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: 2f81acdfc052234279195fd9bb2946e70bc6addf
Parent: 76753d1d501c15b6284fc68958c7bf8d910a4e23
Author: Pablo <pablo-escobar@riseup.net>
Date: Fri, 13 Jan 2023 01:32:31 +0000

Extracted a result to a couple of examples

Extracted the result on the fact that all finite-dimensional irreducible representations of the direct sum are tensor products o irreducible representations of each factor to a separate example

Diffstat

3 files changed, 53 insertions, 34 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/complete-reducibility.tex	28	24	4
Modified	sections/introduction.tex	10	8	2
Modified	sections/mathieu.tex	49	21	28

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -48,6 +48,15 @@ smaller pieces. This leads us to the following definitions.
   \(K\)-subspaces, let alone \(\mathfrak{g}\)-submodules.
 \end{example}
 
+\begin{example}\label{ex:all-simple-reps-are-tensor-prod}
+  Given a finite-dimensional simple \(\mathfrak{g}\)-module \(M\) and a
+  finite-dimensional simple \(\mathfrak{h}\)-module \(N\), the tensor product
+  \(M \otimes N\) is a simple \(\mathfrak{g} \oplus \mathfrak{h}\)-module. All
+  finite-dimensional simple \(\mathfrak{g} \oplus \mathfrak{h}\)-modules have
+  the form \(M \otimes N\) for unique (up to isomorphism) \(M\) and \(N\) --
+  see \cite[ch.~3]{etingof}.
+\end{example}
+
 The general strategy for classifying finite-dimensional modules over an algebra
 is to classify the indecomposable modules. This is because\dots
 
@@ -940,12 +949,23 @@ This sequence always splits, which implies we can deduce information about
 part'' \(\mfrac{\mathfrak{g}}{\mathfrak{rad}(\mathfrak{g})}\) -- see
 Proposition~\ref{thm:quotients-by-rads}. In practice this translates to\dots
 
-\begin{proposition}
-  Every simple \(\mathfrak{g}\)-module is the tensor product of
-  a simple \(\mfrac{\mathfrak{g}}{\mathfrak{rad}(\mathfrak{g})}\)-module 
-  and a \(1\)-dimensional \(\mathfrak{rad}(\mathfrak{g})\)-module.
+\begin{proposition}[Lie]\label{thm:lie-thm-solvable-reps}
+  Let \(\mathfrak{g}\) be a solvable Lie algebra. Every finite-dimensional
+  simple \(\mathfrak{g}\)-module is \(1\)-dimensional.
 \end{proposition}
 
+\begin{corollary}
+  Let \(\mathfrak{g}\) be a Lie algebra. Every finite-dimensional simple
+  \(\mathfrak{g}\)-module is the tensor product of a simple
+  \(\mfrac{\mathfrak{g}}{\mathfrak{rad}(\mathfrak{g})}\)-module and a
+  \(1\)-dimensional \(\mathfrak{rad}(\mathfrak{g})\)-module.
+\end{corollary}
+
+\begin{proof}
+  This follows at once from Proposition~\ref{thm:lie-thm-solvable-reps} and
+  Example~\ref{ex:all-simple-reps-are-tensor-prod}.
+\end{proof}
+
 Having finally reduced our initial classification problem to that of
 classifying the finite-dimensional simple \(\mathfrak{g}\)-modules, we can now
 focus exclusively in this particular class of \(\mathfrak{g}\)-modules.

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -1070,10 +1070,10 @@ This last proposition is known as \emph{Frobenius reciprocity}, and was first
 proved by Frobenius himself in the context of finite groups. Another
 interesting construction is\dots
 
-\begin{example}\index{\(\mathfrak{g}\)-module!tensor product}
+\begin{example}\label{ex:tensor-prod-separate-algs}\index{\(\mathfrak{g}\)-module!tensor product}
   Let \(\mathfrak{g}\) and \(\mathfrak{h}\) be Lie algebras. Given a
   \(\mathfrak{g}\)-module \(M\) and a \(\mathfrak{h}\)-module \(N\), the space
-  \(M \boxtimes N = M \otimes_K N\) has the natural structure of a
+  \(M \otimes N = M \otimes_K N\) has the natural structure of a
   \(\mathfrak{g} \oplus \mathfrak{h}\)-module, where the action of
   \(\mathfrak{g} \oplus \mathfrak{h}\) is given by
   \[
@@ -1081,6 +1081,12 @@ interesting construction is\dots
   \]
 \end{example}
 
