Jean Avan [pdf]

Title: Integrability structures for peakons

Abstract: Classical peakons are singular solutions of continuous integrable fluid equations, yielding N-body dynamical systems exhibiting integrabilty features. We discuss a number of issues related to the Lax matrices and their associated classical r-matrix structures. We focus on the multi-parameter extensions of the Camassa-Holm peakon dynamics, and comment on some  challenging issues regarding these new N-body systems and De Gasperis-Procesi peakons.

Martina Balagovic

Title: Towards bases for representations of QSP coideal subalgebras

Abstract: This is a continuation of Stefan Kolb's talk on finite dimensional representations of certain QSP coideal subalgebras of quantum groups. I will describe desired properties of bases for these representations that one could hope to construct in analogy with classical Lie theory, list some questions about structures which control these bases, and partially answer some of these questions. Joint work in progress with Stefan Kolb. 

Pascal Baseilhac [pdf]

Title: On the R-matrix and Drinfeld type presentations of the q-Onsager algebra

Abstract: The R-matrix presentation  [Reshetikhin-Semenov-Tian-Shansky,'90] of Drinfeld second realization ^{Dr}LU_qsl_2  [Ding-Frenkel,'92] has found numerous applications ranging from representation theory to mathematical physics.In the first part of the talk, I will present the corresponding structure for the q-Onsager algebra - the simplest example of quantum symmetric pair coideal subalgebra - from different perspectives (alternating generators and root vectors). In terms of certain `half-currents' for `Drinfeld type' generators ^{Dr}O_q [Lu-Wang,'20], the R-matrix presentation takes the form of a reflection algebra and a q-determinant condition satisfied by a K-operator with `twisted LDU' decomposition. In the second part,  I will discuss the `standard' injective homomorphism ^{Dr}O_q \rightarrow ^{Dr}LU_qsl_2. In particular, a factorized formula for the image of Lu-Wang's current for imaginary root vectors of O_q (and its coproduct) is proven. As a corollary, independent proofs of two recent theorems by T. Przezdziecki are obtained. Some perspectives will be briefly presented. Based on a joint work with A. Gainutdinov and G. Lemarthe.

Tomasz Brzeziński [pdf]

Title: Lie brackets on affine spaces vs Lie algebras

Abstract: Vector space valued Lie brackets on affine spaces or Lie affgebras were introduced by Grabowska, Grabowski and Ubrański in early 2000s. With the recent developments of heaps  and heaps of modules, it has become clear how to extend this definition to  affine space valued Lie brackets on affine spaces. In this talk I will present basic elements of the Lie affgebra theory, focusing on the two-way relationship between Lie affgebras and Lie algebras. The results presented in this talk have been obtained in collaboration with Ryszard Andruszkiewicz, James Papworth and Krzysztof Radziszewski.

Nigel Byott [pdf]

Title: Counting Quaternion and Dihedral Braces and the Associated Hopf-Galois Structures

Abstract: In the 2017 paper where they introduced skew braces, Guarnieri and Vendramin presented the results of computer calculations enumerating braces, and on the basis of these made several conjectures. One of these predicts for all m > 2 the number q(4m) of isomorphism classes of braces of size 4m whose multiplicative group is a generalised quaternion group. Rump (2020) gave a proof of this in the case m = 2^n , n ≥ 3. I will outline a proof of this conjecture for all m, together with the analogous result for dihedral braces. The methods used have their origin in Hopf-Galois theory, and we also obtain results on the number of Hopf-Galois structures on a Galois extension of fields with quaternion or dihedral Galois group.

Vincent Caudrelier [pdf]

Title: Yang-Baxter and reflection maps: soliton interactions and factorisation in loop groups

Abstract: I will describe how the dressing method for the construction of multisoliton solutions of certain integrable PDEs leads to solutions of the set-theoretical Yang-Baxter equation whose physical interpretation relates to soliton interactions. I will also describe how the investigation of integrable boundary conditions in those PDEs allowed us to introduce the set-theoretical reflection equation and some solutions which represent the soliton-boundary interaction. This provides a striking classical counterpart of well-known structures and phenonema in quantum integrable systems. I will present briefly the general underlying mathematical mechanism (factorisation in certain loops groups and involutions) and, time permitting, the Poisson properties of Yang-Baxter and reflection maps. 

