lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -409,120 +409,6 @@ ready to dive deeper into them.
   \mathfrak{sl}_n(K) \oplus K\).
 \end{example}
 
-\section{Representations}
-
-\begin{definition}
-  Given a Lie algebra \(\mathfrak{g}\) over \(K\), \emph{a representation \(V\)
-  of \(\mathfrak{g}\)}, or \emph{\(\mathfrak{g}\)-module}, is a \(K\)-vector
-  space endowed with a homomorphism of Lie algebras \(\rho : \mathfrak{g} \to
-  \mathfrak{gl}(V)\).
-\end{definition}
-
-\begin{example}
-  Given a Lie algebra \(\mathfrak{g}\), consider the homomorphism
-  \(\operatorname{ad} : \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})\) given by
-  \(\operatorname{ad}(X) Y = [X, Y]\). This gives \(\mathfrak{g}\) the
-  structure of a representation of \(\mathfrak{g}\), known as \emph{the adjoint
-  representation}.
-\end{example}
-
-\begin{example}
-  Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{g}\)-modules \(V\) and
-  \(W\), the the spaces \(V \oplus W\), \(V^*\), \(V \otimes W\) and
-  \(\operatorname{Hom}(V, W)\) are all \(\mathfrak{g}\)-modules -- where the
-  action of \(\mathfrak{g}\) is given by
-  \begin{align*}
-    X (v + w)       & = X v + X w                     &
-    X \cdot f       & = - f \circ X                   \\
-    X (v \otimes w) & = X v \otimes w + v \otimes X w &
-    (X \cdot T) v   & = X T v - T X v,
-  \end{align*}
-  respectively.
-\end{example}
-
-\begin{definition}
-  Given a Lie algebra \(\mathfrak{g}\) and two representations \(V\) and \(W\)
-  of \(\mathfrak{g}\), we call a linear map \(T : V \to W\) \emph{an
-  intertwiner} or \emph{a homomorphism of representations} if it cummutes with
-  the action of \(\mathfrak{g}\) in \(V\) and \(W\), in the sence that the
-  diagram
-  \begin{center}
-    \begin{tikzcd}
-      V \rar{T} \dar[swap]{X} & W \dar{X} \\
-      V \rar[swap]{T}         & W
-    \end{tikzcd}
-  \end{center}
-  commutes for all \(X \in \mathfrak{g}\). We denote the space of all
-  intertwiners \(V \to W\) by \(\operatorname{Hom}_{\mathfrak{g}}(V, W)\).
-\end{definition}
-
-% TODO: Point out g-Mod is indeed a category
-
-\begin{example}
-  Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{h}\) be a subalgebra.
-  Given a representation \(V\) of \(\mathfrak{g}\), denote by
-  \(\operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} V = V\) the representation
-  of \(\mathfrak{h}\) where the action of \(\mathfrak{h}\) is given by
-  restricting the map \(\mathfrak{g} \to \mathfrak{gl}(V)\) to
-  \(\mathfrak{g}\). Any homomorphism of \(\mathfrak{g}\)-modules \(V \to W\) is
-  also a homomorphism of \(\mathfrak{h}\)-modules and this construction is
-  clearly functorial.
-  \[
-    \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} :
-    \mathfrak{g}\text{-}\mathbf{Mod} \to
-    \mathfrak{h}\text{-}\mathbf{Mod}
-  \]
-\end{example}
-
-\begin{definition}
-  Given a Lie algebra \(\mathfrak{g}\) and a representation \(V\) of
-  \(\mathfrak{g}\), a subspace \(W \subset V\) is called \emph{a
-  subrepresentation} if it is stable under the action of \(\mathfrak{g}\) --
-  i.e. \(X w \in W\) for all \(w \in W\) and \(X \in \mathfrak{g}\).
