# QA Seminar

## 2022

- 14/11/2022: Alfilgen N. Sebandal (Iligan), Lie Bracket algebras over Leavitt path algebras
- 18/05/2022: Lorenzo Stefanello (Pisa), On bi-skew braces and related topics
- 18/05/2022: Dora Puljic (Edinburgh), Braces and Classification Efforts
- 18/05/2022: Ilaria Colazzo (Exeter), Set-theoretic solution of the Pentagon Equation: the involutive case

## Abstracts

November 14

Alfilgen N. Sebandal (Iligan) Lie Bracket algebras over Leavitt path algebras and Classificiations of graphs with disjoint cycles: a Talented monoid view

Abstract

Given a directed graph, one can associate two algebraic entities: the Leavitt path algebra and the talented monoid which has interesting relationship between them. The talented monoid is isomorphic to the positive cone of the graded K0-group of the Leavitt path algebra which is naturally equipped with a Z-action. In this talk, we see the Jordan-Hölder theorem for refinement monoids with group action. This gives rise to the use of certain composition series in the talented monoid which is then utilized to compute the Gelfand-Kirillov Dimension of a Leavitt path algebra when a graph is composed of disjoint cycles. Furthermore, we characterize simplicity for Lie bracket algebras arising from Leavitt path algebras, based on the talented monoid of the underlying graph. We show that graded simplicity and simplicity of the Leavitt path algebra can be connected via the Lie bracket algebra.

May 18

Lorenzo Stefanello (Pisa)

On bi-skew braces and related topics

Abstract

The main goal of the talk is to discuss some new results on bi-skew braces, which are examples of skew braces recently introduced by L. N. Childs. The talk is divided into three parts. In the first part, after recalling the definition and some known characterisations, we present new structural results on bi-skew braces, showing that Byott’s conjecture holds in this setting. In the second part, we introduce a new way to obtain certain families of bi-skew braces, explaining the relation with other known constructions in the literature. In the final part, we explore the role of bi-skew braces in the well-known connections between skew braces and solutions of the Yang–Baxter equation.

Dora Puljic (Edinburgh) Braces and Classification Efforts

Abstract

A brace consists of a set with two operations, one forming an abelian group and the other a group, along with a certain distribution law. Braces were introduced by Wolfgang Rump in 2007 to help the study of non-degenerate, involutive, set-theoretic solutions to the Yang-Baxter equation. Connections to other objects have been found since - braid groups with an involutive braiding operator, bijective 1-cocycles, quantum groups etc. Recently, braces have been studied in relation to Hopf-Galois theory as skew braces parameterise Hopf-Galois extensions through regular subgroups of the holomorph. The aim of this talk is to give an overview of the field and the classification efforts made focusing on how braces relate to pre-Lie rings, and further how we can compute corresponding Hopf-Galois extensions.

Ilaria Colazzo (Exeter) Set-theoretic solution of the Pentagon Equation: the involutive case

Abstract

The pentagon equation appears in various contexts: For example, any finite-dimensional Hopf algebra is characterised by an invertible solution of the Pentagon Equation, or an arrow is a fusion operator for a fixed braided monoidal category if it satisfies the Pentagon Equation. This talk, based on joint work with E.Jespers and Ł. Kubat, will introduce the basic properties of set-theoretic solutions of the Pentagon Equation. Furthermore, we will look at bijective solutions, focusing on the involutive case. In the latter case, we provide a complete description of all involutive solutions and discuss when two involutive solutions are isomorphic.