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: 3eebca52c020e8dae3207aef16c93b5fb74d8021
Parent: 896a555b6440d2ead5110c6de05094ecf83a5de0
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 15 May 2023 13:59:09 +0000

Added further comments on simple modules of the direct sum

Added further explanations of the fact that finite-dimensional simple 𝔤⊕𝔥-modules are given by tensor products of simple 𝔤-modules and simple 𝔥-modules

Also added further examples of applications of the PBW theorem to the introduction

Diffstat

2 files changed, 52 insertions, 13 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/complete-reducibility.tex	8	6	2
Modified	sections/introduction.tex	57	46	11

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -53,8 +53,12 @@ smaller pieces. This leads us to the following definitions.
   finite-dimensional simple \(\mathfrak{h}\)-module \(N\), the tensor product
   \(M \otimes N\) is a simple \(\mathfrak{g} \oplus \mathfrak{h}\)-module. All
   finite-dimensional simple \(\mathfrak{g} \oplus \mathfrak{h}\)-modules have
-  the form \(M \otimes N\) for unique (up to isomorphism) \(M\) and \(N\) --
-  see \cite[ch.~3]{etingof}.
+  the form \(M \otimes N\) for unique (up to isomorphism) \(M\) and \(N\). In
+  light of Example~\ref{ex:univ-enveloping-of-sum-is-tensor}, this is a
+  particular case of the fact that, given \(K\)-algebras \(A\) and \(B\), all
+  finite-dimensional simple \(A \otimes_K B\)-modules are given tensor products
+  of simple \(A\)-modules with simple \(B\)-modules -- see
+  \cite[ch.~3]{etingof}.
 \end{example}
 
 The general strategy for classifying finite-dimensional modules over an algebra

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -691,15 +691,47 @@ The Poincaré-Birkoff-Witt Theorem 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 the inclusion
-  \(\mathfrak{g} \to \mathcal{U}(\mathfrak{g})\) is injective.
-\end{corollary}
-
-\begin{corollary}
   Let \(\mathfrak{g}\) be a Lie algebra over \(K\). Then
-  \(\mathcal{U}(\mathfrak{g})\) is a domain.
+  \(\mathcal{U}(\mathfrak{g})\) is a domain and the inclusion \(\mathfrak{g}
+  \to \mathcal{U}(\mathfrak{g})\) is injective.
 \end{corollary}
 
+The Poincaré-Birkoff-Witt 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
+  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 \(A \to
+  \mathcal{U}(A)\) do not satisfy the same relations as the elements of \(A\).
+\end{example}
+
+\begin{example}
+  Let \(\mathfrak{g}\) be an Abelian Lie algebra. As previously stated, any
+  choice of basis \(\{X_i\}_i \subset \mathfrak{g}\) induces an isomorphism of
+  algebras \(\mathcal{U}(\mathfrak{g}) \isoto K[x_1, x_2, \ldots, x_i,
+  \ldots]\) which takes \(X_i \in \mathfrak{g}\) to the variable \(x_i \in
+  K[x_1, x_2, \ldots, x_i, \ldots]\).
+\end{example}
+
+\begin{example}\label{ex:univ-enveloping-of-sum-is-tensor}
+  Let \(\mathfrak{g}\) and \(\mathfrak{h}\) be Lie algebras over \(K\). We
+  claim that the natural map
+  \begin{align*}
+    f: \mathcal{U}(\mathfrak{g}) \otimes_K \mathcal{U}(\mathfrak{h}) &
+    \to \mathcal{U}(\mathfrak{g} \oplus \mathfrak{h}) \\
+    u \otimes v & \mapsto u \cdot v
+  \end{align*}
+  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
+  implies that \(f\) is a linear isomorphism.
+\end{example}
+
 The construction of \(\mathcal{U}(\mathfrak{g})\) may seem like a purely
 algebraic affair, but the universal enveloping algebra of the Lie algebra of a
 Lie group \(G\) is in fact intimately related with the algebra
@@ -1098,11 +1130,14 @@ proved by Frobenius himself in the context of finite groups. Another
 interesting construction is\dots
 
 \begin{example}\label{ex:tensor-prod-separate-algs}\index{\(\mathfrak{g}\)-module!tensor product}
-  Let \(\mathfrak{g}\) and \(\mathfrak{h}\) be Lie algebras. Given a
-  \(\mathfrak{g}\)-module \(M\) and a \(\mathfrak{h}\)-module \(N\), the space
-  \(M \otimes N = M \otimes_K N\) has the natural structure of a
-  \(\mathfrak{g} \oplus \mathfrak{h}\)-module, where the action of
-  \(\mathfrak{g} \oplus \mathfrak{h}\) is given by
+  Given two \(K\)-algebras \(A\) and \(B\), an \(A\)-module \(M\) and a
+  \(B\)-module \(N\), \(M \otimes B = M \otimes_K B\) has the natural structure
+  of an \(A \otimes_K B\)-module. In light of
+  Example~\ref{ex:univ-enveloping-of-sum-is-tensor}, this implies that given
+  Lie algebras \(\mathfrak{g}\) and \(\mathfrak{h}\), a \(\mathfrak{g}\)-module
+  \(M\) and a \(\mathfrak{h}\)-module \(N\), the space \(M \otimes N\) has the
+  natural structure of a \(\mathfrak{g} \oplus \mathfrak{h}\)-module, where the
+  action of \(\mathfrak{g} \oplus \mathfrak{h}\) is given by
   \[
     (X + Y) \cdot (m \otimes n) = X \cdot m \otimes n + m \otimes Y \cdot n
   \]