Fabienne Chouraqui [pdf]

Title: The Yang-Baxter equation, quantum computing and quantum entanglement

Abstract: Every unitary solution of the Yang-Baxter equation (YBE), R: V \otimes V \rightarrow V \otimes V, where V is a complex vector space of dimension d, can be described by R: \mathbb{C}^d\otimes\mathbb{C}^d \rightarrow \mathbb{C}^d\otimes\mathbb{C}^d and R is a 2-qudit gate. In a series of papers, L.H. Kaufman and S.J. Lomonaco Jr initiate the study of 2-qubit gates, R:\mathbb{C}^2\otimes\mathbb{C}^2  \rightarrow \mathbb{C}^2\otimes\mathbb{C}^2, which satisfy the Yang-Baxter equation. In particular, using the classification of unitary solutions with d=2 from Dye's paper, they investigate this special class of 2-qubit gates and determine which are entangling using the criteria from the paper of J.L. Brylinski and R. Brylinski. In this paper, I provide a tool to construct infinite families of entangling 2-qudit gates, and amongst them entangling 2-qudit gates which satisfy the Yang-Baxter equation, relying on the results of L.H. Kaufman and S.J. Lomonaco Jr and of J.L. Brylinski and R. Brylinski. This relies on two main results on the Tracy-Singh product of matrices (or with abuse of notation of operators). The first result I prove states that the Tracy-Singh product of two (unitary) solutions of the Yang-Baxter equation, with a specific block partition, is also a (unitary) solution of the YBE. The second result I prove states that the Tracy-Singh product of two operators, with a specific block partition, where at least one of them is entangling, is an entangling operator.

Ilaria Colazzo [pdf]

Title: Bijective solutions to the pentagon equation

Abstract: This talk will focus on the set-theoretic version of the pentagon equation. After discussing some basic properties of such solutions, we will focus on bijective solutions and provide a complete description of all bijective solutions. As a surprising corollary, we will see that skew braces produce solutions to the pentagon equation. Based on a joint work with E.Jespers and Ł. Kubat and on a joint work with J. Okninksi and A. Van Antwerpen.

Alexander Cooper [pdf]

Title: A Perspective on the Rational Limit of the XXZ Chain

Abstract: In this talk, I will review a procedure for recovering solutions to operator relations, such as the (boundary) Yang-Baxter equation, fusion-, factorisation-, and TQ-relations in the context of the XXX Heisenberg spin chain, through taking the rational (q \to 1) limit of their XXZ counterparts.  In doing so, I will highlight the useful role of a free parameter in representations of the Borel subalgebras of the underlying quantum affine algebra U_q(\hat{sl}_2).

Nicolas Crampe [pdf]

Title: Absorption or emission of solitons at a boundary

Abstract: We want to outline the possibility that for particular choices of boundary in classical integrable models, there exist solutions where a soliton can be emitted or absorbed by the boundary. In this talk, we focus on the Ablowitz-Ladik model which is a well-known integrable system for which we study the more general integrable boundary and we obtain, by the inverse scattering method, the desired solutions.

Andrew Darlington [pdf]

Title: Hopf-Galois Structures on Parallel Extensions

Abstract: Given a separable but non-normal field extension L/K of degree n with normal closure E, there may be other degree n sub-extensions L’/K of E/K (we say that L’/K is parallel to L/K) which can be related to L/K in many different ways. It is then an interesting question to ask whether, given an extension L/K admitting a Hopf-Galois structure, can we say anything about the Hopf-Galois structures on all of the extensions L’/K  parallel to L/K? This talk will take a first look at answering this question, approaching the problem from a group-theoretical perspective, outlining some interesting results along the way.

Luc Frappat [pdf]

Title: The higher-rank Askey–Wilson algebra aw(n) and its braid group automorphisms.

Abstract: The Askey-Wilson algebra aw(3) is introduced. The generalization to higher rank Askey-Wilson algebra aw(n) is then defined. It is shown that there exist a natural braid group action on it, as well as a coproduct morphism that maps aw(n) on aw(n+1). Casimir elements are determined and the link with the centralizer of the diagonal action of Uq(sl_2) in the n-fold tensor product is made explicit.