+Example~\ref{ex:tensor-prod-separate-algs} thus provides a way to describe
+representations of \(\mathfrak{g} \oplus \mathfrak{h}\) in terms of the
+representations of \(\mathfrak{g}\) and \(\mathfrak{h}\). We will soon see that
+in many cases \emph{all} (simple) \(\mathfrak{g} \oplus \mathfrak{h}\)-modules
+can be constructed in such a manner.
+
 This concludes our initial remarks on \(\mathfrak{g}\)-modules. In the
 following chapters we will explore the fundamental problem of representation
 theory: that of classifying all representations of a given algebra up to

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -105,6 +105,20 @@ to the case it holds. This brings us to the following definition.
   \left(\mfrac{M}{N}\right)_\lambda\) is surjective.
 \end{example}
 
+\begin{example}\label{thm:simple-weight-mod-is-tensor-prod}
+  Let \(\mathfrak{g} = \mathfrak{z} \oplus \mathfrak{s}_1 \oplus \cdots \oplus
+  \mathfrak{s}_r\) be a reductive Lie algebra, where \(\mathfrak{z}\) is the
+  center of \(\mathfrak{g}\) and \(\mathfrak{s}_1, \ldots, \mathfrak{s}_r\) are
+  its simple components. As in
+  Example~\ref{ex:all-simple-reps-are-tensor-prod}, any simple weight
+  \(\mathfrak{g}\)-module \(M\) can be decomposed as
+  \[
+    M \cong Z \otimes M_1 \otimes \cdots \otimes M_r
+  \]
+  where \(Z\) is a \(1\)-dimensional representation of \(\mathfrak{z}\) and
+  \(M_i\) is a simple weight \(\mathfrak{s}_i\)-module.
+\end{example}
+
 A particularly well behaved class of examples are the so called
 \emph{admissible} weight modules.
 
@@ -1409,22 +1423,12 @@ coherent \(\mathfrak{g}\)-families themselves, which is the subject of sections
 We unfortunately do not have the necessary space to discuss these results in
 detail, but we will now provide a brief overview.
 
-First and foremost, the problem of classifying \(\mathfrak{g}\)-family can be
-reduced to that of classifying only \(\mathfrak{sl}_n(K)\)-families and
-coherent \(\mathfrak{sp}_{2 n}(K)\)-families. This is because of the following
-results.
-
-\begin{proposition}
-  If \(\mathfrak{g} = \mathfrak{z} \oplus \mathfrak{s}_1 \oplus \cdots \oplus
-  \mathfrak{s}_r\), where \(\mathfrak{z}\) is the center of \(\mathfrak{g}\)
-  and \(\mathfrak{s}_1, \ldots, \mathfrak{s}_r\) are its simple components,
-  then any simple weight \(\mathfrak{g}\)-module \(M\) can be decomposed as
-  \[
-    M \cong Z \boxtimes M_1 \boxtimes \cdots \boxtimes M_r
-  \]
-  where \(Z\) is a \(1\)-dimensional representation of \(\mathfrak{z}\) and
-  \(M_i\) is an irreducible weight \(\mathfrak{s}_i\)-module.
-\end{proposition}
+First and foremost, notice that because of
+Example~\ref{thm:simple-weight-mod-is-tensor-prod} the problem of classifying
+the simple weight \(\mathfrak{g}\)-modules can be reduced to that of
+classifying the simple weight modules of its simple components. In addition, it
+turns out that very few simple Lie algebras admit cuspidal modules at all.
+Specifically\dots
 
 \begin{proposition}[Fernando]\label{thm:only-sl-n-sp-have-cuspidal}
   Let \(\mathfrak{s}\) be a finite-dimensional simple Lie algebra. Suppose
@@ -1432,18 +1436,7 @@ results.
   \cong \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong \mathfrak{sp}_{2 n}(K)\).
 \end{proposition}
 
-We have previously seen that the representations of Abelian Lie algebras,
-particularly the \(1\)-dimensional ones, are well understood. Hence to classify
-the simple modules of an arbitrary reductive algebra it suffices to classify
-those of its simple components. To classify these module we can apply
-Fernando's results and reduce the problem to constructing the cuspidal modules
-of the simple Lie algebras. But by
-Proposition~\ref{thm:only-sl-n-sp-have-cuspidal} only \(\mathfrak{sl}_n(K)\)
-and \(\mathfrak{sp}_{2 n}(K)\) admit cuspidal modules, so it suffices to
-consider these two cases.
-
-Finally, we apply Mathieu's results to further reduce the problem to that of
-classifying the irreducible semisimple coherent families of
+Hence it suffices to classify the irreducible semisimple coherent families of
 \(\mathfrak{sl}_n(K)\) and \(\mathfrak{sp}_{2 n}(K)\). These can be described
 either algebraically, using combinatorial invariants -- which Mathieu does in
 sections 7, 8 and 9 of his paper -- or geometrically, using algebraic varieties