lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
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Commit: e4156ebd9234c44a44269d8af459075a532634eb
Parent: 984fec64c5331590ec2a177a4d4c086501b5c50e
Author: Pablo <pablo-escobar@riseup.net>
Date: Fri, 25 Nov 2022 19:07:25 +0000

Revised the second chapter

Diffstat

2 files changed, 53 insertions, 44 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/complete-reducibility.tex	88	44	44
Modified	sections/introduction.tex	9	9	0

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -1,13 +1,14 @@
 \chapter{Semisimplicity \& Complete Reducibility}
 
+% TODO: Automate the "47 pages" thing
 Having hopefully established in the previous chapter that Lie algebras and
 their representations are indeed useful, we are now faced with the Herculean
-task of trying to understand them. We have seen that representations are a
-remarkably effective way to derive information about groups -- and therefore
-algebras -- but the question remains: how to we go about classifying the
-representations of a given Lie algebra? This is a question that have sparked an
-entire field of research, and we cannot hope to provide a comprehensive answer
-the 47 pages we have left. Nevertheless, we can work on particular cases.
+task of trying to understand them. We have seen that representations can be
+used to derive geometric information about groups, but the question remains:
+how to we go about classifying the representations of a given Lie algebra? This
+is a question that have sparked an entire field of research, and we cannot hope
+to provide a comprehensive answer the 47 pages we have left. Nevertheless, we
+can work on particular cases.
 
 For instance, one can readily check that a representation \(V\) of the
 \(n\)-dimensional Abelian Lie algebra \(K^n\) is nothing more than a choice of
@@ -16,14 +17,13 @@ canonical basis elements \(e_1, \ldots, e_n \in K^n\). In other words,
 classifying the representations of Abelian algebras is a trivial affair.
 Instead, we focus on the the finite-dimensional representations of a
 finite-dimensional Lie algebra \(\mathfrak{g}\) over an algebraically closed
-field of characteristic \(0\). But why are the representations semisimple
+field \(K\) of characteristic \(0\). But why are the representations semisimple
 algebras simpler -- or perhaps \emph{semisimpler} -- to understand than those
 of any old Lie algebra?
 
-We will come back to this question in a moment, but for now we simply note
-that, when solving a classification problem, it is often profitable to break
-down our structure is smaller peaces. This leads us to the following
-definitions.
+We will get back to this question in a moment, but for now we simply note that,
+when solving a classification problem, it is often profitable to break down our
+structure is smaller peaces. This leads us to the following definitions.
 
 \begin{definition}
   A representation of \(\mathfrak{g}\) is called \emph{indecomposable} if it is
@@ -32,7 +32,7 @@ definitions.
 
 \begin{definition}
   A representation of \(\mathfrak{g}\) is called \emph{irreducible} if it has
-  no nonzero subrepresentations.
+  no nonzero proper subrepresentations.
 \end{definition}
 
 \begin{example}
@@ -54,13 +54,13 @@ algebra is to classify the indecomposable representations. This is because\dots
 Hence finding the indecomposable representations suffices to find \emph{all}
 finite-dimensional representations: they are the direct sum of indecomposable
 representations. The existence of the decomposition should be clear from the
-definitions. Indeed, if \(V\) is representation of \(\mathfrak{g}\) a simple
-argument via induction in \(\dim V\) suffices to prove the existence: if \(V\)
-is indecomposable then there is nothing to prove, and if \(V\) is not
-indecomposable then \(V = W \oplus U\) for some \(W, U \subsetneq V\) nonzero
-subrepresentations, so that their dimensions are both strictly smaller than
-\(\dim V\) and the existence follows from the induction hypothesis. For a proof
-of uniqueness please refer to \cite{etingof}.
+definitions. Indeed, if \(V\) is a finite-dimensional representation of
+\(\mathfrak{g}\) a simple argument via induction in \(\dim V\) suffices to
+prove the existence: if \(V\) is indecomposable then there is nothing to prove,
+and if \(V\) is not indecomposable then \(V = W \oplus U\) for some \(W, U
+\subsetneq V\) nonzero subrepresentations, so that their dimensions are both
+strictly smaller than \(\dim V\) and the existence follows from the induction
+hypothesis. For a proof of uniqueness please refer to \cite{etingof}.
 