Tatiana Gateva-Ivanova [pdf]

Title: Quadratic algebras associated to set-theoretic solutions of YBE

Abstract:  We study the quadratic algebras A(K,X,r) associated to various classes of set-theoretic solutions (X,r) of the Yang-Baxter or braid relations. These ‘Yang-Baxter algebras’ are braided-symmetric and behave like ‘quantum affine spaces’  particularly in cases when  r is involutive (r^2= 1), but have very different properties in the non-involutive case when r is idempotent (r^2 = r). We prove that if (X,r) is a finite nondegenerate involutive solution then the Yang-Baxter algebra A(K,X,r) is PBW if and only if (X,r) is square-free. In contrast, all algebras A(K,X,r) with r of a certain permutation idempotent type are isomorphic for a given n=|X|, leading to canonical PBW algebras A(K,n). We study the properties of various types of Yang-Baxter algebras via Veronese subalgebras and Segre products and Veronese and Segre morphisms (first steps in noncommutative algebraic geometry on these spaces) and in terms of noncommutative differential geometry. We also obtain new results on general PBW algebras which we apply in the permutation idempotent case. Some of the results in this talk are joint with Shahn Majid.

Stefan Kolb

Title: Representations of very non-standard quantum so(2N-1)

Abstract: Letzter's theory of quantum symmetric pairs provides new quantum deformations of the Lie algebra so(n-1) considered as a Lie subalgebra of so(n). These deformations are realized as coideal subalgebras B of the Drinfeld-Jimbo quantum enveloping algebra U=Uq(so(n)). For even n=2N the algebra B has an obvious Cartan subalgebra which makes it possible to mimic quantum group constructions. In this talk I will discuss this example as an illustration of Letzter's theory. I will outline a Poincare-Birkhoff-Witt Theorem and the classification of finite- dimensional irreducible representations of B in this case. The talk is based on work with Jake Stephens.

Christian Korff

Title: Hirzebruch's formal group law and 2-parameter quantum groups

Abstract: Formal group laws first appeared in algebraic geometry, bordism theory, and are connected with Hirzebruch genera. In this talk I will discuss their use in the definition of formal Demazure operators and connect them via quantum affine Schur-Weyl duality with 2-parameter quantum groups originally introduced by Takeuchi and subsequently discussed in the work of Benkart and Witherspoon. I will address some of their possible applications from the point of view of quantum integrable systems.

Theodoros Kouloukas [pdf]

Title: Integrability of Deformed Cluster Maps

Abstract: We present deformations of cluster mutations with covariant log-canonical presymplectic forms defined by the corresponding exchange matrix. By considering periodic sequences of cluster mutations, we define deformed cluster maps and investigate their Liouville integrability. We study examples of parametric deformations of A4 cluster map and of affine Dynkin quivers associated with periodic reductions of integrable lattice equations and Yang-Baxter maps.

Łukasz Kubat [pdf]

Title: On simple solutions of the Yang-Baxter equation

Abstract: Recall a solution (X,r) of the Yang--Baxter equation is called simple provided any epimorphic image of (X,r) is isomorphic either to (X,r) or to a solution of size 1. The aim of this talk is to present a characterisation of finite simple non-degenerate bijective solutions in terms of algebraic properties of skew left braces associated to such solutions.

Jules Lamers [pdf]

Title: Long-range quantum integrability

Abstract: Systems with long-range spin interactions are important because of applications in experiments, quantum information and computing, and high-energy theory. Yet they are notoriously difficult to solve, except for isolated integrable examples that in particular include the Haldane--Shastry chain. In recent years, this area has been reinvigorated, and there are now several new integrable long-range spin chains. The key to their integrability are their connections to quantum many-body systems of Calogero--Sutherland and Ruijsenaars--Macdonald type, which provide underlying affine Hecke-algebraic structures. In this talk I will introduce this area of integrability, give an overview of the state of the art, and identify some key open problems. Based on joint works with Rob Klabbers (Humboldt U Berlin) and A Ben Moussa and D Serban (IPhT Saclay).

