Projects per year
Abstract / Description of output
Let A be a brace of cardinality pn where p>n+1 is prime, and let ann(p2) be the set of elements of additive order at most p2 in this brace. We construct a preLie ring related to the brace A/ann(p2).
In the case of strongly nilpotent braces of nilpotency index k<p the brace A/ann(p2) can be recovered by applying the construction of the group of flows to the resulting preLie ring. We don't know whether, when applied to braces which are not right nilpotent, our construction is related to the group of flows. We use powerful Lie rings associated with finite pgroups in the study of brace automorphisms with few fixed points. As an application we bound the number of elements which commute with a given element in a brace, as well as the number of elements which multiplied from left by a given element give zero.
We also study various Lie rings associated to powerful groups and braces whose adjoint groups are powerful, and show that the obtained Lie and preLie rings are also powerful.
We also show that braces whose adjoint groups are powerful and powerful left nilpotent preLie rings are in onetoone correspondence and that they are left and right nilpotent under some cardinality assumptions.
In the case of strongly nilpotent braces of nilpotency index k<p the brace A/ann(p2) can be recovered by applying the construction of the group of flows to the resulting preLie ring. We don't know whether, when applied to braces which are not right nilpotent, our construction is related to the group of flows. We use powerful Lie rings associated with finite pgroups in the study of brace automorphisms with few fixed points. As an application we bound the number of elements which commute with a given element in a brace, as well as the number of elements which multiplied from left by a given element give zero.
We also study various Lie rings associated to powerful groups and braces whose adjoint groups are powerful, and show that the obtained Lie and preLie rings are also powerful.
We also show that braces whose adjoint groups are powerful and powerful left nilpotent preLie rings are in onetoone correspondence and that they are left and right nilpotent under some cardinality assumptions.
Original language  English 

Journal  Proceedings of the american mathematical society 
Publication status  Accepted/In press  9 Oct 2023 
Fingerprint
Dive into the research topics of 'From braces to preLie rings'. Together they form a unique fingerprint.Projects
 1 Finished

Enhancing Representation Theory, Noncommutative Algebra And Geometry Through Moduli, Stability And Deformations
Gordon, I., Bayer, A. & Smoktunowicz, A.
1/05/18 → 30/04/24
Project: Research