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

Commit
c2a9394c0558a6f2aed221195d0cea8911fef779
Parent
7116b6fbd0d4e0dd5585bf4faa0937649a56bc69
Author
Pablo <pablo-escobar@riseup.net>
Date

Changed the notation of the last chapter

Changed the notation of the section on sl-families to work with sl(n) instead of sl(n+1)

Also changed the notation for sl(n)-sequences

Diffstat

1 file changed, 51 insertions, 53 deletions

Status File Name NĀ° Changes Insertions Deletions
Modified sections/coherent-families.tex 104 51 53
diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -162,17 +162,17 @@ We can find an orthonormal basis \(\{\epsilon_1, \ldots, \epsilon_n\}\) for
   with \(m_1 > m_2 > \cdots > m_n > 0\).
 \end{theorem}
 
-% TODOO: Change the notation in here to work with sl(n) (n >= 3) instead of
-% sl(n + 1) (n >= 2)
-\section{Coherent \(\mathfrak{sl}_{n + 1}(K)\)-families}
+\section{Coherent \(\mathfrak{sl}_n(K)\)-families}
+
+% TODO: Fix n >= 3
 
 % TODOO: Add notes about this basis beforehand
-Take the standard Cartan subalgebra \(\mathfrak{h} = \{ X \in \mathfrak{sl}_{n
-+ 1}(K) : X \ \text{is diagonal}\}\) of \(\mathfrak{sl}_{n + 1}(K)\) as in
+Take the standard Cartan subalgebra \(\mathfrak{h} = \{ X \in
+\mathfrak{sl}_n(K) : X \ \text{is diagonal}\}\) of \(\mathfrak{sl}_n(K)\) as in
 Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\epsilon_i,
-\ldots, \epsilon_{n + 1} \in \mathfrak{h}^*\) such that \(\epsilon_i(H)\) is the
+\ldots, \epsilon_n \in \mathfrak{h}^*\) such that \(\epsilon_i(H)\) is the
 \(i\)-th entry of the diagonal of \(H\). Let \(\Sigma = \{ \beta_1, \ldots,
-\beta_n \}\) be the basis for \(\Delta\) given by \(\beta_i = \epsilon_i -
+\beta_{n-1} \}\) be the basis for \(\Delta\) given by \(\beta_i = \epsilon_i -
 \epsilon_{i + 1}\).
 
 % TODO: Add some comments on the proof of this: while the proof that these
@@ -184,52 +184,50 @@ Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\epsilon_i,
 % there really isn't any reasonable name for it
 \begin{lemma}\label{thm:sl-bounded-weights}
   Let \(\lambda \notin P^+\) and \(A(\lambda) = \{ i : \lambda(H_{\beta_i})\
-  \text{is not a non-negative integer}\}\). Then \(L(\lambda)\) is bounded if,
-  and only if one of the following assertions holds.
+  \text{is \emph{not} a non-negative integer}\}\). Then \(L(\lambda)\) is
+  bounded if, and only if one of the following assertions holds.
   \begin{enumerate}
-    \item \(A(\lambda) = \{1\}\) or \(A(\lambda) = \{n\}\).
-    \item \(A(\lambda) = \{i\}\) for some \(1 < i < n\) and \((\lambda +
+    \item \(A(\lambda) = \{1\}\) or \(A(\lambda) = \{n - 1\}\).
+    \item \(A(\lambda) = \{i\}\) for some \(1 < i < n - 1\) and \((\lambda +
       \rho)(H_{\beta_{i - 1}} + H_{\beta_i})\) or \((\lambda +
       \rho)(H_{\beta_i} + H_{\beta_{i + 1}})\) is a positive integer.
-    \item \(A(\lambda) = \{i, i + 1\}\) for some \(1 \le i < n\) and
+    \item \(A(\lambda) = \{i, i + 1\}\) for some \(1 \le i < n - 1\) and
       \((\lambda + \rho)(H_{\beta_i} + H_{\beta_{i + 1}})\) is a positive
       integer.
   \end{enumerate}
 \end{lemma}
 
-% TODO: Change the notation: these should be called "sl_n+1-sequences", not
-% "sl_n+1(K)-sequences"
 \begin{definition}
-  A \emph{\(\mathfrak{sl}_{n + 1}(K)\)-sequence} \(m\) is a \(n + 1\)-tuple \(m
-  \in K^{n + 1}\) such that \(m_1 + \cdots + m_{n + 1} = 0\).
+  A \emph{\(\mathfrak{sl}_n\)-sequence} \(m\) is a \(n\)-tuple \(m \in K^n\)
+  such that \(m_1 + \cdots + m_n = 0\).
 \end{definition}
 