 Finding the indecomposable representations of an arbitrary Lie algebra,
 however, turns out to be a bit of a circular problem: the indecomposable
@@ -119,49 +119,49 @@ clear things up.
 \end{proposition}
 
 \begin{proof}
-  We begin by \(\textbf{(i)} \Rightarrow \textbf{(ii)}\). Let
+  We begin by \(\textbf{(i)} \implies \textbf{(ii)}\). Let
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} &
-      W \arrow{r}{f} &
-      V \arrow{r}{g} &
-      U \arrow{r} &
+      0 \arrow{r}      &
+      W \arrow{r}{i}   &
+      V \arrow{r}{\pi} &
+      U \arrow{r}      &
       0
     \end{tikzcd}
   \end{center}
   be an exact sequence of representations of \(\mathfrak{g}\). We can suppose
   without any loss of generality that \(W \subset V\) is a subrepresentation
-  and \(f\) is the usual inclusion, for if this is not the case there is an
+  and \(i\) is its inclusion in \(V\), for if this is not the case there is an
   isomorphism of sequences
   \begin{center}
     \begin{tikzcd}
-         0 \arrow{r} &
-         W \arrow{r}{f} \arrow[swap]{d}{f} &
-         V \arrow{r}{g} \arrow[Rightarrow, no head]{d} &
-         U \arrow{r}    \arrow[Rightarrow, no head]{d} &
-         0 \\
-         0 \arrow{r} &
-      f(W) \arrow{r} &
-         V \arrow[swap]{r}{g} &
-         U \arrow{r} &
+         0 \arrow{r}                                     &
+         W \arrow{r}{i}   \arrow[swap]{d}{i}             &
+         V \arrow{r}{\pi} \arrow[Rightarrow, no head]{d} &
+         U \arrow{r}      \arrow[Rightarrow, no head]{d} &
+         0                                               \\
+         0 \arrow{r}                                     &
+      i(W) \arrow{r}                                     &
+         V \arrow[swap]{r}{\pi}                          &
+         U \arrow{r}                                     &
          0
     \end{tikzcd}
   \end{center}
 
   It then follows from \textbf{(i)} that there exists a subrepresentation \(U'
-  \subset V\) such that \(V = W \oplus U'\). Finally, the projection \(\pi : V
+  \subset V\) such that \(V = W \oplus U'\). Finally, the projection \(T : V
   \to W\) is an intertwiner satisfying
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} &
-      W \arrow{r}{f} &
-      V \arrow{r}{g} \arrow[bend left=30]{l}{\pi} &
-      U \arrow{r} &
+      0 \arrow{r}                                 &
+      W \arrow{r}{i}                              &
+      V \arrow{r}{\pi} \arrow[bend left=30]{l}{T} &
+      U \arrow{r}                                 &
       0
     \end{tikzcd}
   \end{center}
 
-  Next is \(\textbf{(ii)} \Rightarrow \textbf{(iii)}\). If \(V\) is an
+  Next is \(\textbf{(ii)} \implies \textbf{(iii)}\). If \(V\) is an
   indecomposable \(\mathfrak{g}\)-module and \(W \subset V\) is a
   subrepresentation, we have an exact sequence
   \begin{center}
@@ -178,11 +178,11 @@ clear things up.
   Since our sequence splits, we must have \(V \cong W \oplus \mfrac{V}{W}\).
   But \(V\) is indecomposable, so that either \(W = V\) or \(W = 0\). Since
   this holds for all \(W \subset V\), \(V\) is irreducible. For
-  \(\textbf{(iii)} \Rightarrow \textbf{(iv)}\) it suffices to apply
+  \(\textbf{(iii)} \implies \textbf{(iv)}\) it suffices to apply
   theorem~\ref{thm:krull-schmidt}.
 
-  Finally, for \(\textbf{(iv)} \Rightarrow \textbf{(i)}\), if we assume
-  \(\textbf{(iii)}\) and let \(V\) be a representation of \(\mathfrak{g}\) with
+  Finally, for \(\textbf{(iv)} \implies \textbf{(i)}\), if we assume
+  \(\textbf{(iv)}\) and let \(V\) be a representation of \(\mathfrak{g}\) with
   decomposition into irreducible subrepresentations
   \[
     V = \bigoplus_i V_i
@@ -457,7 +457,7 @@ basic}. In fact, all we need to know is\dots
   a more modern account using derived categories.
 \end{note}
 
-We are particular interested in the case where \(S = K\) is the trivial
+We are particularly interested in the case where \(S = K\) is the trivial
 representation of \(\mathfrak{g}\). Namely, we may define\dots
 
 \begin{definition}

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -360,6 +360,15 @@ also share structural features with groups. For example\dots
   for all \(X, Y \in \mathfrak{g}\).
 \end{definition}
 
+\begin{note}
+  Notice that an Abelian Lie algebra is determined by its dimension. Indeed,
+  any linear map \(\mathfrak{g} \to \mathfrak{h}\) between Abelian Lie algebras
+  \(\mathfrak{g}\) and \(\mathfrak{h}\) is a homomorphism of Lie algebras. In
+  particular, any linear isomorphism \(\mathfrak{g} \isoto K^n\) -- where
+  \(K^n\) is endowed with the trivial brackets \([v, w] = 0 \, \forall v, w \in
+  K^n\) -- is an isomorphism of Lie algebras for Abelian \(\mathfrak{g}\).
+\end{note}
+
 \begin{example}
   Let \(G\) be a connected algebraic \(K\)-group and \(\mathfrak{g}\) be its
   Lie algebra. Then \(G\) is Abelian if, and only if \(\mathfrak{g}\) is