lie-algebras-and-their-representations

01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77

\chapter*{About These Notes}

\thispagestyle{empty}

\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}.

\newpage

\thispagestyle{empty}