 % TODO: Revise the notation for this? I don't really like calling this
 % bijection m
 % TODO: Note that this prove is similar to the previous one, and that the
-% equivariance of the map follows from the nature of the isomorphism W ā‰… S_n+1
+% equivariance of the map follows from the nature of the isomorphism W ā‰… S_n
 % TODOO: Describe this isomorphism beforehand
 \begin{proposition}
   The map
   \begin{align*}
-    m : \mathfrak{h}^* & \to K^{n + 1} \\
+    m : \mathfrak{h}^* & \to K^n \\
         \lambda &
         \mapsto
         (
           \kappa(\epsilon_1, \lambda + \rho),
           \cdots,
-          \kappa(\epsilon_{n + 1}, \lambda + \rho)
+          \kappa(\epsilon_n, \lambda + \rho)
         )
   \end{align*}
-  is \(W\)-equivariant bijection onto the space of all \(\mathfrak{sl}_{n +
-  1}(K)\)-sequences, where the action \(W \cong S_{n + 1}\) on
+  is \(W\)-equivariant bijection onto the space of all
+  \(\mathfrak{sl}_n\)-sequences, where the action \(W \cong S_n\) on
   \(\mathfrak{h}^*\) is given by the dot-action and the action of \(W\) on
   \(K^n\) is given my permuting coordinates. A weight \(\lambda \in
   \mathfrak{h}^*\) satisfies the conditions of
   Lemma~\ref{thm:sl-bounded-weights} if, and only if 
   the diferences between all but one consecutive coordinates of \(m(\lambda)\)
-  are positive integers -- i.e. there is some unique \(i \le n\) such that
-  \(m(\lambda)_i - m(\lambda)_{i + 1}\) is not a positive integer.
+  are positive integers -- i.e. there is some unique \(i < n\) such that
+  \(m(\lambda)_i - m(\lambda)_{i + 1}\) is \emph{not} a positive integer.
 \end{proposition}
 
 % TODO: Change the notation for š“Ÿ ? The paper on affine vertex algebras calls
@@ -239,11 +237,11 @@ Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\epsilon_i,
 % implies L(Ī¼) is contained in š“”š”š“½(L(Ī»)) - so that L(Ī¼) is also bounded and
 % š“”š”š“½(L(Ī»)) ā‰… š“”š”š“½(L(Ī¼))
 \begin{definition}
-  Denote by \(\mathscr{P}\) the set of \(\mathfrak{sl}_{n + 1}(K)\)-sequences
-  \(m\) such that \(m_i - m_{i + 1}\) is a positive integer for all but a
-  single \(i \le n\). Given \(m, m' \in \mathscr{P}\), say there is an arrow
-  \(m \to m'\) if the unique \(i\) such that \(m_i - m_{i + 1}\) is not a
-  positive integer is such that \(m' = \sigma_i \cdot m\).
+  Denote by \(\mathscr{P}\) the set of \(\mathfrak{sl}_n\)-sequences \(m\) such
+  that \(m_i - m_{i + 1}\) is a positive integer for all but a single \(i <
+  n\). Given \(m, m' \in \mathscr{P}\), say there is an arrow \(m \to m'\) if
+  the unique \(i\) such that \(m_i - m_{i + 1}\) is not a positive integer is
+  such that \(m' = \sigma_i \cdot m\).
 \end{definition}
 
 It should then be obvious that\dots
@@ -257,21 +255,21 @@ It should then be obvious that\dots
 \end{proposition}
 
 \begin{definition}
-  Let \(\mathscr{P}^+ = \{m \in \mathscr{P} : m_1 - m_2 \ \text{is not a
-  positive integer}\}\) and \(\mathscr{P}^- = \{m \in \mathscr{P} : m_n - m_{n
-  + 1} \ \text{is not a positive integer}\}\).
+  Let \(\mathscr{P}^+ = \{m \in \mathscr{P} : m_1 - m_2 \ \text{is \emph{not} a
+  positive integer}\}\) and \(\mathscr{P}^- = \{m \in \mathscr{P} : m_{n-1} -
+  m_n \ \text{is \emph{not} a positive integer}\}\).
 \end{definition}
 
 \begin{definition}
-  A \(\mathfrak{sl}_{n + 1}(K)\)-sequence \(m\) is called \emph{integral} if
-  \(m_i - m_j \in \mathbb{Z}\) for all \(i\) and \(j\).
+  A \(\mathfrak{sl}_n\)-sequence \(m\) is called \emph{integral} if \(m_i - m_j
+  \in \mathbb{Z}\) for all \(i\) and \(j\).
 \end{definition}
 
 % TODOO: Discuss the notion of a regular weight beforehand
 \begin{definition}
-  A \(\mathfrak{sl}_{n + 1}(K)\)-sequence \(m\) is called \emph{regular} if
-  \(m_i \ne m_j\) for all \(i \ne j\). A sequence is called \emph{singular} if
-  it is not regular.
+  A \(\mathfrak{sl}_n\)-sequence \(m\) is called \emph{regular} if \(m_i \ne
+  m_j\) for all \(i \ne j\). A sequence is called \emph{singular} if it is not
+  regular.
 \end{definition}
 