Victoria Lebed

Title: Trying to compute the structure group of at least some YBE solutions

Abstract: To any set-theoretic solution (X,r) to the Yang-Baxter equation, one can associate a group Ass(X,r). This yields nicely behaved groups, which are however very difficult to compute in practice. In this talk will almost compute Ass(G,r) for solutions of the form r(x,y)=(y,y^{-1}xy) (conjugation quandles for those who know), where G is what we call a conjugation group. This means a group admitting a presentation with only conjugation relations of type y^{-1}xy=z and power relations of type x^d=1. This vast class of groups includes free abelian groups, symmetric and braid groups and various generalizations thereof, Thompson’s group F, cactus groups, knot groups, structure groups and their Coxeter-like finite quotients, and many more. In addition, we characterize conjugation groups without power relations as structure groups of quandles.

Gandalf Lechner [pdf]

Title: Quantum Field Theory on a Nichols Algebra: From Solutions to the YBE to Local Quantum Physics

Abstract: In this talk, I will explain how a solution S of the Yang-Baxter equation on a Hilbert space can be used to define a quantum field theory. This QFT will be defined on the Nichols algebra over the Hilbert space. In the cases of immediate interest to physics, S needs to be involutive and infinite- dimensional, and subject to further conditions including crossing symmetry. This on the one hand opens new questions and classification ideas for solutions to the YBE with links to representation theory and functional analysis, and on the other hand provides non-perturbative solutions to the relativistic elastic inverse scattering problem in two dimensions.

Shahn Majid

Title: Braided-racks and non-commutative differential structures

Abstract: This talk will be about interactions between the YBE and noncommutative differential geometry. One of the points of interaction is the braided-Lie algebra dual to the left-invariant 1-forms for a bicovariant calculus on a quantum group, which I will illustrate for the coordinate algebra of the quantum double D(G) of a finite group G and where the corresponding braided-Lie algebras generalise the notion of a rack. Moreover, coirreducible such calculi correspond to irreps of  D(G), which play the role of quasi-particles in the Kitaev model of topologically fault-tolerant quantum computing. These irreps can be seen as fully analogous to the Wigner construction of irreps of the Poincare group but with the role of spacetime played by the group algebra of G. If time, I will also explain how the braiding associated with a bicovariant calculus is a special case of a more general construction of a solution of the YBE on bimodules over a differential algebra from a connection on the bimodule. The main part of the talk is based on joint work with L. McCormack. 

Paul Martin

Title: Approaches to classification of braid representations (constant Yang-Baxter)

Abstract: We discuss the classification of representations of the monoidal braid category, reviewing the complete classification in the `charge-conserving' case.

Marzia Mazzotta [pdf]

Title: The algebraic structure of dual weak braces

Abstract: A weak brace is an algebraic structure that gives rise to a set-theoretic solution of the Yang-Baxter equation, which is close to being bijective and non-degenerate. We show that every dual weak brace is a strong semilattice of some skew braces. Consequently, even its solution is the strong semilattice of the bijective non-degenerate solutions associated with each of these skew braces. The talk is based on a recent joint work in collaboration with Francesco Catino and Paola Stefanelli.

Leo Sean McCormack

Title: Quantum geometry on a group algebra and the quantum double in the role of its Poincare group

Abstract: For G is a finite group, we consider its group algebra as a noncommutative spacetime. We show how `quasiparticle’  irreps of the quantum double D(G) can be viewed  as certain sections of bundles over the group algebra in a manner fully analogous to the Wigner construction for irreps of the Poincare group as fields on spacetime subject to wave equations. This requires a certain quantum differential structure on the group algebra and the framework of quantum principal bundles. Some background to the work will be covered in the talk of S. Majid.

Jan Okniński [pdf]

Title:  Constructing non-degenerate involutive solutions of the YBE

Abstract: Various constructions of (mostly finite) set-theoretic non-degenerate involutive solutions of the Yang-Baxter equation are presented. These include constructions based: on systems of imprimitivity of the action of the permutation group, on simple left braces, and on asymmetric products of braces. Our focus is on indecomposable solutions and on simple solutions. The talk is based on a recent joint work with Ferran Cedo.

