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.