1,38,49] are presented in [7]. ...

Reflection equation as a tool for studying solutions to the Yang–Baxter equation
  • Citing Article
  • February 2021

Journal of Algebra