- Commit
- e066394ba65cd9a80b9e069037148334bf1b9c91
- Parent
- f72a9a6c98cd1c4f1c1cdf71134b8aeae10d6b4f
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Fixed a sign in some equations
Also clarified an inequality
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Fixed a sign in some equations
Also clarified an inequality
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 \[ \dim W - = \sum_k \dim V_{\lambda + k (\alpha_i - \alpha_j)} + = \sum_k \dim V_{\lambda - k (\alpha_i - \alpha_j)} < \sum_k \dim W_{\lambda([E_{i j}, E_{j i}]) - 2k} = \dim W, \]