George Papamikos [pdf]

Title: Birational solutions of the set theoretic Yang-Baxter equation and their integrability properties

Abstract: I will present certain solutions of the set-theoretic parametric Yang-Baxter equation and explore some of the integrability properties. This class of solutions are birational maps with several invariants manifolds and a Lax representation. We show that we can use these maps to construct higher dimensional birational maps which admit nice properties and we prove their integrability in the Liouville sense. If time is permitted, I will present several generalisations in different directions.

Agata Pilitowska [pdf]

Title: Permutational biracks and their relations with multipermutation solutions of the YBE

Abstract: A birack is an algebra  having structure of two one-sided quasigroups related by some additional conditions. Biracks appear in low-dimensional topology. They are associated with a link diagram and they are invariant (up to isomorphism) under the generalized Reidemeister moves for virtual knots and links. On the other hand, it is known that there is a one-to-one correspondence between such algebras and non-degenerate  solutions of the Yang-Baxter equation. In particular, biracks corresponding to finite injective solutions or solutions which originate from skew braces are in fact biquandles.  This fact allows one to characterize solutions of the Yang-Baxter equation applying the universal algebra tools. During the talk I will show that permutational biracks give equational characterization of multipermutation solutions. I will also present an algorithm how to obtain, in effective way,  all solutions of the Yang-Baxter equation of multipermutation level 2, up to isomorphism. The talk is based on a joint work with P. Jedlicka.

Dora Puljic [pdf]

Title: Pre-  and post-Lie algebras and braces

Abstract: The connection between R-braces of abelian type and pre-Lie algebras over R was first descibed by Rump in 2014. The correspondence between strongly nilpotent braces of abelian type of prime power order and nilpotent pre-Lie rings of the same order was explicitly established by Smoktunowicz, subsequently allowing for the classification of braces of abelian type of order p^4.  A natural question that arises is whether there exists a generalisation of a pre-Lie algebra from which we can construct a skew brace. In 2024, Bai, Guo, Sheng and Tang offer post-Lie algebras as an affirmative answer to that question. In this talk, we give exposition on the topic of the connection between braces and pre- and post-Lie algebras, and give the classification of braces of abelian type of order p^4 as an application.

Eric Ragoucy [pdf]

Title: Algebraic Bethe Ansatz for models based on orthogonal algebras

Abstract: We consider integrable models with o_{2n+1} symmetry. Within the framework of Algebraic Bethe Ansatz, we construct their Bethe vectors and compute their scalar products.  The calculation is done using the current presentation of Yangians.

Vidas Regelskis 

Title: Shifted Yangians and functional relations of the XXX spin chain

Abstract: Transfer matrices and Baxter Q-operators are fundamental objects in the study of integrable spin chains. They are known to satisfy certain families of functional relations, known as TT- and TQ-relations. In the case of the XXZ spin chain, the origin of these relations can be traced back to representation theory of quantum affine algebras and their Borel subalgebras. The origin of functional relations of the XXX spin chain is less well understood. In this talk I will explain the role of shifted Yangians in the latter case.

Alexander Shapiro [pdf]

Title: Ruijsenaars wavefunctions as modular group matrix coefficients

Abstract: The phase space of the Ruijsenaars integrable system can be identified with (a Poisson reduction of) the moduli space of GL_n local systems on a punctured torus. The latter admits a structure of a cluster Poisson variety. On the algebraic level, this leads to an injective homomorphism from a spherical subalgebra of the double affine Hecke algebra into the quantized algebra of global functions on the named cluster variety. From an analytic point of view, it allows for a unitary equivalence between Toda and Ruijsenaars quantum integrable systems. This in turn allows one to present the eigenfunctions of Macdonald operators as a matrix coefficient of an order 4 element in the mapping class group of a punctured torus. During this talk we will focus on the n=2 case when no Hamiltonian reduction is required. It will be based on a joint work with P. DiFrancesco, R. Kedem, S. Khoroshkin, and G. Schrader.

Evgeny Sklyanin [pdf]

Title: New linearisation formula for q-ultraspherical polynomials

Abstract: For the product C_m(x;b,q) C_n(x;b',q) of two continuous q-ultraspherical polynomials with different parameters b and b', explicit formulae are presented for the coefficients of its expansion in the basis C_k(x;b,q) in the case b'=q/b. We also discuss a generalisation to multivariate Jack and Macdonald polynomials and its relation to Pieri formulas and Q-operators.

