Schedule for: 24w5201 - Skew Braces, Braids and the Yang-Baxter Equation

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Informal Meet and Greet at the BIRS Lounge (PDC 2nd Floor).

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Recently, several algebraic structures have emerged for studying the set-theoretic version of the Yang–Baxter equation. Skew braces are particularly important among them. This talk will give a brief introduction to skew braces and discuss their properties, as well as their connection to solutions. Additionally, we'll explore computer tools for constructing different families of finite solutions and skew braces.

Describing the self-homeomorphisms of a surface with marked points in its interior. Giving a group presentation to families of changing configurations of links in a 3-dimensional space. Characterising the space of embeddings of a submanifold in an ambient manifold. All these constructions use different formalisms to carry out the task of describing the various ways to continuously move a subspace within a larger space over a certain time, ultimately returning the subspace to its original position. The theory of motion groups began in the 1960s as a programme to provide a general framework for addressing this task, drawing inspiration from Artin braid groups. In this talk, we will provide an overview of known results and explore questions, approaches, and mathematical languages for a modern theory of motions.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Define a braid representation to be a strict monoidal functor from the braid category into Mat - the former has objects the natural numbers and endomorphisms at each n the n strand braid group; the latter has the same objects and matrices as morphisms. Each braid representation gives braid group representations for all n in a coherent way. Classification of braid representations is in general hard. However, progress has been made by restricting the target category, and by extending the source category.

Subfactor planar algebras first were constructed by Vaughan Jones as a diagrammatic axiomatization of the standard invariant of a subfactor. Planar algebras can be conveniently encoded by diagrams in the plane. These diagrams satisfy some skein relations and have an invariant called an index. The Kuperberg Program asks to find all diagrammatic presentations of subfactor planar algebras. This program has been completed for index less than 4. In this talk, I will introduce subfactor planar algebras and find presentations for subfactor planar algebras of index 4 associated with the affine A Dynkin diagram.

Given an object X, a natural approach to understanding some of its intrinsic properties is to decompose X as a "union" of suitable "disjoint" subobjects. For example, we find here set partitions, direct sum decompositions of modules, or even free product decompositions of groups. Of particular interest is the poset of those subobjects which arise as factors of such decompositions (with some "inclusion" ordering between them), and its associated set of decompositions ordered by refinement. The topology and combinatorics of these posets are related to group cohomology stability and the determination of dualising modules. In this talk, I will show how all these constructions can be regarded as a special case of a unified categorial framework. On the way, we will encounter and address appealing algebraic, combinatorial and topological problems.

I will report on recent work with I. Colazzo, E. Jespers and L. Kubat. Let $(X,r)$ be a left non-degenerate set-theoretic solution of the Yang-Baxter equation. Denote $M(X,r)$ its Yang-Baxter monoid, i.e. the monoid generated by $X$ where the defining relation is given by $r$. We discuss the finiteness conditions of both $M(X,r)$ and the monoid algebra $KM(X,r)$. In particular, we present a novel ideal chain in $M(X,r)$ and present the role the divisibility structure plays in the ring theoretical and homological properties of $KM(X,r)$. Finally, we present an explicit form for the cancellative congruence of $M(X,r)$, which allows an algebraic method to connect results between bijective non-degenerate solutions and their injectivization.

This talk is based on joint work with Carsten Dietzel and Senne Trappeniers. We study indecomposable involutive non-degenerate set-theoretic solutions to the Yang-Baxter equation. Building upon the works of Jedlička-Pilitowska and Cedó-Okniński and through the study of their associated permutation skew brace, we fully classify such solutions of prime-squared size.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A skew left brace $(B, +, \cdot)$ consists of a set $B$ with two operations $+$ and $\cdot$ such that $(B, +)$ and $(B,\cdot)$ are groups linked by the compatibility $a(b+c)=ab-a+ac$ for $a,b,c\in B$. For every $b\in B$, we define $\lambda_b \in Aut(B,+)$ by $\lambda_b(a) = -b+ba$ for $a \in A$. Then the set $H =\{(b,\lambda_b) \colon b \in B\}$ is a regular subgroup of $Hol(B,+)$ isomorphic to $(B,\cdot)$. It is well known that the skew left braces with additive group $(B, +)$ are in correspondence with the regular subgroups of the holomorph $Hol(B, +)$ of $(B, +)$. Given a regular subgroup $H$ of $Hol(B,+)$, we can consider the subgroup $S$ generated by $K = (B,+)$ and $H$. If $\pi$ denotes the projection of $Hol(B,+)$ onto $Aut(B,+)$ and $E = \pi(H)$, we obtain that $S = KH = KE = HE$, $K \trianglelefteq S$, and $K cap E = H \cap E = 1$, that is, the group $S$ possesses a triple factorisation. Conversely, a triply factorised group of this kind induces a skew left brace with additive group isomorphic to $K$ and multiplicative group isomorphic to $H$. Representation theory studies algebraic structures by expressing their elements as linear transformations of a vector space. One of the handicaps we find when trying to define representations of skew left braces is the necessity of giving a skew left brace structure to some subgroups of the general linear groups, where only a binary operation is defined. We present a proposal for a definition of representations of skew left braces and we compare it with other definitions in the literature. We use representations of triply factorised groups as a tool to obtain representations of skew left braces.

