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Project
The combinatorics of the Yang-Baxter equation and related algebraic structures (FWOTM1085)
One of the fundamental equations of mathematical physics is the
Yang-Baxter equation (YBE), but a complete description of all its
solutions remains an open problem. Therefore, in 1992, Drinfel'd
proposed the subproblem of describing the so called set-theoretic
solutions of the YBE. In 2006, Rump introduced a new algebraic
structure called a left brace, which was later generalized to the notion
of a skew left brace. More precisely these are sets with two group
operations and a distributive type law relating them, and turned out to
be inherently connected to set-theoretic solutions. A prime example
of this connection was recently described by Bachiller, who proved
that a complete classification of skew braces gives a complete
classification of set-theoretic solutions of the YBE. This result, among
others, motivates the study of skew braces in its own right.
My research project has three main objectives. The first one
concerns a study of the structure group and monoid associated to a
solution, more specifically the divisibility structure of the structure
monoid and the Garside structure of the structure group will be
studied. The second research objective focuses on the structure of
finite skew left braces, e.g. a novel approach will be to use
representation theory of groups. At last, I will use the results of the
previous objectives in the classification of set-theoretic solutions and
try to obtain estimates on the set-theoretic solutions of a given size
Yang-Baxter equation (YBE), but a complete description of all its
solutions remains an open problem. Therefore, in 1992, Drinfel'd
proposed the subproblem of describing the so called set-theoretic
solutions of the YBE. In 2006, Rump introduced a new algebraic
structure called a left brace, which was later generalized to the notion
of a skew left brace. More precisely these are sets with two group
operations and a distributive type law relating them, and turned out to
be inherently connected to set-theoretic solutions. A prime example
of this connection was recently described by Bachiller, who proved
that a complete classification of skew braces gives a complete
classification of set-theoretic solutions of the YBE. This result, among
others, motivates the study of skew braces in its own right.
My research project has three main objectives. The first one
concerns a study of the structure group and monoid associated to a
solution, more specifically the divisibility structure of the structure
monoid and the Garside structure of the structure group will be
studied. The second research objective focuses on the structure of
finite skew left braces, e.g. a novel approach will be to use
representation theory of groups. At last, I will use the results of the
previous objectives in the classification of set-theoretic solutions and
try to obtain estimates on the set-theoretic solutions of a given size
Date:1 Nov 2021 → Today
Keywords:Yang-Baxter equation, skew braces, groups
Disciplines:Associative rings and algebras, Group theory and generalisations