lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -23,7 +23,7 @@
   for all \(X, Y \in \mathfrak{g}\).
 \end{definition}
 
-\begin{example}
+\begin{example}\label{ex:inclusion-alg-in-lie-alg}
   Given an associatice \(K\)-algebra \(A\), we can view \(A\) as a Lie algebra
   over \(K\) with the Lie brackets given by the commutator \([a, b] = ab -
   ba\). In particular, given a \(K\)-vector space \(V\) we may view the
@@ -323,6 +323,24 @@
   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
@@ -334,9 +352,8 @@
   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
-  \[
-    X (v + W) = X v + W
-  \]
+  \(X (v + W) = X v + W\). The projection \(V \to \mfrac{V}{W}\) is an
+  intertwiner.
 \end{example}
 
 \begin{example}
@@ -364,3 +381,130 @@
 \end{lemma}
 
 \section{The Universal Enveloping Algebra}
+
+\begin{definition}
+  Let \(\mathfrak{g}\) be a Lie algebra and \(T \mathfrak{g} = \bigoplus_n
+  \mathfrak{g}^{\otimes n}\) be its tensor algebra -- i.e. the free
+  \(K\)-algebra generated by the elements of \(\mathfrak{g}\). We call the
+  \(K\)-algebra \(\mathcal{U}(\mathfrak{g}) = \mfrac{T \mathfrak{g}}{I}\)
+  \emph{the universal enveloping algebra of \(\mathfrak{g}\)}, where \(I = ([X,
+  Y] - (X \otimes Y - Y \otimes X) : X, Y \in \mathfrak{g})\) is the left ideal
+  of \(T \mathfrak{g}\) generated by the elements \([X, Y] - (X \otimes Y - Y
+  \otimes X)\).
+\end{definition}
+
+\begin{proposition}
+  Let \(\mathfrak{g}\) be a Lie algebra and \(A\) be an associative
+  \(K\)-algebra. Then every homomorphism of Lie algebras \(f : \mathfrak{g} \to
+  A\) -- where \(A\) is endowed with the structure of a Lie algebra as in
+  example~\ref{ex:inclusion-alg-in-lie-alg} -- can be uniquely extended to a
+  homomorphism of algebras \(\mathcal{U}(\mathfrak{g}) \to A\).
+  \begin{center}
+    \begin{tikzcd}
+      \mathcal{U}(\mathfrak{g})  \rar[dotted] & A \dar[Rightarrow, no head] \\
+      \mathfrak{g} \rar[swap]{f} \uar         & A
+    \end{tikzcd}
+  \end{center}
+\end{proposition}
+
+% TODO: Remark this construction is functorial
+
+\begin{center}
+  \begin{tikzcd}
+    \mathcal{U}(\mathfrak{g}) \arrow[dotted]{rr}{\mathcal{U}(f)} & &
+    \mathcal{U}(\mathfrak{h}) \dar[Rightarrow, no head]          \\
+    \mathfrak{g} \rar[swap]{f} \uar                              &
+    \mathfrak{h} \rar                                            &
+    \mathcal{U}(\mathfrak{h})
+  \end{tikzcd}
+\end{center}
+
+% TODO: Point out U is not the "inverse" of K-Alg -> K-LieAlg, but there is an
+% adjunction
+
+\begin{corollary}
+  If \(\operatorname{Lie} : K\text{-}\mathbf{Alg} \to
+  K\text{-}\mathbf{LieAlg}\) is the functor described in
+  example~\ref{ex:inclusion-alg-in-lie-alg}, there is an adjunction
+  \(\operatorname{Lie} \vdash \mathcal{U}\).
+\end{corollary}
+
+% TODO: Point out Hom(U(g), End(V)) ≃ Hom(g, gl(V))
+
+\begin{proposition}
+  There is a natural equivalence of categories
+  \(\mathfrak{g}\text{-}\mathbf{Mod} \isoto
+  \mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\), which takes
+  finite-dimensional repesentations to finitely generated modules.
+\end{proposition}
+
+\begin{example}
+  Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{h}\) be a subalgebra.
+  Given a representation \(V\) of \(\mathfrak{h}\), denote by
+  \(\operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V\) the representation of
+  \(\mathfrak{h}\) corresponding to the \(\mathcal{U}(\mathfrak{h})\)-module
+  \(\mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(h)} V\) -- where the action
+  of \(\mathfrak{h}\) in \(\mathcal{U}(\mathfrak{g})\) is given by left
+  multiplication. Any homomorphism of \(\mathfrak{h}\)-modules \(T : V \to W\)
+  induces a homomorphism \(\operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} T =
+  \operatorname{Id} \otimes T :
+  \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V \to
+  \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} W\) and this construction is
+  clearly functorial.
+  \[
+    \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} :
+    \mathfrak{h}\text{-}\mathbf{Mod} \to \mathfrak{g}\text{-}\mathbf{Mod}
+  \]
+\end{example}
+
+\begin{proposition}
+  Given a Lie algebra\(\mathfrak{g}\), a subalgebra \(\mathfrak{h} \subset
+  \mathfrak{g}\), a representation \(V\) of \(\mathfrak{h}\) and a
+  representation \(W\) of \(\mathfrak{g}\), the map
+  \[
+    \arraycolsep=1.4pt
+    \begin{array}[t]{rl}
+      \Phi :
+      \operatorname{Hom}_{\mathfrak{g}}(
+        \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V,
+        W
+      ) & \to
+      \operatorname{Hom}_{\mathfrak{h}}(
+        V,
+        \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} W
+      ) \\
+      T & \mapsto
+      \begin{array}[t]{rl}
+        \Phi(T) : V & \to \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} W \\
+                  v & \mapsto T (1 \otimes v)
+      \end{array}
+    \end{array}
+  \]
+  is a \(K\)-linear isomorphism. In other words, there is an adjunction
+  \(\operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} \vdash
+  \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}}\).
+\end{proposition}
+
+\begin{theorem}
+  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{theorem}
+
+% 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 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: The analytic proof of PBW only works for finite-dimensional algebras