In a tensor category one can take direct sums, tensor products, and duals of objects and morphisms. Of particular interest are braided fusion categories — semisimple tensor categories in which the tensor product is subject to a commutativity constraint, called braiding. This name is justified by the fact that tensor powers of objects in such a category admit actions of braid groups. In this talk, I will give basic definitions and discuss the structural theory and examples of braided fusion categories. I will also provide an overview of some of the machinery used to address current problems in this field.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We give a review on the subject of topological quantum computing (TQC). In the narrowest sense, TQC refers to encoding qubits in global degrees of non-Abelian anyons/quasi-particles and executing quantum gates by manipulating anyons. In the broadest sense, TQC includes any fault-tolerant quantum computing that relies on topological properties of the underlying mechanism. The toric code, for example, has no use in conventional TQC since it only hosts Abelian anyons, but it provides one of the best error correcting codes for fault-tolerant quantum computing which has been implemented experimentally. We explore the relations between topological order, anyons, and quantum error correcting codes. We also address some questions regarding universality and leakage-free entangling gates in anyon models.

We study the quadratic algebras $A(K,X,r)$ associated to a class of strictly braided but idempotent set-theoretic solutions $(X,r)$ of the Yang-Baxter or braid relations. In the invertible case, these algebras would be analogues of braided-symmetric algebras or `quantum affine spaces' but due to $r$ being idempotent they have very different properties. We show that all $A(K,X,r)$ for $r$ of a certain permutation idempotent type are isomorphic for a given $n=|X|$, leading to canonical algebras $A(K,n)$. We study the properties of these both via Veronese subalgebras and Segre products and in terms of noncommutative differential geometry. We also obtain new results on general PBW algebras which we apply in the permutation idempotent case.

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. This talk will focus on solutions of the form $r(x,y)=(y,y^{-1}xy)$, where X has a group structure. In particular we show that for conjugation groups, $Ass(X,r)$ is a subgroup of $X\times {\mathbb Z}^n$ for certain $n$. Here a conjugation group is a group admitting a presentation with only conjugation relations of type $y^{-1}xy=z$ and power relations $x^d=1$. 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, and certain general conjugation groups as the finite quotients of structure groups of quandles.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Already known as quasi-braid groups, Henriques and Kamnitzer defined cactus groups in terms of actions on tensor products in cobordism categories. If group presentations are known for these groups, their combinatorial properties and the relationships with braid groups are not yet fully understood. In this talk we will explore some first results of a combinatorial nature and we will present some generalizations of such groups. Based on joint works with Hugo Chemin & Victoria Lebed and with Eddy Godelle & Luis Paris.

I will introduce the family of trickle groups, which includes right-angled artin groups, cactus groups and many other interesting groups. I will explain why we think this family is of interest. As an example, I will define dual cactus groups, related to the dual structure of Artin groups. In particular I will explain how to solve the word problem for these groups and why they are closed to Artin and Coxeter groups. This is a join work with Paolo Bellingeri and Luis Paris.

We endow a category of representations of the unrolled quantum group for sl(2) with a Hermitian structure. This leads to a non-degenerate Hermitian pairing on the state spaces for surfaces coming from a non-semisimple TQFT. We study the resulting unitary representations of the braid group and show that the representations sometimes have dense image. This is joint with Nathan Geer, Aaron Lauda, and Bertrand Patureau-Mirand.

Not every monoidal structure on a category yields a braiding. However in certain instances the monoidal category is still ’generically braided’. In the first half of the talk we will make the latter notion precise and present some examples. By doing so we will encounter in the second half of the talk the concept of a ’system of renormalized R-matrices’ (following Cautis-Williams). We will briefly explain our interest in such categories, namely the wish of a certain combinatorial structure on the Grothendieck ring of the monoidal category. Finally, we will propose the question of finding more instances.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.