Skew braces and the YBE
April 2
Paolo Saracco
Hopf envelopes of finite-dimensional bialgebras and beyond
The Hopf envelope of a bialgebra B is a certain universal Hopf algebra that we can associate with B and that plays for it the same role that the universal enveloping group plays for a monoid. In categorical terms, the Hopf envelope is the left adjoint to the forgetful functor from Hopf algebras to bialgebras, hence it may be legitimately called the free Hopf algebra generated by B. Its existence is a well-known fact in Hopf algebra theory, but its construction is very technical and not always helpful.
Our main aim will be to see how, at least in the finite-dimensional setting, we can realize the Hopf envelope of a bialgebra B as a suitable quotient of B itself. The proof of this result relies on the fact that any such bialgebra comes equipped with a (one-sided) $n$-Hopf algebra structure, a notion that we will introduce contextually.
Time permitting, we will discuss why our construction can be extended to Artinians and to certain cocommutative bialgebras.
This talk is based on an ongoing project with A. Ardizzoni and C. Menini.
March 26
Senne Trappeniers
A Lazard correspondence between post-Lie rings and skew braces
Skew braces are algebraic structures that arise naturally in the study of solutions of the set-theoretic Yang-Baxter equation and are related to regular affine actions of groups. Post-Lie rings appear, among other places, in the study of regular affine actions of Lie groups, already hinting at a connection with skew braces. Indeed, results by Burde-Dekimpe-Deschamps, Rump, Smoktunowicz and Bai-Guo-Sheng-Tang give concrete conditions for when a (partial) correspondence exists between skew braces and post-Lie rings.
In this talk, we first recall and explain the statement of the classical Lazard correspondence. Next, we discuss how skew braces are the natural group theoretic counterpart of post-Lie rings. This will provide us with the necessary tools and notions to formulate a precise statement of a Lazard correspondence between skew braces and post-Lie rings and give a sketch of its proof. Special emphasis is put on the crucial role played by the classical Lazard correspondence and how the novel notion of L-nilpotency appears as a natural condition.
The slides are here.
March 19
Daimy Van Caudenberg
SAT-Based Enumeration of Solutions to the Yang-Baxter Equation
We tackle the problem of enumerating set-theoretic solutions to the Yang-Baxter equation. Non-degenerate, involutive solutions have been enumerated for sets up to size 10 using constraint programming with partial static symmetry breaking by Ö. Akgun, M. Mereb and L. Vendramin (2022); for general non-involutive solutions, a similar approach was used to enumerate solutions for sets up to size 8. In this talk, 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 lexicographic 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. For example, we constructed all isomorphism classes of non-degenerate involutive solutions of size 11. The talk is based on a joint work with B. Bogaerts and L. Vendramin (arXiv:2501.14363).
The slides are here.
March 12
Mahender Singh
Idempotents in quandle rings
Quandles are non-associative algebraic structures that arise from the algebraic formulation of the Reidemeister moves of planar diagrams of knots and links in the 3-sphere. Along with their weaker counterparts called racks, quandles also provide bijective non-degenerate set-theoretical solutions to the Yang-Baxter Equation. Beyond knot theory, these structures appear in a diverse spectrum of mathematics including surface theory, Riemannian symmetric spaces, Lie algebras and groups. Quandle rings, introduced as analogues of group rings, aim to incorporate ring-theoretic techniques into the study of quandles. Just as group elements serve as units in a group ring, the elements of a quandle act as idempotents in a quandle ring. In this talk, we will investigate the idempotents in quandle rings and their connection to quandle coverings. We will show that integral quandle rings of finite-type quandles, which are non-trivial coverings over well-behaved base quandles, possess infinitely many non-trivial idempotents, and provide a complete characterization of these idempotents. Furthermore, we show that integral quandle rings of free quandles contain only trivial idempotents, thereby identifying an infinite family of quandles with this property. If time permits, we will discuss the application of these idempotents to knot theory.
The slides are here.
February 28
Teresa Krick
An Effective Positivstellensatz over the Rational Numbers for Finite Semialgebraic Sets
I will present recent joint results with Lorenzo Baldi and Bernard Mourrain, on the exact representation of multivariate integer polynomials that are nonnegative on finite semialgebraic sets, described by means of a zero-dimensional ideal, as sums of squares of rational polynomials. We provide existential results for the strictly positive case and a sufficient condition for the nonnegative case, and degree and height (i.e. bit-size) bounds.
The slides are here.
February 21
Andrew Darlington
Some problems in Hopf-Galois theory
Hopf-Galois theory gives a Galois theory for more general field (and even ring) extensions, where one now lets a Hopf algebra play the role of the Galois group. In the case that the extensions involved are separable (but not necessarily normal), the problem of finding Hopf-Galois structures (the analogue of the Galois group) can be approached entirely group theoretically. This talk will begin by exploring some basic definitions, examples and results that link Hopf-Galois structures on separable extensions to transitive subgroups of the holomorph. I will then present some recent results which involve different ways to classify and study these structures using group theory. Finally, we will discuss a few problems that I am currently interested in, focusing in particular on the connection between Hopf-Galois structures and skew braces. The talk will be self-contained, so no need to brush-up on any of the specific terms mentioned above!
The slides are here.