 % TODO: Add notes on what are the sets W . m āˆ© š“Ÿ  : the connected component of
@@ -288,8 +286,8 @@ It should then be obvious that\dots
   \begin{enumerate}
     \item If \(m\) is regular and integral then there exists\footnote{Notice
       that in this case $m' \notin \mathscr{P}$, however.} a unique \(m' \in W
-      \cdot m\) such that \(m_1' > m_2' > \cdots > m_{n + 1}'\), in which case
-      the connected component of \(m\) is given by
+      \cdot m\) such that \(m_1' > m_2' > \cdots > m_n'\), in which case the
+      connected component of \(m\) is given by
       \[
         \begin{tikzcd}[cramped, sep=small]
           \sigma_1 \sigma_2 \cdots \sigma_i \cdot m'           \rar &
@@ -304,16 +302,16 @@ It should then be obvious that\dots
           \sigma_i \cdot m'                                         &
           \sigma_{i+1} \sigma_i \cdot m'                       \lar &
           \cdots                                               \lar &
-          \sigma_n \cdots \sigma_i \cdot m'                    \lar &
+          \sigma_{n-1} \cdots \sigma_i \cdot m'                \lar &
         \end{tikzcd}
       \]
       for some unique \(i\), with \(\sigma_1 \cdots \sigma_i \cdot m' \in
-      \mathscr{P}^+\) and \(\sigma_n \cdots \sigma_i \cdot m' \in
+      \mathscr{P}^+\) and \(\sigma_{n-1} \cdots \sigma_i \cdot m' \in
       \mathscr{P}^-\).
 
     \item If \(m\) is singular then there exists unique \(m' \in W \cdot m\)
       and \(i\) such that \(m_1' > m_2' > \cdots > m_i' = m_{i + 1}' > \cdots >
-      m_{n + 1}'\), in which case the connected component of \(m\) is given by
+      m_n'\), in which case the connected component of \(m\) is given by
       \[
         \begin{tikzcd}[cramped, sep=small]
           \sigma_1 \sigma_2 \cdots \sigma_{i-1} \cdot m'       \rar &
@@ -328,26 +326,26 @@ It should then be obvious that\dots
           m'                                                        &
           \sigma_{i+1} \cdot m'                                \lar &
           \cdots                                               \lar &
-          \sigma_n \cdots \sigma_{i+1} \cdot m'                \lar &
+          \sigma_{n-1} \cdots \sigma_{i+1} \cdot m'            \lar &
         \end{tikzcd}
       \]
       with \(\sigma_1 \cdots \sigma_{i-1} \cdot m' \in \mathscr{P}^+\) and
-      \(\sigma_n \cdots \sigma_{i+1} \cdot m' \in \mathscr{P}^-\).
+      \(\sigma_{n-1} \cdots \sigma_{i+1} \cdot m' \in \mathscr{P}^-\).
 
     \item If \(m\) is non-integral then there exists unique \(m' \in W \cdot
-      m\) such that \(m_2' > m_3' > \cdots > m_{n + 1}'\), in which case the
+      m\) such that \(m_2' > m_3' > \cdots > m_n'\), in which case the
       connected component of \(m\) is given by
       \[
         \begin{tikzcd}[cramped]
-          m'                                \rar      &
-          \sigma_1 \cdot m'                 \rar \lar &
-          \sigma_2 \sigma_1 \cdot m'        \rar \lar &
-          \cdots                            \rar \lar &
-          \sigma_n \cdots \sigma_1 \cdot m'      \lar &
+          m'                                    \rar      &
+          \sigma_1 \cdot m'                     \rar \lar &
+          \sigma_2 \sigma_1 \cdot m'            \rar \lar &
+          \cdots                                \rar \lar &
+          \sigma_{n-1} \cdots \sigma_1 \cdot m'      \lar &
         \end{tikzcd}
       \]
-      with \(m' \in \mathscr{P}^+\) and \(\sigma_n \cdots \sigma_1 \cdot m' \in
-      \mathscr{P}^-\).
+      with \(m' \in \mathscr{P}^+\) and \(\sigma_{n-1} \cdots \sigma_1 \cdot m'
+      \in \mathscr{P}^-\).
   \end{enumerate}
 \end{proposition}
 
@@ -357,7 +355,7 @@ It should then be obvious that\dots
 % āˆˆ š“Ÿ + āˆŖ š“Ÿ - we stablish this by hand, and for the general case it suffices to
 % notice that there is always some path Ī¼ ā†’ ... ā†’ Ī» with Ī¼ āˆˆ š“Ÿ + āˆŖ š“Ÿ -
 % TODO: Perhaps this could be incorporated into the discussion of the lemma
-% that characterizes the weights of sl(n + 1) whose L is bounded
+% that characterizes the weights of sl(n) whose L is bounded
 
 % TODO: Prove this
 \begin{theorem}[Mathieu]
@@ -365,7 +363,7 @@ It should then be obvious that\dots
   \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\) if, and only if \(m(\lambda)\) and
   \(m(\mu)\) lie in the same connected component of \(\mathscr{P}\). In
   particular, the isomorphism classes of semisimple irreducible coherent
-  \(\mathfrak{sl}_{n + 1}(K)\)-families are parameterized by the set
+  \(\mathfrak{sl}_n(K)\)-families are parameterized by the set
   \(\pi_0(\mathscr{P})\) of the connected components of \(\mathscr{P}\).
 \end{theorem}