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Project
OZR backup mandate: Skew braces, quivers, and applications to the Yang-Baxter equation. (OZR4014)
The following research proposal aims to study solutions of the Yang- Baxter equations and certain ring-like algebraic structures known as skew braces. This research will follow two directions.
The first one is to study the structure theory of skew braces. The project is to deal with the classification of skew braces of prime- power size. The plan is to start analysing those of size a power of 2. Then developing group-theoretical techniques to be applied to the general case. Of course, the structure of skew braces is related to their automorphism groups. That is why other assignments of this project will be to study automorphism groups of skew braces and to classify skew braces with trivial automorphism group. Computer- generated examples will be helpful here to better understand the
latter mentioned phenomenon. Finally, the search for the right analog
of Köethe conjecture for skew braces is another quest of this research project. Here the intention is to develop algorithms to test which nil-related notion for skew braces is the most suitable. Then there will be a need to develop algebraic techniques to prove the conjecture in this context.
The second direction is to investigate the quiver-theoretical Yang- Baxter equations and their associated algebraic structures. This task requires studying the already known YBE in the context of quivers and then finding out what structures like skew braces appear in this setting. This piece of work opens an entirely new research direction.
The first one is to study the structure theory of skew braces. The project is to deal with the classification of skew braces of prime- power size. The plan is to start analysing those of size a power of 2. Then developing group-theoretical techniques to be applied to the general case. Of course, the structure of skew braces is related to their automorphism groups. That is why other assignments of this project will be to study automorphism groups of skew braces and to classify skew braces with trivial automorphism group. Computer- generated examples will be helpful here to better understand the
latter mentioned phenomenon. Finally, the search for the right analog
of Köethe conjecture for skew braces is another quest of this research project. Here the intention is to develop algorithms to test which nil-related notion for skew braces is the most suitable. Then there will be a need to develop algebraic techniques to prove the conjecture in this context.
The second direction is to investigate the quiver-theoretical Yang- Baxter equations and their associated algebraic structures. This task requires studying the already known YBE in the context of quivers and then finding out what structures like skew braces appear in this setting. This piece of work opens an entirely new research direction.
Date:1 Nov 2022 → 31 Oct 2023
Keywords:Skew braces, Yang-Baxter equation, Quivers
Disciplines:Associative rings and algebras, Group theory and generalisations, Non-associative rings and algebras, Algebra not elsewhere classified