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: 578d7a44d93cd41a10e02dd46e042a452ee2eb49
Parent: 12047085f6e751cae156e4fb690b3f7ebeace90c
Author: Pablo <pablo-escobar@riseup.net>
Date: Sun, 27 Nov 2022 14:32:28 +0000

Added a proof of a technical result

Diffstat

2 files changed, 131 insertions, 19 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/semisimple-algebras.tex	8	8	0
Modified	sections/sl2-sl3.tex	142	123	19

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -651,6 +651,14 @@ the argument used for \(\mathfrak{sl}_3(K)\). Namely\dots
   irreducible.
 \end{proposition}
 
+The proof of proposition~\ref{thm:irr-subrep-generated-by-vec} is very similar
+to that of proposition~\ref{thm:sl3-positive-roots-span-all-irr-rep} in
+spirit: we use the commutator relations of \(\mathfrak{g}\) to inductively show
+that the subspace spanned by the images of a highest weight vector under
+successive applications of negative root vectors is invariant under the action
+of \(\mathfrak{g}\) -- please refer to \cite{fulton-harris} for further
+details. Of course, what we are really interested in is\dots
+
 \begin{corollary}
   Let \(V\) and \(W\) be finite-dimensional irreducible
   \(\mathfrak{g}\)-modules with highest weight given by some common \(\lambda

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -908,11 +908,11 @@ weight of \(V\) we started with.
     \begin{rootSystem}{A}
       \setlength{\weightRadius}{2pt}
       \weightLattice{5}
-      \draw[thick] \weight{3}{1} -- \weight{-3}{4};
-      \draw[thick] \weight{3}{1} -- \weight{4}{-1};
-      \draw[thick] \weight{-3}{4} -- \weight{-4}{3};
-      \draw[thick] \weight{-4}{3} -- \weight{-1}{-3};
-      \draw[thick] \weight{1}{-4} -- \weight{4}{-1};
+      \draw[thick] \weight{3}{1}   -- \weight{-3}{4};
+      \draw[thick] \weight{3}{1}   -- \weight{4}{-1};
+      \draw[thick] \weight{-3}{4}  -- \weight{-4}{3};
+      \draw[thick] \weight{-4}{3}  -- \weight{-1}{-3};
+      \draw[thick] \weight{1}{-4}  -- \weight{4}{-1};
       \draw[thick] \weight{-1}{-3} -- \weight{1}{-4};
       \wt[black]{-4}{3}
       \wt[black]{-3}{1}
@@ -992,8 +992,8 @@ establishing\dots
 
 \begin{theorem}\label{thm:sl3-existence-uniqueness}
   For each pair of positive integers \(n\) and \(m\), there exists precisely
-  one irreducible representation \(V\) of \(\mathfrak{sl}_3(K)\) whose highest
-  weight is \(n \alpha_1 - m \alpha_3\).
+  one finite-dimensional irreducible representation \(V\) of
+  \(\mathfrak{sl}_3(K)\) whose highest weight is \(n \alpha_1 - m \alpha_3\).
 \end{theorem}
 
 To proceed further we once again refer to the approach we employed in the case
@@ -1006,20 +1006,124 @@ highest weight vector under successive applications by half of the root spaces
 of \(\mathfrak{sl}_2(K)\). The advantage of this alternative formulation is, of
 course, that the same holds for \(\mathfrak{sl}_3(K)\). Specifically\dots
 
