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: d1b2663123f5a5828cad19d9022059d8527ec475
Parent: 491346ac0d57fd72dd8538362dde464788b16ee4
Author: Pablo <pablo-escobar@riseup.net>
Date: Wed, 28 Dec 2022 21:26:22 +0000

Changed the notation for the Weyl group

Diffstat

2 files changed, 17 insertions, 17 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	10	5	5
Modified	sections/semisimple-algebras.tex	24	12	12

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -282,23 +282,23 @@ is well understood. Namely, Fernando himself established\dots
   \mathfrak{p}^\sigma\) and \(M \cong \sigma N\) for some\footnote{Here
   $\mathfrak{p}^\sigma$ denotes the image of $\mathfrak{p}$ under the
   automorphism of $\sigma : \mathfrak{g} \to \mathfrak{g}$ given by the
-  canonical action of $\mathcal{W}$ on $\mathfrak{g}$ and $\sigma N$ is the
+  canonical action of $W$ on $\mathfrak{g}$ and $\sigma N$ is the
   $\mathfrak{p}$-module given by composing the map $\mathfrak{p}' \to
   \mathfrak{gl}(N)$ with the restriction $\sigma\!\restriction_{\mathfrak{p}} :
-  \mathfrak{p} \to \mathfrak{p}'$.} \(\sigma \in \mathcal{W}_M\), where
+  \mathfrak{p} \to \mathfrak{p}'$.} \(\sigma \in W_M\), where
   \[
-    \mathcal{W}_M
+    W_M
     = \langle
       \sigma_\beta : \beta \in \Sigma, H_\beta + \mathfrak{nil}(\mathfrak{p})
       \ \text{is central in}\ \mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}
       \ \text{and}\ H_\beta\ \text{acts as a positive integer in}\ M
       \rangle
-    \subset \mathcal{W}
+    \subset W
   \]
 \end{proposition}
 
 \begin{note}
-  The definition of the subgroup \(\mathcal{W}_M \subset \mathcal{W}\) is
+  The definition of the subgroup \(W_M \subset W\) is
   independent of the choice of basis \(\Sigma\).
 \end{note}

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -573,7 +573,7 @@ continuous strings symmetric with respect to the lines \(K \alpha\) with
 conclusion\dots
 
 \begin{definition}
-  We refer to the group \(\mathcal{W} = \langle \sigma_\alpha : \alpha \in
+  We refer to the group \(W = \langle \sigma_\alpha : \alpha \in
   \Delta^+ \rangle \subset \operatorname{O}(\mathfrak{h}^*)\) as \emph{the Weyl
   group of \(\mathfrak{g}\)}.
 \end{definition}
@@ -582,15 +582,15 @@ conclusion\dots
   The weights of a simple \(\mathfrak{g}\)-module \(M\) with highest weight
   \(\lambda\) are precisely the elements of the weight lattice \(P\) congruent
   to \(\lambda\) modulo the root lattice \(Q\) lying inside the convex hull of
-  the orbit of \(\lambda\) under the action of the Weyl group \(\mathcal{W}\).
+  the orbit of \(\lambda\) under the action of the Weyl group \(W\).
 \end{theorem}
 
 Aside from showing up in the previous theorem, the Weyl group will also play an
 important role in chapter~\ref{ch:mathieu} by virtue of the existence of a
-canonical action of \(\mathcal{W}\) on \(\mathfrak{h}\). By its very nature,
-\(\mathcal{W}\) acts in \(\mathfrak{h}^*\). If we conjugate the action
+canonical action of \(W\) on \(\mathfrak{h}\). By its very nature,
+\(W\) acts in \(\mathfrak{h}^*\). If we conjugate the action
 \(\sigma\!\restriction_{\mathfrak{h}^*} : \mathfrak{h}^* \isoto
-\mathfrak{h}^*\) of some \(\sigma \in \mathcal{W}\) by the isomorphism
+\mathfrak{h}^*\) of some \(\sigma \in W\) by the isomorphism
 \(\mathfrak{h}^* \isoto \mathfrak{h}\) afforded by the restriction of the
 Killing for to \(\mathfrak{h}\) we get a linear automorphism
 \(\sigma\!\restriction_{\mathfrak{h}} : \mathfrak{h} \isoto \mathfrak{h}\). As
@@ -609,17 +609,17 @@ translates into the following results, which we do not prove -- but see
   e^{\operatorname{ad}(E_\alpha)} e^{- \operatorname{ad}(F_\alpha)}
   e^{\operatorname{ad}(E_\alpha)} : \mathfrak{g} \isoto \mathfrak{g}\). Then
   \(\tilde\sigma_\alpha\) is an automorphism of Lie algebras, and this defines
-  an action of \(\mathcal{W}\) on \(\mathfrak{g}\) which is compatible with the
-  canonical action of \(\mathcal{W}\) on \(\mathfrak{h}\) -- i.e.
+  an action of \(W\) on \(\mathfrak{g}\) which is compatible with the
+  canonical action of \(W\) on \(\mathfrak{h}\) -- i.e.
   \(\tilde\sigma\!\restriction_{\mathfrak{h}} =
-  \sigma\!\restriction_{\mathfrak{h}}\) for all \(\sigma \in \mathcal{W}\).
+  \sigma\!\restriction_{\mathfrak{h}}\) for all \(\sigma \in W\).
 \end{proposition}
 
 \begin{note}
-  Notice that the action of \(\mathcal{W}\) on \(\mathfrak{g}\) from
+  Notice that the action of \(W\) on \(\mathfrak{g}\) from
   Proposition~\ref{thm:weyl-group-action} is not canonical, since it depends on
   the choice of \(E_\alpha\) and \(F_\alpha\). Nevertheless, \(\mathfrak{h}\)
-  is stable under the action of \(\mathcal{W}\) -- i.e. \(\mathcal{W} \cdot
+  is stable under the action of \(W\) -- i.e. \(W \cdot
   \mathfrak{h} \subset \mathfrak{h}\) -- and the restriction of this action to
   \(\mathfrak{h}\) is independent of any choices.
 \end{note}
@@ -914,10 +914,10 @@ this is not a coincidence.
 The proof of Proposition~\ref{thm:verma-is-finite-dim} is very technical and we
 won't include it here, but the idea behind it is to show that the set of
 weights of \(L(\lambda)\) is stable under the natural action of the Weyl group
-\(\mathcal{W}\) on \(\mathfrak{h}^*\). One can then show that the every weight
+\(W\) on \(\mathfrak{h}^*\). One can then show that the every weight
 of \(L(\lambda)\) is conjugate to a single dominant integral weight of
 \(L(\lambda)\), and that the set of dominant integral weights of \(L(\lambda)\)
-is finite. Since \(\mathcal{W}\) is finitely generated, this implies the set of
+is finite. Since \(W\) is finitely generated, this implies the set of
 weights of the unique simple quotient of \(M(\lambda)\) is finite. But
 each weight space is finite-dimensional. Hence so is the simple quotient
 \(L(\lambda)\).