-\end{definition}
-
-\begin{example}
-  Given a Lie algebra \(\mathfrak{g}\), a representation \(V\) of
-  \(\mathfrak{g}\) and a subrepresentation \(W \subset V\), the space
-  \(\mfrac{V}{W}\) has the natural structure of a \(\mathfrak{g}\)-module where
-  \(X (v + W) = X v + W\). The projection \(V \to \mfrac{V}{W}\) is an
-  intertwiner.
-\end{example}
-
-\begin{example}
-  Given a Lie algebra \(\mathfrak{g}\) and representations \(V\) and \(W\) of
-  \(\mathfrak{g}\), the spaces \(V \wedge W\) and \(V \odot W\) are both
-  representations of \(G\): they are both quotients of \(V \otimes W\).
-\end{example}
-
-\begin{definition}
-  A representation of \(\mathfrak{g}\) is called \emph{indecomposable} if it is
-  not isomorphic to the direct sum of two non-zero representations.
-\end{definition}
-
-\begin{definition}
-  A representation of \(\mathfrak{g}\) is called \emph{irreducible} if it has
-  no non-zero subrepresentations.
-\end{definition}
-
-\begin{lemma}[Schur]
-  Let \(\mathfrak{g}\) be a Lie algebra over a field \(K\). If \(V\) and \(W\)
-  irreducible representations of \(\mathfrak{g}\). and \(T : V \to W\) be an
-  intertwiner then \(T\) is either \(0\) or an isomorphism. Furtheremore, if
-  \(K\) is algebraicly closed and \(V = W\) then \(T\) is a scalar operator.
-\end{lemma}
-
-\begin{proof}
-  For the first statement, it suffices to notice that \(\ker T\) and
-  \(\operatorname{im} T\) are both subrepresentations. In particular, either
-  \(\ker T = 0\) and \(\operatorname{im} T = W\) or \(\ker T = V\) and
-  \(\operatorname{im} T = 0\). Now suppose \(K\) is algebraicly closed and \(V
-  = W\). Let \(\lambda \in K\) be an eigenvalue of \(T\) and \(V_\lambda\) be
-  its corresponding eigenspace. Given \(v \in V_\lambda\), \(T X v = X T v =
-  \lambda \cdot X v\). In other words, \(V_\lambda\) is a subrepresentation.
-  It then follows \(V_\lambda = V\), given that \(V_\lambda \ne 0\).
-\end{proof}
-
 \section{The Universal Enveloping Algebra}
 
 \begin{definition}
@@ -607,6 +493,107 @@ ready to dive deeper into them.
   \(\operatorname{Lie} \vdash \mathcal{U}\).
 \end{corollary}
 
+% TODO: Define the algebra of differential operators of a given algebra
+\begin{proposition}
+  Let \(G\) be a Lie group and \(\mathfrak{g}\) be its Lie algebra. Denote by
+  \(\operatorname{Diff}(G)^G\) the algebra of \(G\)-invariant differential
+  operators in \(G\) -- i.e. the algebra of all differential operators \(L :
+  C^\infty(G) \to C^\infty(G)\) such that \((L(f \circ \ell_g)) \circ
+  \ell_{g^{-1}} = L f\) for all \(f \in C^\infty(G)\) and \(g \in G\). There is
+  a canonical isomorphism of algebras \(\mathcal{U}(\mathfrak{g}) \isoto
+  \operatorname{Diff}(G)^G\).
+\end{proposition}
+
+\begin{proof}
+  An order \(1\) \(G\)-invariant differential operator in \(G\) is simply a
+  left invariant derivation \(C^\infty(G) \to C^\infty(G)\). All other
+  \(G\)-invariant differetial operators are generated by such derivations. Now
+  recall that there is a canonical isomorphism of Lie algebras
+  \(\mathfrak{X}(G) \isoto \operatorname{Der}(G)\). This isomorphism takes left
+  invariant fields to left invariant derivations, so it restricts to an
+  isomorphism \(f : \mathfrak{g} \isoto \operatorname{Der}(G)^G \subset
+  \operatorname{Diff}(G)^G\) -- here \(\operatorname{Der}(G)^G \subset
+  \operatorname{Der}(G)\) denotes the Lie subalgebra of invariant derivations.
