Ternary Day Edinburgh 2022

Heriot-Watt University, 17 June 2022

A workshop on ternary and higher-arity structures in algebra, geometry and category theory bringing together mathematicians, physicists and computer scientists. This meeting is intended to communicate recent research on non-binary, non-sequential or otherwise exotic structures in multiple areas of mathematical sciences and to create the opportunity for cross-pollination between different areas of specialized research.

Logistics

The meeting will take place in the Heriot-Watt University campus (map) during Friday 17 June 2022. We will gather in the morning for coffee at the Colin Maclaurin Building (map) Common Room (1st floor) and then move to Lecture Theatre 2, easy to find from the campus Main Reception (map). Each 1-hour session will have a speaker presenting for approximately 30 minutes and the rest of the hour will be used for questions, discussions and coffee breaks. We will follow the following schedule, although exact timings may slightly vary since we would like to encourage informal discussions to take place organically:

10:30 - 11:00 Reception and coffee at Colin Maclaurin Building
11:00 - 12:00 Opening, talk by Carlos Zapata-Carratalá and discussion
12:00 - 13:00 Talk by José Figueroa-O'Farrill and discussion
13:00 - 14:00 Lunch break
14:00 - 15:00 Talk by Bernard Rybolowicz and discussion
15:00 - 16:00 Talk by Tomasz Brzezinski, coffee and discussion
16:00 - 17:00 Talk by Anastasia Doikou and discussion

Talks

Chemistries and Heapoids
Carlos Zapata-Carratalá

I will introduce a class of partial algebraic structures called 'chemistries' which I have recently defined as a parsimonious higher-arity generalization of categories where commutative diagrams are given by hypergraphs (instead of the usual directed graphs in ordinary categories). I will show that this new notion encompasses some well-known examples such as (semi)categories, groupoids and operads in a very natural way. I will then show that (semi)heapoids are naturally expressed as chemistries with a ternary composition operation, thus connecting with long-forgotten research by V.V. Wagner on partial (semi)heaps.

Metric 3-Leibniz algebras in Physics
José Figueroa-O'Farrill

I will describe some old work of mine in collaboration with Paul de Medeiros, Elena Méndez-Escobar and Patricia Ritter on a certain class of 3-Leibniz algebras admitting an “ad”-invariant inner product which played a decisive rôle in our understanding of the AdS_4/CFT_3 correspondence associated to M2 branes in M-theory. The talk will concentrate on the algebraic structure and its intimate relationship with Lie (super)algebras, but I’d be happy to answer questions about the Physics during the discussion. These metric 3-Leibniz algebras include a large class of ternary algebras: 3-Lie, (anti) Lie triple systems, (anti) Jordan triple systems, quaternionic triple systems,… and can be constructed from a metric Lie algebra and a unitary representation. Some of them are Lie embedabble and the 3-bracket can be understood as nested brackets in a Lie (super)algebra.

Introduction to the Theory of Trusses
Bernard Rybolowicz

I will present a definition of a truss and discuss its connections to rings and braces. Next, I will introduce a module over a truss and discuss its link to modules over a ring. The main goal of the talk will be to consider and discuss the quotients of previously introduced structures. That will provide us with an interpretation of what a truss is in the ring theory.

Towards Hochschild Cohomology of Trussess
Tomasz Brzezinski

An attempt to define Hochschild cohomology of trusses is presented.

Non-involutive solutions of the Yang-Baxter equation from (skew) braces
Anastasia Doikou

I will discuss novel non-involutive solutions of the Yang-Baxter equation coming from (skew) braces. These solutions are generalisations of the known ones coming from the standard braces and skew braces and are related to certain ternary operations. In the case of two-sided (skew) braces one can assign such solutions to every element of the set. Novel bijective maps associated to the inverse solutions are also introduced.