lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -158,7 +158,7 @@ this last construction.
     f & = \begin{pmatrix} 0 & 0 \\ 1 &  0 \end{pmatrix} &
     h & = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
   \end{align*}
-  form a basis for \(\mathfrak{sl}_2(K)\) and subject to the following
+  form a basis for \(\mathfrak{sl}_2(K)\) and are subject to the following
   relations.
   \begin{align*}
     [e, f] & = h & [h, f] & = -2 f & [h, e] = 2 e
@@ -186,8 +186,7 @@ this last construction.
   \[
     \mathfrak{sp}_{2 n}(K) =
     \left\{
-      X \in \mathfrak{gl}_{2 n}(K) :
-      X^{\operatorname{T}}
+      X \in \mathfrak{gl}_{2 n}(K) : X^\top
       \begin{pmatrix}
         0                     & \operatorname{Id}_n \\
         - \operatorname{Id}_n & 0
@@ -397,8 +396,8 @@ also share structural features with groups. For example\dots
 \end{example}
 
 \begin{definition}
-  A Lie algebra \(\mathfrak{g}\) is called \emph{nilpotent} if its derived
-  series
+  A Lie algebra \(\mathfrak{g}\) is called \emph{nilpotent} if its lower
+  central series
   \[
     \mathfrak{g}
     \supseteq [\mathfrak{g}, \mathfrak{g}]
@@ -436,7 +435,7 @@ Other interesting classes of Lie algebras are the so called \emph{simple} and
   \mathfrak{a}\), so again \(\mathfrak{a} = \mathfrak{sl}_2(K)\). Similarly, if
   \(f \in \mathfrak{a}\) then \([e, f] = h \in \mathfrak{a}\) and
   \(\mathfrak{a} = \mathfrak{sl}_2(K)\). More generally, the Lie algebra
-  \(\mathfrak{sl}_n(K)\) is simple for each \(n > 0\) -- see the section of
+  \(\mathfrak{sl}_n(K)\) is simple for each \(n > 1\) -- see the section of
   \cite[ch. 6]{kirillov} on invariant bilinear forms and the semisimplicity of
   classical Lie algebras.
 \end{example}
@@ -461,7 +460,7 @@ A slight generalization is\dots
 \end{definition}
 
 \begin{example}
-  The Lie algebra \(\mathfrak{gl}_n(K)\) is reducible. Indeed,
+  The Lie algebra \(\mathfrak{gl}_n(K)\) is reductive. Indeed,
   \[
     X
     =
@@ -514,8 +513,6 @@ semisimple and reductive algebras by modding out by certain ideals, known as
   \]
 \end{definition}
 
-As promised, we finds\dots
-
 \begin{proposition}\label{thm:quotients-by-rads}
   Let \(\mathfrak{g}\) be a Lie algebra. Then
   \(\mfrac{\mathfrak{g}}{\mathfrak{rad}(\mathfrak{g})}\) is semisimple and
@@ -671,9 +668,9 @@ 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
 \(\operatorname{Diff}(G)\) of differential operators \(C^\infty(G) \to
-C^\infty(G)\) -- \(\mathbb{R}\)-linear endomorphisms \(C^\infty(G) \to
-C^\infty(G)\) of finite order as defined in Coutinho's \citetitle{coutinho}
-\cite[ch.~3]{coutinho}, for example. Algebras of differential operators and
+C^\infty(G)\) -- i.e. \(\mathbb{R}\)-linear endomorphisms \(C^\infty(G) \to
+C^\infty(G)\) of finite order, as defined in Coutinho's \citetitle{coutinho}
+\cite[ch.~3]{coutinho} for example. Algebras of differential operators and
 their modules are the subject of the theory of \(D\)-modules, which has seen
 remarkable progress in the past century. Specifically, we find\dots
 
@@ -689,12 +686,12 @@ remarkable progress in the past century. Specifically, we find\dots
 
 \begin{proof}
   An order \(0\) \(G\)-invariant differential operator in \(G\) is simply
-  multiplication by a constant in \(\mathbb{R}\). An order \(1\)
+  multiplication by a constant in \(\mathbb{R}\). A homogeneous 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
   differential operators are generated by invariant operators of order \(0\)
   and \(1\). Hence \(\operatorname{Diff}(G)^G\) is generated by
-  \(\operatorname{Der}(G)^G\) -- here \(\operatorname{Der}(G)^G \subset
+  \(\operatorname{Der}(G)^G + K\) -- here \(\operatorname{Der}(G)^G \subset
   \operatorname{Der}(G)\) denotes the Lie subalgebra of invariant derivations.
 
