lie-algebras-and-their-representations

 \chapter*{About These Notes}
 
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 \begin{note}
   Under construction!
 \end{note}
 
 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. 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
 complete proofs. We will assume basic knowledge of abstract algebra. In
 particular, we assume that the reader is familiarized with multi-linear
 algebra, the theory of modules over an algebra and exact sequences. We also
 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 notes as a whole. Additional topics will be covered in the
 notes as needed.
 
 \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 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
 \href{https://creativecommons.org/licenses/by/4.0/}{Creative Commons
 Attribution 4.0 license}.
 
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