Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Restructured the preface
Added dedicated "Acknowledgements" and "License" sections to the preface
Also expanded the comments on my creative process and the contents of the notes
1 file changed, 38 insertions, 18 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/preface.tex | 56 | 38 | 18 |
diff --git a/sections/preface.tex b/sections/preface.tex @@ -2,20 +2,36 @@ \thispagestyle{empty} -These notes were originally based on my undergraduate dissertation, produced in -2022 under the supervision of professor Iryna Kashuba of the department of -mathematics of the Institute of Mathematics and Statistics of the University of -São Paulo (IME-USP), Brazil. +These notes are perhaps better understood as a coming-of-age tale. They were +originally part of some notes of mine on representations of Lie groups, written +in early 2021 as part of my second scientific internship project with professor +Iryna Kashuba of the department of mathematics of the Institute of Mathematics +and Statistics of the University of São Paulo (IME-USP), Brazil. These were +later adapted and expanded into my undergraduate dissertation, produced in late +2022 under the supervision of professor Kashuba. In mid 2023, after the +publication of my undergraduate dissertation, the notes were once again +expanded with the addition of their final chapter. All in all, I have been +working on the prose that follows for the better part of my early higher +education. + +As they currently stand, the subject of these notes is a select topic in the +representation theory of semisimple Lie algebras: Olivier Mathieu's +classification of simple weight modules. Its first four chapters consist of a +pretty standard account of the basic theory of semisimple Lie algebras and +their finite-dimensional representations, providing a concise exposition of the +background required for understanding the classification. On the other hand, +the last two chapter of the notes should be understood as a reading guide for +Mathieu's original paper \cite{mathieu}, with an emphasis on the intuition +behind its major results. Throughout these notes we will follow some guiding principles. First, lengthy proofs are favored as opposed to collections of smaller lemmas. This is a deliberate effort to emphasize the relevant results. Secondly, and this is more important, we are primarily interested in the broad strokes of the theory -highlighted in the following chapters. The topic of the dissertation at hand, -the representation theory of Lie algebras, is a profoundly technical one. In -particular, certain proofs can sometimes feel like an unmotivated pile of -technical arguments. Instead, we prefer to focus on the intuition behind the -relevant results. +highlighted in the following chapters. This is because the topic of the +dissertation at hand is a profoundly technical one. In particular, certain +proofs can sometimes feel like an unmotivated pile of technical arguments. +Instead, we prefer to focus on the intuition behind the relevant results. Hence some results are left unproved. Nevertheless, we include numerous references throughout the text to other materials where the reader can find @@ -26,24 +42,28 @@ assume familiarity with the language of categories, functors and adjunctions. Understanding some examples in the introductory chapter requires basic knowledge of differential and algebraic geometry, as well as rings of differential operators, but these examples are not necessary to the -comprehension of the following chapters. Additional topics will be covered in -the notes as needed. +comprehension of the notes as a whole. Additional topics will be covered in the +notes as needed. -This document was typeset and compiled using free software. Its \LaTeX~source -code is freely available at -\url{https://git.pablopie.xyz/lie-algebras-and-their-representations}, for -distribution under the terms of the Creative Commons Attribution 4.0 license. +\section*{Acknowledgments} I would like to thank my family for their tireless love and support. I would specially like to thank professor Kashuba and Eduardo Monteiro Mendonça for -their support. Their guidance over the past two years has profoundly shaped my -passion for mathematics, and I could not have gotten to where I am now without -their help. I suppose I should also thank them for enduring my chaotic +their support. Their guidance over the past three years has profoundly shaped +my passion for mathematics, and I could not have gotten to where I am now +without their help. I suppose I should also thank them for enduring my chaotic digressions on the relationship between representation theory and geometry. Finally, I would like to thank my dear friend Lucas Dias Schiezari for somehow convincing me to apply for a bachelors degree in pure mathematics, as well as the moments of joy we shared. May he rest in peace. +\section*{License} + +This document was typeset and compiled using free software. Its \LaTeX~source +code is freely available at +\url{https://git.pablopie.xyz/lie-algebras-and-their-representations}, for +distribution under the terms of the Creative Commons Attribution 4.0 license. + \newpage \thispagestyle{empty}