+
+  Since \(f\) is a homomorphism of Lie algebras, it can be extended to an
+  algebra homomorphism \(g : \mathcal{U}(\mathfrak{g}) \to
+  \operatorname{Diff}(G)^G\). We claim \(g\) is an isomorphim. To see that
+  \(g\) is injective, it suffices to notice
+  \[
+    g(X_1 \cdots X_n)
+    = g(X_1) \cdots g(X_n)
+    = f(X_1) \cdots f(X_n)
+    \ne 0
+  \]
+  for all nonzero \(X_1, \cdots, X_n \in \mathfrak{g}\) --
+  \(\operatorname{Diff}(G)^G\) is a domain. Since \(\mathcal{U}(\mathfrak{g})\)
+  is generated by the image of the inclusion \(\mathfrak{g} \to
+  \mathcal{U}(\mathfrak{g})\), this implies \(\ker g = 0\). Given that
+  \(\operatorname{Diff}(G)^G\) is generated by \(\operatorname{Der}(G)^G\),
+  this also goes to show \(g\) is surjective.
+\end{proof}
+
+% TODO: Comment on the fact this holds for algebraic groups too
+
+\begin{theorem}[Poincaré-Birkoff-Witt]
+  Let \(\mathfrak{g}\) be a finite-dimensional Lie algebra over \(K\) and
+  \(\{X_i\}_i \subset \mathfrak{g}\) be a basis for \(\mathfrak{g}\). Then
+  \(\{X_{i_1} \cdot X_{i_2} \cdots X_{i_n} : n \ge 0, i_1 \le i_2 \le \cdots
+  \le i_n\}\) is a basis for \(\mathcal{U}(\mathfrak{g})\).
+\end{theorem}
+
+% TODO: Comment on the fact that modules of invariant differential operators
+% over G are precisely the same as representations of g
+
+\section{Representations}
+
+\begin{definition}
+  Given a Lie algebra \(\mathfrak{g}\) over \(K\), \emph{a representation \(V\)
+  of \(\mathfrak{g}\)}, or \emph{\(\mathfrak{g}\)-module}, is a \(K\)-vector
+  space endowed with a homomorphism of Lie algebras \(\rho : \mathfrak{g} \to
+  \mathfrak{gl}(V)\).
+\end{definition}
+
+\begin{example}
+  Given a Lie algebra \(\mathfrak{g}\), consider the homomorphism
+  \(\operatorname{ad} : \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})\) given by
+  \(\operatorname{ad}(X) Y = [X, Y]\). This gives \(\mathfrak{g}\) the
+  structure of a representation of \(\mathfrak{g}\), known as \emph{the adjoint
+  representation}.
+\end{example}
+
+\begin{example}
+  Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{g}\)-modules \(V\) and
+  \(W\), the the spaces \(V \oplus W\), \(V^*\), \(V \otimes W\) and
+  \(\operatorname{Hom}(V, W)\) are all \(\mathfrak{g}\)-modules -- where the
+  action of \(\mathfrak{g}\) is given by
+  \begin{align*}
+    X (v + w)       & = X v + X w                     &
+    X \cdot f       & = - f \circ X                   \\
+    X (v \otimes w) & = X v \otimes w + v \otimes X w &
+    (X \cdot T) v   & = X T v - T X v,
+  \end{align*}
+  respectively.
+\end{example}
+
+\begin{definition}
+  Given a Lie algebra \(\mathfrak{g}\) and two representations \(V\) and \(W\)
+  of \(\mathfrak{g}\), we call a linear map \(T : V \to W\) \emph{an
+  intertwiner} or \emph{a homomorphism of representations} if it cummutes with
+  the action of \(\mathfrak{g}\) in \(V\) and \(W\), in the sence that the
+  diagram
+  \begin{center}
+    \begin{tikzcd}
+      V \rar{T} \dar[swap]{X} & W \dar{X} \\
+      V \rar[swap]{T}         & W
+    \end{tikzcd}
+  \end{center}
+  commutes for all \(X \in \mathfrak{g}\). We denote the space of all
+  intertwiners \(V \to W\) by \(\operatorname{Hom}_{\mathfrak{g}}(V, W)\).
