MAGIC111: Introduction to set-theoretic solutions to the Yang-Baxter equation and skew braces

Course details

A specialist MAGIC course


Spring 2023
Monday, January 23rd to Friday, March 31st


Live lecture hours
Recorded lecture hours
Total advised study hours


12:05 - 12:55 (UK)


The Yang-Baxter equation originates from papers by Yang and Baxter on statistical mechanics, and the search for solutions has attracted numerous studies both in mathematical physics and pure mathematics. As the study of arbitrary solutions is extremely hard, Drinfeld initiated the investigations of set-theoretic solutions of the Yang-Baxter equation, i.e. solutions induced by a linear extension of a bijective map defined on a basis. New algebraic structures, called skew braces, have been introduced to describe such solutions.

In the course, we will introduce the Yang-Baxter equation and the algebraic theory of skew braces. We will focus on radical rings as a concrete example of skew braces. We build the connection between set-theoretical solutions to the Yang-Baxter equation and skew braces: we show that any skew brace gives rise to a solution of the Yang-Baxter equation and that this construction is universal. We will finish the course by introducing fascinating connections among skew braces, regular subgroups and bijective 1-cocycles.


  • Essential: undergraduate group theory. 
  • Advantageous: undergraduate ring theory. 
We will introduce all the necessary definitions in the course.


  • The Yang-Baxter equation.
  • Radical rings and set-theoretic solutions to the Yang-Baxter equation.
  • Skew braces.
  • Ideals.
  • Skew braces and set-theoretic solutions to the Yang-Baxter equation.
  • Skew braces and bijective 1-cocycles (optional).
  • Skew braces and regular subgroups (optional).


  • Dr Ilaria Colazzo

    Dr Ilaria Colazzo

    University of Leeds


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The assessment for this course will be released on Monday 1st May 2023 at 00:00 and is due in before Monday 15th May 2023 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.


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