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Project
p-local properties of Skew Braces and solutions to the Yang-Baxter equations (FWOTM1118)
The Yang-Baxter Equation (YBE) is an important problem both in
physics and mathematics, whose general solutions are not classified
so far.
It is the main goal of this project to give further understanding of the
properties of some groups constructed from particular solutions to
the YBE. More concretely, given a solution, we consider the structure
group, introduced by Etingof, Schedler and Soloviev, which is defined
in terms of a suitable group presentation derived from a fixed solution
to the YBE. This group, among many interesting properties, has the
structure of a skew brace which, roughly, is a set with two group
structures and a compatibility rule between them.
In this project, we propose to study the p-local properties of skew
braces via p-subgroup complexes and fusion systems. This will allow
us to apply not only purely algebraic techniques, but also geometric
combinatoric methods to yield consequences on the structure of
skew braes (and hence on solutions to the YBE). We will also
address the problem of understanding the relation between the
simple composition factors that appear in the two groups defining a
skew brace, with the aim of studying a conjecture raised by Byott on
solvable skew braces. Finally, for not necessarily finite skew braces,
we will focus on understanding some problems from the
combinatorial group theory viewpoint, such as the relation between
the properties UP and diffuse
physics and mathematics, whose general solutions are not classified
so far.
It is the main goal of this project to give further understanding of the
properties of some groups constructed from particular solutions to
the YBE. More concretely, given a solution, we consider the structure
group, introduced by Etingof, Schedler and Soloviev, which is defined
in terms of a suitable group presentation derived from a fixed solution
to the YBE. This group, among many interesting properties, has the
structure of a skew brace which, roughly, is a set with two group
structures and a compatibility rule between them.
In this project, we propose to study the p-local properties of skew
braces via p-subgroup complexes and fusion systems. This will allow
us to apply not only purely algebraic techniques, but also geometric
combinatoric methods to yield consequences on the structure of
skew braes (and hence on solutions to the YBE). We will also
address the problem of understanding the relation between the
simple composition factors that appear in the two groups defining a
skew brace, with the aim of studying a conjecture raised by Byott on
solvable skew braces. Finally, for not necessarily finite skew braces,
we will focus on understanding some problems from the
combinatorial group theory viewpoint, such as the relation between
the properties UP and diffuse
Date:1 Nov 2022 → Today
Keywords:Mathematics
Disciplines:Group theory and generalisations, Algebraic topology, Algebra not elsewhere classified, Order, lattices, ordered algebraic structures, General mathematics not elsewhere classified