Tag - Yang-Baxter equations

Marzia Mazzotta: Classification of set-theoretical solutions to the pentagon equation

17th April, 2023

The pentagon equation classically originates from the field of Mathematical Physics. Our attention is placed on the study of set-theoretical solutions of this equation, namely, maps s: X×X → X×X given by s(x, y)=(xy, θx(y)), where X is a semigroup and θx: X → X is a map satisfying two laws. In this talk, we give some recent descriptions of some classes of solutions achieved starting from particular semigroups. Into the specific, we provide a characterization of idempotent-invariant solutions on a Clifford semigroup X, that are those for which θa remains invariant on the set of idempotents E(X). In addition, we will focus on the classes of involutive and idempotent solutions, which are solutions fulfilling s2=idX×X and s2=s, respectively.

Přemysl Jedlička: Non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation of multipermutation level 2

3rd April, 2023

A set-theoretic solution of the Yang-Baxter equation is a mapping r : X×X → X×X satisfying (r×1)(1×r)(r×1) = (1×r)(r×1)(1×r). A solution r : (x, y) → (σx(y), τy (x)) is called non-degenerate if the mappings σx and τy are permutations, for all x, y ∈ X. A solution is called involutive if r2 = 1. If (X, r) is a non-degenerate involutive solution (X, r) then the relation ∼ defined by x ∼ y ≡ σx = σy is a congruence. A solution is of multipermutation level 2 if |(X/ ∼)/ ∼ | = 1. In our talk, we focus on these solutions and we present several constructions and properties.

Dietrich Burde: Pre-Lie algebra structures on reductive Lie algebras and étale affine representations

27th February, 2023

Étale affine representations of Lie algebras and algebraic groups arise in the context of affine geometry on Lie groups, operad theory, deformation theory and Yang-Baxter equations. For reductive groups, every étale affine representation is equivalent to a linear representation and we obtain a special case of a prehomogeneous representation. Such representations have been classified by Sato and Kimura in some cases. The induced representation on the Lie algebra level gives rise to a pre-Lie algebra structure on the Lie algebra 𝔤 of G. For a Lie group G, a pre-Lie algebra structure on 𝔤 corresponds to a left-invariant affine structure on G. This refers to a well-known question by John Milnor from 1977 on the existence of complete left-invariant affine structures on solvable Lie groups. We present results on the existence of étale affine representations of reductive groups and Lie algebras and discuss a related conjecture of V. Popov concerning flattenable groups and linearizable subgroups of the affine Cremona group.

Alberto Facchini: Multiplicative lattices, skew braces

29th September, 2022

The multiplicative lattices we will consider are those defined in A. Facchini, C. A. Finocchiaro and G. Janelidze, Abstractly constructed prime spectra, Algebra universalis 83(1) (2022). Multiplicative lattices yield the natural setting in which several basic mathematical questions concerning algebraic structures find their answer (Zariski spectrum, nilpotency, solvability, abelian algebraic structures,...) We will consider the particular case of skew braces, which appear in connection to the study of the Yang-Baxter equation.

Vladimir Bazhanov: Quantum geometry of 3-dimensional lattices

26th October, 2021

In this lecture I will explain a relationship between incidence theorems in elementary geometry and the theory of integrable systems, both classical and quantum. We will study geometric consistency relations between angles of 3-dimensional (3D) circular quadrilateral lattices, lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable 'ultra-local' Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure allowed us to obtain new solutions of the tetrahedron equation (the 3D analogue of the Yang-Baxter equation) as well as reproduce all those that were previously known. These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry.

Petr Vojtechovsky: Quandles and other classes of set-theoretic solutions of the Yang-Baxter equation

16th September, 2021

Quandles are algebraic structures designed to mesh with the Reidemeister moves of knot theory. Joyce and Matveev showed that quandles give rise to a complete invariant of oriented knots. Since the Yang-Baxter equation resembles the third Reidemeister move, it is not surprising that quandles also form a class of set-theoretic solutions of the Yang-Baxter equation. In this talk I will explain how quandles and connected quandles can be enumerated up to isomorphism and list a few open problems. I will also present two additional classes (involutive and idempotent) of set-theoretic solutions of the Yang-Baxter equation with rich algebraic theory.

Agata Smoktunowicz: Some questions related to nilpotent rings and braces

27th February, 2021

In around 2005, Wolfgang Rump introduced braces, a generalization of nilpotent rings to describe all involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation. This formulation then rapidly found application in other research areas. This talk will review these applications.

Definition. A set A with binary operations of addition +, and multiplication ∘ is a brace if (A, +) is an abelian group, (A, ∘) is a group and a ∘ (b+c)+a = a ∘ b+a ∘ c for every a, b, c ∈ A. It follows from this definition that every nilpotent ring with the usual addition and with multiplication a ∘ b = ab + a + b is a brace.

Braces have been shown to be equivalent to several concepts in group theory such as groups with bijective 1-cocycles and regular subgroups of the holomorph of abelian groups. In algebraic number theory there is a correspondence between braces and Hopf-Galois extensions of abelian type first observed by David Bachiller. There is also connection between R-braces and pre-Lie algebras discovered by Wolfgang Rump in 2014. One generator braces have been shown to describe indecomposable, involutive solutions of the Yang-Baxter equation.

On the other hand, Anastasia Doikou and Robert Weston have recently discovered some fascinating connections between braces and quantum integrable systems. In particular, to find solutions of the set-theoretic reflection equation it is needed to solve problems on some polynomial identities in nilpotent rings. Because previously the theory of polynomial identities was mainly developed for prime rings, and for the reflection equation we only consider nilpotent rings, there are no known methods for solving such problems. We will mention some open problems on polynomial identities in nilpotent rings which appear in this situation.

Alina Vdovina: Buildings, quaternions and Drinfeld-Manin solutions of Yang-Baxter equations

11th February, 2021

We will give a brief introduction to the theory of buildings and present their geometric, algebraic and arithmetic aspects. In particular, we present explicit constructions of infinite families of quaternionic cube complexes, covered by buildings. We will introduce new connections of geometric group theory and theoretical physics by using quaternionic lattices to find new infinite families of Drinfeld-Manin solutions of Yang-Baxter equations.