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
e066394ba65cd9a80b9e069037148334bf1b9c91
Parent
f72a9a6c98cd1c4f1c1cdf71134b8aeae10d6b4f
Author
Pablo <pablo-escobar@riseup.net>
Date

Fixed a sign in some equations

Also clarified an inequality

Diffstat

1 file changed, 16 insertions, 15 deletions

Status File Name N° Changes Insertions Deletions
Modified sections/sl2-sl3.tex 31 16 15
diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -509,14 +509,14 @@ example~\ref{ex:gln-inclusions} -- restricts to an injective homomorphism
 Our first observation is that, since the root spaces act by translation, the
 subspace
 \[
-  \bigoplus_{k \in \mathbb{Z}} V_{\lambda + k (\alpha_1 - \alpha_2)},
+  \bigoplus_{k \in \mathbb{Z}} V_{\lambda - k (\alpha_1 - \alpha_2)},
 \]
 must be invariant under the action of \(E_{1 2}\) and \(E_{2 1}\) for all
-\(\lambda \in \mathfrak{h}^*\). This goes to show \(\bigoplus_k V_{\lambda + k
+\(\lambda \in \mathfrak{h}^*\). This goes to show \(\bigoplus_k V_{\lambda - k
 (\alpha_1 - \alpha_2)}\) is a \(\mathfrak{sl}_2(K)\)-submodule of \(V\) for all
 weights \(\lambda\) of \(V\). Furtheremore, one can easily see that the
 eigenspace of the eigenspace \(\lambda(H) - 2k\) of \(h\) in \(W\) is precisely
-the weight space \(V_{\lambda + k (\alpha_2 - \alpha_1)}\).
+the weight space \(V_{\lambda - k (\alpha_2 - \alpha_1)}\).
 
 Visually,
 \begin{center}
@@ -549,15 +549,16 @@ Visually,
 In general, we find\dots
 
 \begin{proposition}
-  The subalgebra \(\mathfrak{s}_{\alpha_i - \alpha_j} = K \langle E_{i j}, E_{j
-  i}, [E_{i j}, E_{j i}] \rangle\) is isomorphic to \(\mathfrak{sl}_2(K)\).
-  In addition, given weight \(\lambda \in \mathfrak{h}^*\) of \(V\), the space
+  Given \(i < j\), the subalgebra \(\mathfrak{s}_{\alpha_i - \alpha_j} = K
+  \langle E_{i j}, E_{j i}, [E_{i j}, E_{j i}] \rangle\) is isomorphic to
+  \(\mathfrak{sl}_2(K)\). In addition, given weight \(\lambda \in
+  \mathfrak{h}^*\) of \(V\), the space
   \[
-    W = \bigoplus_{k \in \mathbb{Z}} V_{\lambda + k (\alpha_i - \alpha_j)}
+    W = \bigoplus_{k \in \mathbb{Z}} V_{\lambda - k (\alpha_i - \alpha_j)}
   \]
   is invariant under the action of \(\mathfrak{s}_{i j}\) and
   \[
-    V_{\lambda + k (\alpha_i - \alpha_j)}
+    V_{\lambda - k (\alpha_i - \alpha_j)}
     = W_{\lambda([E_{i j}, E_{j i}]) - 2k}
   \]
 \end{proposition}
@@ -574,21 +575,21 @@ In general, we find\dots
 
   To see that \(W\) is invariant under the action of \(\mathfrak{s}_{i j}\), it
   suffices to notice \(E_{i j}\) and \(E_{j i}\) map \(v \in V_{\lambda + k
-  (\alpha_i - \alpha_j)}\) to \(E_{i j} v \in V_{\lambda + (k + 1) (\alpha_i -
-  \alpha_j)}\) and \(E_{j i} v \in V_{\lambda + (k - 1) (\alpha_i -
+  (\alpha_i - \alpha_j)}\) to \(E_{i j} v \in V_{\lambda - (k - 1) (\alpha_i -
+  \alpha_j)}\) and \(E_{j i} v \in V_{\lambda - (k + 1) (\alpha_i -
   \alpha_j)}\). Moreover,
   \[
-    (\lambda + k (\alpha_i - \alpha_j))(H)
-    = \lambda([E_{i j}, E_{j i}]) + k (-1 - 1)
+    (\lambda - k (\alpha_i - \alpha_j))([E_{i j}, E_{j i}])
+    = \lambda([E_{i j}, E_{j i}]) - k (1 - (-1))
     = \lambda([E_{i j}, E_{j i}]) - 2 k,
   \]
-  which goes to show \(V_{\lambda + k (\alpha_i - \alpha_j)} \subset
+  which goes to show \(V_{\lambda - k (\alpha_i - \alpha_j)} \subset
   W_{\lambda([E_{i j}, E_{j i}]) - 2k}\). On the other hand, if we suppose \(0
-  < \dim V_{\lambda + k (\alpha_i - \alpha_j)} < \dim W_{\lambda([E_{i j}, E_{j
+  < \dim V_{\lambda - k (\alpha_i - \alpha_j)} < \dim W_{\lambda([E_{i j}, E_{j
   i}]) - 2 k}\) for some \(k\) we arrive at
   \[
     \dim W
-    = \sum_k \dim V_{\lambda + k (\alpha_i - \alpha_j)}
+    = \sum_k \dim V_{\lambda - k (\alpha_i - \alpha_j)}
     < \sum_k \dim W_{\lambda([E_{i j}, E_{j i}]) - 2k}
     = \dim W,
   \]