Paola Stefanelli [pdf]

Title: Drinfel’d isomorphic solutions of the Yang-Baxter equation and shelves

Abstract: The main aim of this talk is to illustrate recent results obtained in a joint work with Anastasia Doikou and Bernard Rybołowicz. The common thread throughout the presentation will be the algebraic structures of left shelves and their connection with left non-degenerate solutions of the Yang-Baxter equation. In particular, we discuss the Yang-Baxter algebra for solutions of the braid equation associated with a given rack/quandle and, motivated by the notion of pre-Lie algebras, introduce a novel algebraic structure called the pre-Lie skew brace. Finally, a brief part will concern an application to the set-theoretical solutions of the reflection equation contained in an ongoing work with Andrea Albano and Marzia Mazzotta.

Lorenzo Stefanello [pdf]

Title: Classifying Galois extensions with Childs’s property 

Abstract: The transition from Galois theory to Hopf–Galois theory introduces some complexities. One example is the potential lack of bijectivity of the Hopf–Galois correspondence, which implies that in principle the fundamental theorem of Galois theory can only be recast in a weaker version for Hopf–Galois structures. In 2017, Childs proved that every Hopf–Galois structure on a cyclic Galois extension of odd prime power degree has a bijective Hopf–Galois correspondence. The aim of this talk is to provide a complete classification of Galois extensions that exhibit this property. We achieve this by employing a description of Hopf–Galois structures via skew braces, leveraging on the connection between these topics developed by Byott and Vendramin, as outlined in a recent joint work with Trappen.

Jasper Stokman [pdf]

Title: Quasi-polynomial and vector-valued Macdonald polynomials.

Abstract: Cherednik used representation theory of the double affine Hecke algebra (DAHA) to construct commuting matrix-valued Macdonald operators in terms of certain solutions of root system analogs of quantum Yang-Baxter equations with spectral parameter. I will discuss the diagonalisation of these operators using quasi-polynomial analogs of Macdonald polynomials, which were recently introduced  in joint work with S. Sahi and V. Venkateswaran.

Fiona Torzewska [pdf]

Title: Classification of charge conserving loop-braid representations.

Abstract: Progress has been made in classifying monoidal braid representations by restricting the target category, and by extending the source category. The loop braid category LB is a generalisation of the braid category whose morphisms correspond to motions of unknotted, unlinked loops in 3-dimensional space. A loop-braid representation gives a braid representation, but not every braid representation lifts to a loop braid representation. A rank-N charge-conserving loop braid representation is a strict monoidal functor from LB into the category MatchN of rank-N charge-conserving matrices. Here I will discuss the construction of all charge-conserving loop braid representations, and their classification up to suitable notions of isomorphism.

Arne Van Antwerpen [pdf]

Title: Cabling non-involutive non-degenerate solution of the Yang-Baxter equation

Abstract: Recently, Lebed, Ramirez and Vendramin introduced the technique of cabling of involutive non-degenerate solution of the Yang-Baxter equation and show it unifies most known results on the decomposability of solutions. We extend this technique to non-involutive non-degenerate solutions. We will discuss how cabling can be used to study indecomposability of non-involutive solutions. In particular, we show a non-involutive version of Rump’s well-known decomposition theorem, i.e. every finite square-free involutive solution is decomposable, and the recent theorem of Camp-Mora and Sastriques. Finally, it is shown that cabling can be utilized to construct new simple solutions, i.e. solutions with no non-trivial homomorphic images.

Leandro Vendramin [pdf]

Title: Nichols algebras over groups

Abstract: Nichols algebras appear in several areas of mathematics, from Hopf algebras and quantum groups to Schubert calculus and conformal field theories. In this talk, I will review the main problems related to Nichols algebras (over groups) and discuss some recent classification theorems.

Bart Vlaar [pdf]

Title: Reflections on quantum integrability

Abstract: We will survey some recent developments and open problems in the universal approach to solutions of reflection equations via quasitriangular Hopf algebras, as well as its applications to quantum integrable systems. The main class of examples is provided by (untwisted) quantum affine algebras and the associated trigonometric K-matrices.

Paul Zinn-Justin

Title: The exceptional series and the Yang–Baxter equation