   Now recall that there is a canonical isomorphism of Lie algebras
@@ -718,7 +715,7 @@ remarkable progress in the past century. Specifically, we find\dots
   Since \(\mathcal{U}(\mathfrak{g})\) is generated by the image of the
   inclusion \(\mathfrak{g} \to \mathcal{U}(\mathfrak{g})\), this implies \(\ker
   \tilde f = 0\). Given that \(\operatorname{Diff}(G)^G\) is generated by
-  \(\operatorname{Der}(G)^G\), this also goes to show \(\tilde f\) is
+  \(\operatorname{Der}(G)^G + K\), this also goes to show \(\tilde f\) is
   surjective.
 \end{proof}
 
@@ -745,21 +742,23 @@ First introduced in 1896 by Georg Frobenius in his paper \citetitle{frobenius}
 one of the cornerstones of modern mathematics. In this section we provide a
 brief overview of basic concepts of the representation theory of Lie algebras.
 We should stress, however, that the representation theory of Lie algebras is
-only a small fragment of what is today known as representation theory, which is
-in general concerned with a diverse spectrum of algebraic and combinatorial
-structures -- such as groups, quivers and associative algebras. An introductory
-exploration of some of this structures can be found in \cite{etingof}.
+only a small fragment of what is today known as ``representation theory'',
+which is in general concerned with a diverse spectrum of algebraic and
+combinatorial structures -- such as groups, quivers and associative algebras.
+An introductory exploration of some of this structures can be found in
+\cite{etingof}.
 
 We begin by noting that any \(\mathcal{U}(\mathfrak{g})\)-module \(V\) may be
 regarded as a \(K\)-vector space endowed with a ``linear action'' of
 \(\mathfrak{g}\). Indeed, by restricting the action map
-\(\mathcal{U}(\mathfrak{g}) \to \operatorname{End}(V)\) to \(\mathfrak{g}\)
-yields a homomorphism of Lie algebras \(\mathfrak{g} \to \mathfrak{gl}(V) =
-\operatorname{End}(V)\). In fact proposition~\ref{thm:universal-env-uni-prop}
-implies that given a vector space \(V\) there is a one-to-one correspondence
-between \(\mathcal{U}(\mathfrak{g})\)-module structures for \(V\) and
-homomorphisms \(\mathfrak{g} \to \mathfrak{gl}(V)\). This leads us to the
-following definition.
+\(\mathcal{U}(\mathfrak{g}) \to \operatorname{End}(V)\) to \(\mathfrak{g}
+\subset \mathcal{U}(\mathfrak{g})\) yields a homomorphism of Lie algebras
+\(\mathfrak{g} \to \mathfrak{gl}(V) = \operatorname{End}(V)\). In fact
+proposition~\ref{thm:universal-env-uni-prop} implies that given a vector space
+\(V\) there is a one-to-one correspondence between
+\(\mathcal{U}(\mathfrak{g})\)-module structures for \(V\) and homomorphisms
+\(\mathfrak{g} \to \mathfrak{gl}(V)\). This leads us to the following
+definition.
 
 \begin{definition}
   Given a Lie algebra \(\mathfrak{g}\) over \(K\), \emph{a representation \(V\)
@@ -927,7 +926,8 @@ define\dots
   Given a Lie algebra \(\mathfrak{g}\) and representations \(V\) and \(W\) of
   \(\mathfrak{g}\), the exterior and symmetric products \(V \wedge W\) and \(V
   \odot W\) are both representations of \(\mathfrak{g}\): they are quotients of
-  \(V \otimes W\).
+  \(V \otimes W\). In particular, the exterior and symmetric powers \(\wedge^n
+  V\) and \(\operatorname{Sym}^n V\) are \(\mathfrak{g}\)-modules.
 \end{example}
 
 \begin{example}
@@ -946,7 +946,7 @@ separate algebras. In particular, we may define\dots
   \(\operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} V = V\) 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
+  \(\mathfrak{h}\). Any homomorphism of \(\mathfrak{g}\)-modules \(V \to W\) is
   also a homomorphism of \(\mathfrak{h}\)-modules and this construction is
   clearly functorial.
   \[
@@ -958,12 +958,11 @@ separate algebras. In particular, we may define\dots
 
 \begin{example}
   Given a Lie algebra \(\mathfrak{g}\), the adjoint representation of
-  \(\mathfrak{g}\) is a subrepresentation of
-  \(\operatorname{Res}_{\mathfrak{g}}^{\mathcal{U}(\mathfrak{g})}
-  \mathcal{U}(\mathfrak{g})\).
+  \(\mathfrak{g}\) is a subrepresentation of the restriction of the adjoint
+  representation of \(\mathcal{U}(\mathfrak{g})\) to \(\mathfrak{g}\).
 \end{example}
 
-Surprisingly, this functor has right adjoint.
+Surprisingly, this functor has a right adjoint.
 
 \begin{example}
   Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{h}\) be a subalgebra.
@@ -971,10 +970,11 @@ Surprisingly, this functor has right adjoint.
   \(\operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V\) the representation of
   \(\mathfrak{h}\) corresponding to the \(\mathcal{U}(\mathfrak{g})\)-module
   \(\mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{h})} V\) -- 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 :
+  the right \(\mathcal{U}(\mathfrak{h})\)-module structure of
+  \(\mathcal{U}(\mathfrak{g})\) is given by right 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.
@@ -1037,7 +1037,7 @@ Surprisingly, this functor has right adjoint.
 \end{proof}
 
 This last proposition is known as \emph{Frobenius reciprocity}, and was first
-proved by Frobenius himself in the context of finite-groups. Another
+proved by Frobenius himself in the context of finite groups. Another
 interesting construction is\dots
 
 \begin{example}
@@ -1053,4 +1053,4 @@ interesting construction is\dots
 
 This concludes our initial remarks on representations. In the following
 chapters we will explore the fundamental problem of representation theory: that
-of classifying all representations up to isomorphism.
+of classifying all representations of a given algebra up to isomorphism.