The interplay between the Yang-Baxter equation and its associated algebraic structures, with special focus on old conjectures.

Project Details

Description

In this project we propose to study set-theoretic solutions of the
Yang-Baxter equation and their associated algebraic structures with
group and ring theoretical techniques. In particular, we will link
properties of bijective (left) non-degenerate set-theoretic solutions to
algebraic properties of their structure groups/monoids and their
structure algebras .

Secondly, as structure groups of bijective non-degenerate settheoretic solutions carry a natural skew left brace structure, we will develop, using inspiration from group and ring theory, novel techniques to study skew left braces. In particular, we will investigate factorizations of skew left braces, which can be linked to
decompositions of the associated solution, and radicals of skew left
braces, which will provide more information on the build-up of skew
left braces. Moreover, as skew left braces are generalizations of
radical rings and have a flavor of group theory, we propose to
establish and study the notion of a module of a (skew) left brace,
based upon the recently introduced modules of left trusses. This will
allow to study skew left braces from the viewpoint of representation
theory.

Lastly, the new toolbox, designed in the previous objectives, will
provide new methods to attack old conjectures, such as the
Kaplansky conjecture and the Köthe conjecture, which have a link to
left braces. In particular, these methods will provide new positive
partial results and/or inspiration for a possible counterexample
Acronymfwotm992
StatusActive
Effective start/end date1/10/2030/09/23

Flemish discipline codes

  • Associative rings and algebras
  • Group theory and generalisations
  • Algebra not elsewhere classified

Keywords

  • Groups
  • rings
  • Yang-Baxter equation