Copyright

Jan E. Grabowski

Published On

2025-09-22

Page Range

pp. 109–146

Language

  • English

Print Length

38 pages

4. Modules

  • Jan E. Grabowski (author)
Chapter of: Representation Theory: A Categorical Approach (pp. 109–146)

In the last chapter, we first looked at representations of groups on sets, realizing that the minimal assumptions make group actions hard to study collectively. We then linearized, which opens up linear algebra for use in studying linear representations. But again, we were dissatisfied: the constructions one would like to do with an algebraic object are awkward. When we considered quiver representations, something else happened. Now we ended up with a rather different flavour of representation and it was not clear how it related to the cases of groups or algebras. That would mean having to develop two separate lots of theory.

We claim that the answer to all these problems is to study modules instead. We will be able to make all the constructions we wanted, and more, and we will be able to leverage the same theory in many different settings.

Contributors

Jan E. Grabowski

(author)
Professor of Algebra at Lancaster University
https://www.maths.lancs.ac.uk/~grabowsj/

Jan Grabowski is an experienced researcher and educator in mathematics, with over 20 years’ experience of both, and currently holds the position of Professor of Algebra at Lancaster University. Jan has a strong research track record of publications in algebra and related topics, both as a single author and collaboratively. He has taught courses at Oxford and Lancaster across a range of levels and has been teaching abstract algebra for the majority of this time, including covering aspects of the material in the book in courses at both institutions. Jan has had recognition for his teaching, including a teaching prize at Oxford, obtaining a Postgraduate Certificate in Academic Practice and being awarded Senior Fellowship of the Higher Education Academy. He has championed innovation in teaching by presenting a mature exposition of ‘how mathematicians really think about these things’ as compared with other approaches that often defer more advanced ideas or techniques, rather than encouraging students to engage with challenging ideas early and repeatedly.