+\end{definition}
+
+% TODO: Point out g-Mod is indeed a category
+
 % TODO: Point out Hom(U(g), End(V)) ≃ Hom(g, gl(V))
 
 \begin{proposition}
@@ -616,6 +603,43 @@ ready to dive deeper into them.
   finite-dimensional repesentations to finitely generated modules.
 \end{proposition}
 
+\begin{definition}
+  Given a Lie algebra \(\mathfrak{g}\) and a representation \(V\) of
+  \(\mathfrak{g}\), a subspace \(W \subset V\) is called \emph{a
+  subrepresentation} if it is stable under the action of \(\mathfrak{g}\) --
+  i.e. \(X w \in W\) for all \(w \in W\) and \(X \in \mathfrak{g}\).
+\end{definition}
+
+\begin{example}
+  Given a Lie algebra \(\mathfrak{g}\), a representation \(V\) of
+  \(\mathfrak{g}\) and a subrepresentation \(W \subset V\), the space
+  \(\mfrac{V}{W}\) has the natural structure of a \(\mathfrak{g}\)-module where
+  \(X (v + W) = X v + W\). The projection \(V \to \mfrac{V}{W}\) is an
+  intertwiner.
+\end{example}
+
+\begin{example}
+  Given a Lie algebra \(\mathfrak{g}\) and representations \(V\) and \(W\) of
+  \(\mathfrak{g}\), the spaces \(V \wedge W\) and \(V \odot W\) are both
+  representations of \(G\): they are both quotients of \(V \otimes W\).
+\end{example}
+
+\begin{example}
+  Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{h}\) be a subalgebra.
+  Given a representation \(V\) of \(\mathfrak{g}\), denote by
+  \(\operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} V = V\) the representation
+  of \(\mathfrak{h}\) where the action of \(\mathfrak{h}\) is given by
+  restricting the map \(\mathfrak{g} \to \mathfrak{gl}(V)\) to
+  \(\mathfrak{g}\). Any homomorphism of \(\mathfrak{g}\)-modules \(V \to W\) is
+  also a homomorphism of \(\mathfrak{h}\)-modules and this construction is
+  clearly functorial.
+  \[
+    \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} :
+    \mathfrak{g}\text{-}\mathbf{Mod} \to
+    \mathfrak{h}\text{-}\mathbf{Mod}
+  \]
+\end{example}
+
 \begin{example}
   Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{h}\) be a subalgebra.
   Given a representation \(V\) of \(\mathfrak{h}\), denote by
@@ -663,54 +687,31 @@ ready to dive deeper into them.
   \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}}\).
 \end{proposition}
 
-% TODO: Define the algebra of differential operators of a given algebra
-\begin{proposition}
-  Let \(G\) be a Lie group and \(\mathfrak{g}\) be its Lie algebra. Denote by
-  \(\operatorname{Diff}(G)^G\) the algebra of \(G\)-invariant differential
-  operators in \(G\) -- i.e. the algebra of all differential operators \(L :
-  C^\infty(G) \to C^\infty(G)\) such that \((L(f \circ \ell_g)) \circ
-  \ell_{g^{-1}} = L f\) for all \(f \in C^\infty(G)\) and \(g \in G\). There is
-  a canonical isomorphism of algebras \(\mathcal{U}(\mathfrak{g}) \isoto
-  \operatorname{Diff}(G)^G\).
-\end{proposition}
+%\begin{definition}
+%  A representation of \(\mathfrak{g}\) is called \emph{indecomposable} if it is
+%  not isomorphic to the direct sum of two non-zero representations.
+%\end{definition}
+%
+%\begin{definition}
+%  A representation of \(\mathfrak{g}\) is called \emph{irreducible} if it has
+%  no non-zero subrepresentations.