-\begin{theorem}\label{thm:irr-sl3-span}
+\begin{proposition}\label{thm:sl3-positive-roots-span-all-irr-rep}
   Given an irreducible \(\mathfrak{sl}_3(K)\)-representation \(V\) and a
   highest weight vector \(v \in V\), \(V\) is spanned by the images of \(v\)
   under successive applications of \(E_{2 1}\), \(E_{3 1}\) and \(E_{3 2}\).
-\end{theorem}
+\end{proposition}
+
+\begin{proof}
+  Given the fact \(V\) is irreducible, it suffices to show that the subspace
+  \(W\) spanned by successive applications of \(E_{2 1}\), \(E_{3 1}\) and
+  \(E_{3 2}\) to \(v\) is stable under the action of \(\mathfrak{sl}_3(K)\).
+  In addition, since \([E_{2 1}, E_{3 1}] = [E_{3 1}, E_{3 2}] = 0\) and
+  \([E_{2 1}, E_{3 2}] = - E_{3 1}\), all successive product of \(E_{2 1}\),
+  \(E_{3 1}\) and \(E_{3 2}\) in \(\mathcal{U}(\mathfrak{sl}_3(K))\) can be
+  written as \(E_{2 1}^a E_{3 1}^b E_{3 1}^c\) for some \(a\), \(b\) and \(c\),
+  so that \(W\) is spanned by the elements \(E_{2 1}^a E_{3 1}^b E_{3 1}^c v\).
+
+  Recall that \(E_{i j}\) maps \(V_\mu\) to \(V_{\mu + \alpha_i - \alpha_j}\).
+  In particular, \(E_{2 1}^a E_{3 1}^b E_{3 1}^c v \in V_{\lambda - a (\alpha_1
+  - \alpha_2) - b (\alpha_1 - \alpha_3) - c (\alpha_2 - \alpha_3)}\). In other
+  words,
+  \[
+    H E_{2 1}^a E_{3 1}^b E_{3 1}^c v
+    = (\lambda - a (\alpha_1 - \alpha_2)
+               - b (\alpha_1 - \alpha_3)
+               - c (\alpha_2 - \alpha_3))(H)
+      \cdot E_{2 1}^a E_{3 1}^b E_{3 1}^c v
+      \in W
+  \]
+  for all \(H \in \mathfrak{h}\) and \(W\) is stable under the action of
+  \(\mathfrak{h}\). On the other hand, \(W\) is clearly stable under the action
+  of \(E_{2 1}\), \(E_{3 1}\) and \(E_{3 2}\). All it's left is to show \(W\)
+  is stable under the action of \(E_{1 2}\), \(E_{1 3}\) and \(E_{2 3}\).
+
+  We begin by analyzing the case of \(E_{1 2}\). We have
+  \[
+    \begin{split}
+      E_{1 2} E_{2 1}^a E_{3 1}^b E_{3 2}^c v
+      & = ([E_{1 2}, E_{2 1}] + E_{2 1} E_{1 2})
+          E_{2 1}^{a - 1} E_{3 1}^b E_{3 2}^c v \\
+      & = E_{2 1} ([E_{1 2}, E_{2 1}] + E_{2 1} E_{1 2})
+          E_{2 1}^{a - 2} E_{3 1}^b E_{3 2}^c v \\
+      & \phantom{=} \; +
+          (\lambda - (a - 1) (\alpha_1 - \alpha_2)
+                   - b (\alpha_1 - \alpha_3)
+                   - c (\alpha_2 - \alpha_3)) ([E_{1 2}, E_{2 1}])
+          \cdot
+          E_{2 1}^{a - 1} E_{3 1}^b E_{3 2}^c v \\
+      & = E_{2 1}^2 ([E_{1 2}, E_{2 1}] + E_{2 1} E_{1 2})
+          E_{2 1}^{a - 3} E_{3 1}^b E_{3 2}^c v \\
+      & \phantom{=} \; +
+          (\lambda - (a - 1) (\alpha_1 - \alpha_2)
+                   - b (\alpha_1 - \alpha_3)
+                   - c (\alpha_2 - \alpha_3)) ([E_{1 2}, E_{2 1}])
+          \cdot
+          E_{2 1}^{a - 1} E_{3 1}^b E_{3 2}^c v \\
+      & \phantom{=} \; +
+        (\lambda - (a - 2) (\alpha_1 - \alpha_2)
+                   - b (\alpha_1 - \alpha_3)
+                   - c (\alpha_2 - \alpha_3)) ([E_{1 2}, E_{2 1}])
+          \cdot
+          E_{2 1}^{a - 2} E_{3 1}^b E_{3 2}^c v \\
+      & \; \; \vdots \\
+      & = E_{2 1}^a E_{1 2} E_{3 1}^b E_{3 2}^c v \\
+      & \phantom{=} \; +
+          (\lambda - (a - 1) (\alpha_1 - \alpha_2)
+                   - b (\alpha_1 - \alpha_3)
+                   - c (\alpha_2 - \alpha_3)) ([E_{1 2}, E_{2 1}])
+          \cdot
+          E_{2 1}^{a - 1} E_{3 1}^b E_{3 2}^c v \\
+      & \phantom{=} \; +
+        (\lambda - (a - 2) (\alpha_1 - \alpha_2)
+                   - b (\alpha_1 - \alpha_3)
+                   - c (\alpha_2 - \alpha_3)) ([E_{1 2}, E_{2 1}])
+          \cdot
+          E_{2 1}^{a - 2} E_{3 1}^b E_{3 2}^c v \\
+      & \phantom{=} \; + \cdots \\
+      & \phantom{=} \; +
+        (\lambda - (a - a) (\alpha_1 - \alpha_2)
+                   - b (\alpha_1 - \alpha_3)
+                   - c (\alpha_2 - \alpha_3)) ([E_{1 2}, E_{2 1}])
+          \cdot
+          E_{2 1}^{a - a} E_{3 1}^b E_{3 2}^c v \\
+    \end{split}
+  \]
+
+  Since \((\lambda - (a - k) (\alpha_1 - \alpha_2) - b (\alpha_1 - \alpha_3)
+  - c (\alpha_2 - \alpha_3)) ([E_{1 2}, E_{2 1}]) \cdot E_{2 1}^{a - k} E_{3
+  1}^b E_{3 2}^c v \in W\) for all \(k\), it suffices to show \(E_{2 1}^a E_{1
+  2} E_{3 1}^b E_{3 2}^c v \in W\). But
+  \[
+    \begin{split}
+      E_{1 2} E_{3 1}^b
+      & = (E_{3 1} E_{1 2} - E_{3 2}) E_{3 1}^{b - 1} \\
+      & = E_{3 1} E_{1 2} E_{3 1}^{b - 1}
+        - E_{3 1} E_{3 2} E_{3 1}^{b - 1} \\
+      & = E_{3 1} (E_{3 1} E_{1 2} - E_{3 2}) E_{3 1}^{b - 2}
+        - E_{3 2} E_{3 1}^b \\
+      & \; \; \vdots \\
+      & = E_{3 1}^b  E_{1 2} - b E_{3 2} E_{3 1}^b \\
+    \end{split},
+  \]
+  given \([E_{1 2}, E_{3 1}] = - E_{3 2}\) and \([E_{3 2}, E_{3 1}] = 0\).
+  It then follows from the fact \(E_{1 2} v = 0\) that
+  \[
+    E_{2 1}^a E_{1 2} E_{3 1}^b E_{3 2}^c v
+    = E_{2 1}^a E_{3 1}^b E_{3 2}^c E_{1 2} v
+    - b E_{2 1}^a E_{3 1}^b E_{3 2}^{c + 1} v
+    = - b E_{2 1}^a E_{3 1}^b E_{3 2}^{c + 1} v \in W,
+  \]
+  given that \(E_{1 2}\) and \(E_{3 2}\) commute. Hence \(E_{1 2} E_{2 1}^a
+  E_{3 1}^b E_{3 2}^c v \in W\). Similarly,
+  \[
+    E_{1 3} E_{2 1}^a E_{3 1}^b E_{3 2}^c v,
+    E_{2 3} E_{2 1}^a E_{3 1}^b E_{3 2}^c v \in W
+  \]
+\end{proof}
 
