Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Changed the notation for the PBW Theorem
Changed the notation to refer to the PBW theorem to make it standard-conformat
2 files changed, 9 insertions, 8 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/introduction.tex | 15 | 8 | 7 |
| Modified | sections/mathieu.tex | 2 | 1 | 1 |
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -679,7 +679,7 @@ basis \(\{X_i\}_i\) for \(\mathfrak{g}\) induces an isomorphism \(\mathcal{U}(\mathfrak{g}) \cong K[X_1, \ldots, X_i, \ldots]\). More generally, we find\dots -\begin{theorem}[Poincaré-Birkoff-Witt]\index{Poincaré-Birkoff-Witt Theorem} +\begin{theorem}[Poincaré-Birkoff-Witt]\index{PBW Theorem} Let \(\mathfrak{g}\) be a Lie algebra over \(K\) and \(\{X_i\}_i \subset \mathfrak{g}\) be an ordered basis for \(\mathfrak{g}\) -- i.e. a basis indexed by an ordered set. Then \(\{X_{i_1} \cdot X_{i_2} \cdots X_{i_n} : n @@ -687,8 +687,9 @@ we find\dots \(\mathcal{U}(\mathfrak{g})\). \end{theorem} -The Poincaré-Birkoff-Witt Theorem is hugely important and will come up again -and again throughout these notes. Among other things, it implies\dots +This last result is known as \emph{the PBW Theorem}. It is hugely important and +will come up again and again throughout these notes. Among other things, it +implies\dots \begin{corollary} Let \(\mathfrak{g}\) be a Lie algebra over \(K\). Then @@ -696,13 +697,13 @@ and again throughout these notes. Among other things, it implies\dots \to \mathcal{U}(\mathfrak{g})\) is injective. \end{corollary} -The Poincaré-Birkoff-Witt Theorem can also be used to compute a series of +The PBW Theorem can also be used to compute a series of examples. \begin{example} Consider the Lie algebra \(\mathfrak{gl}_n(K)\) and its canonical basis \(\{E_{i j}\}_{i j}\). Even though \(E_{i j} E_{j k} = E_{i k}\) in the - associative algebra \(\operatorname{End}(K^n)\), the Poincaré-Birkoff-Witt + associative algebra \(\operatorname{End}(K^n)\), the PBW Theorem implies \(E_{i j} E_{j k} \ne E_{i k}\) in \(\mathcal{U}(\mathfrak{gl}_n(K))\). In general, if \(A\) is an associative \(K\)-algebra then the elements in the image of the inclusion \(A \to @@ -728,7 +729,7 @@ examples. is an isomorphism of algebras. Since the elements of \(\mathfrak{g}\) commute with the elements of \(\mathfrak{h}\) in \(\mathfrak{g} \oplus \mathfrak{h}\), a simple calculation shows that \(f\) is indeed a - homomorphism of algebras. In addition, the Poincaré-Birkoff-Witt Theorem + homomorphism of algebras. In addition, the PBW Theorem implies that \(f\) is a linear isomorphism. \end{example} @@ -791,7 +792,7 @@ As one would expect, the same holds for complex Lie groups and algebraic groups too -- if we replace \(C^\infty(G)\) by \(\mathcal{O}(G)\) and \(K[G]\), respectively. This last proposition has profound implications. For example, it affords us an analytic proof of certain particular cases of the -Poincaré-Birkoff-Witt Theorem. Most surprising of all, +PBW Theorem. Most surprising of all, Proposition~\ref{thm:geometric-realization-of-uni-env} implies \(\mathcal{U}(\mathfrak{g})\)-modules are \emph{precisely} the same as modules over the ring of \(G\)-invariant differential operators -- i.e.
diff --git a/sections/mathieu.tex b/sections/mathieu.tex @@ -138,7 +138,7 @@ to the case it holds. This brings us to the following definition. computation shows \(K \langle X_1 \cdots X_n H_1 \cdots H_m : X_i \in \mathfrak{g}_{\alpha_i}, H_i \in \mathfrak{h}, \alpha_i \in \Delta, \alpha = \alpha_1 + \cdots + \alpha_n \rangle \subset - \mathcal{U}(\mathfrak{g})_\alpha\). The Poincaré-Birkoff-Witt Theorem and + \mathcal{U}(\mathfrak{g})_\alpha\). The PBW Theorem and Example~\ref{ex:reductive-alg-equivalence} thus imply that \(\mathcal{U}(\mathfrak{g}) = \bigoplus_{\alpha \in Q} \mathcal{U}(\mathfrak{g})_\alpha\) where \(\mathcal{U}(\mathfrak{g})_\alpha =