+%\end{definition}
+%
+%\begin{lemma}[Schur]
+%  Let \(\mathfrak{g}\) be a Lie algebra over a field \(K\). If \(V\) and \(W\)
+%  irreducible representations of \(\mathfrak{g}\). and \(T : V \to W\) be an
+%  intertwiner then \(T\) is either \(0\) or an isomorphism. Furtheremore, if
+%  \(K\) is algebraicly closed and \(V = W\) then \(T\) is a scalar operator.
+%\end{lemma}
+%
+%\begin{proof}
+%  For the first statement, it suffices to notice that \(\ker T\) and
+%  \(\operatorname{im} T\) are both subrepresentations. In particular, either
+%  \(\ker T = 0\) and \(\operatorname{im} T = W\) or \(\ker T = V\) and
+%  \(\operatorname{im} T = 0\). Now suppose \(K\) is algebraicly closed and \(V
+%  = W\). Let \(\lambda \in K\) be an eigenvalue of \(T\) and \(V_\lambda\) be
+%  its corresponding eigenspace. Given \(v \in V_\lambda\), \(T X v = X T v =
+%  \lambda \cdot X v\). In other words, \(V_\lambda\) is a subrepresentation.
+%  It then follows \(V_\lambda = V\), given that \(V_\lambda \ne 0\).
+%\end{proof}
 
-\begin{proof}
-  An order \(1\) \(G\)-invariant differential operator in \(G\) is simply a
-  left invariant derivation \(C^\infty(G) \to C^\infty(G)\). All other
-  \(G\)-invariant differetial operators are generated by such derivations. Now
-  recall that there is a canonical isomorphism of Lie algebras
-  \(\mathfrak{X}(G) \isoto \operatorname{Der}(G)\). This isomorphism takes left
-  invariant fields to left invariant derivations, so it restricts to an
-  isomorphism \(f : \mathfrak{g} \isoto \operatorname{Der}(G)^G \subset
-  \operatorname{Diff}(G)^G\) -- here \(\operatorname{Der}(G)^G \subset
-  \operatorname{Der}(G)\) denotes the Lie subalgebra of invariant derivations.
-
-  Since \(f\) is a homomorphism of Lie algebras, it can be extended to an
-  algebra homomorphism \(g : \mathcal{U}(\mathfrak{g}) \to
-  \operatorname{Diff}(G)^G\). We claim \(g\) is an isomorphim. To see that
-  \(g\) is injective, it suffices to notice
-  \[
-    g(X_1 \cdots X_n)
-    = g(X_1) \cdots g(X_n)
-    = f(X_1) \cdots f(X_n)
-    \ne 0
-  \]
-  for all nonzero \(X_1, \cdots, X_n \in \mathfrak{g}\) --
-  \(\operatorname{Diff}(G)^G\) is a domain. Since \(\mathcal{U}(\mathfrak{g})\)
-  is generated by the image of the inclusion \(\mathfrak{g} \to
-  \mathcal{U}(\mathfrak{g})\), this implies \(\ker g = 0\). Given that
-  \(\operatorname{Diff}(G)^G\) is generated by \(\operatorname{Der}(G)^G\),
-  this also goes to show \(g\) is surjective.
-\end{proof}
-
-% TODO: Comment on the fact this holds for algebraic groups too
-
-% TODO: Comment on the fact that modules of invariant differential operators
-% over G are precisely the same as representations of g
-
-\begin{theorem}[Poincaré-Birkoff-Witt]
-  Let \(\mathfrak{g}\) be a finite-dimensional Lie algebra over \(K\) and
-  \(\{X_i\}_i \subset \mathfrak{g}\) be a basis for \(\mathfrak{g}\). Then
-  \(\{X_{i_1} \cdot X_{i_2} \cdots X_{i_n} : n \ge 0, i_1 \le i_2 \le \cdots
-  \le i_n\}\) is a basis for \(\mathcal{U}(\mathfrak{g})\).
-\end{theorem}