-% TODO: Add a proof? We can hind the induction in "..." fairly easily
-The proof of theorem~\ref{thm:irr-sl3-span} is very similar to that of
-proposition~\ref{thm:basis-of-irr-rep}: we use the commutator relations of
-\(\mathfrak{sl}_3(K)\) to inductively show that the subspace spanned by the
-images of a highest weight vector under successive applications of \(E_{2 1}\),
-\(E_{3 1}\) and \(E_{3 2}\) is invariant under the action of
-\(\mathfrak{sl}_3(K)\) -- please refer to \cite{fulton-harris} for further
-details. The same argument also goes to show\dots
+The same argument also goes to show\dots
 
 \begin{corollary}\label{thm:irr-component-of-high-vec}
   Given a representation \(V\) of \(\mathfrak{sl}_3(K)\) with highest weight
@@ -1029,8 +1133,8 @@ details. The same argument also goes to show\dots
 \end{corollary}
 
 This is very interesting to us since it implies that finding \emph{any}
-representation whose highest weight is \(n \alpha_1 - m \alpha_2\) is enough
-for establishing the ``existence'' part of
+finite-dimensional representation whose highest weight is \(n \alpha_1 - m
+\alpha_2\) is enough for establishing the ``existence'' part of
 theorem~\ref{thm:sl3-existence-uniqueness}. Moreover, constructing such
 representation turns out to be quite simple.