diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -1,12 +1,12 @@
\chapter{Lie Algebras}
-Associative algebras have proven themselves remarkably useful troughout
+Associative algebras have proven themselves remarkably useful throughout
mathematics. There's no lack of natural and interesting examples coming from a
diverse spectrum of different fields: topology, number theory, analysis, you
name it. Associative algebras have thus been studied at length, specially the
commutative ones. On the other hand, non-associative algebras have never
sustained the same degree of scrutiny. To this day, non-associative algebras
-remain remarkably misterious. Many have given up on attempting a sistematic
+remain remarkably mysterious. Many have given up on attempting a systematic
investigation and focus instead on understanding particular classes of
non-associative algebras -- i.e. algebras satisfying
\emph{pseudo-associativity} conditions.
@@ -28,7 +28,7 @@ called \emph{Lie algebras}, and these will be the focus of these notes.
Given two Lie algebras \(\mathfrak{g}\) and \(\mathfrak{h}\) over \(K\), a
homomorphism of Lie algebras \(\mathfrak{g} \to \mathfrak{h}\) is a
\(K\)-linear map \(f : \mathfrak{g} \to \mathfrak{h}\) which \emph{preserves
- brackets} in the sence that
+ brackets} in the sense that
\[
f([X, Y]) = [f(X), f(Y)]
\]
@@ -37,14 +37,14 @@ called \emph{Lie algebras}, and these will be the focus of these notes.
The collection of Lie algebras over a fixed field \(K\) thus form a category,
which we call \(K\text{-}\mathbf{LieAlg}\). We are primarily interested in
-finite-dimensional Lie algebras over algebraicly closed fields of characterist
-\(0\). Hence from now on we assume \(K\) is algebraicly closed and
-\(\operatorname{char} K = 0\) unless explicitely stated otherwise. Ironically,
+finite-dimensional Lie algebras over algebraically closed fields of characteristic
+\(0\). Hence from now on we assume \(K\) is algebraically closed and
+\(\operatorname{char} K = 0\) unless explicitly stated otherwise. Ironically,
perhaps the most basic examples of Lie algebras are derived from associative
algebras.
\begin{example}\label{ex:inclusion-alg-in-lie-alg}
- Given an associatice \(K\)-algebra \(A\), we can view \(A\) as a Lie algebra
+ Given an associative \(K\)-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
\(K\)-algebra \(\operatorname{End}(V)\) as a Lie algebra, which we call
@@ -53,8 +53,8 @@ algebras.
coefficients in \(K\).
\end{example}
-While strainghforward enought, I always found the definition of a Lie algebra
-unconvincing on its own. Specifically, the Jacobi identity can look very alian
+While straightforward enough, I always found the definition of a Lie algebra
+unconvincing on its own. Specifically, the Jacobi identity can look very alien
to someone who has never ventured outside of the realms of associativity.
Traditional abstract algebra courses offer little in the way of a motivation
for studying non-associative algebras in general. Why should we drop the
@@ -63,7 +63,7 @@ ever seen is an associative one? Instead, the most natural examples of Lie
algebras often come from an entirely different field: geometry.
Here the meaning of \emph{geometry} is somewhat vague. Topics such as
-differential and algebraic geometry are proeminantly featured, but examples
+differential and algebraic geometry are prominently featured, but examples
from fields such as the theory of differential operators and \(D\)-modules also
show up a lot in the theory of representations -- which we will soon discuss.
Perhaps one of the most fundamental themes of the study of Lie algebras is
@@ -84,7 +84,7 @@ One specific instance of this last example is\dots
\begin{example}
Given a smooth manifold \(M\), the space \(\mathfrak{X}(M)\) of all smooth
- vector fields is canonically identifyed with \(\operatorname{Der}(M) =
+ vector fields is canonically identified with \(\operatorname{Der}(M) =
\operatorname{Der}(C^\infty(M))\) -- where a field \(X \in \mathfrak{X}(M)\)
is identified with the map \(C^\infty(M) \to C^\infty(M)\) which takes a
function \(f \in C^\infty(M)\) to its derivative in the direction of \(X\).
@@ -203,7 +203,7 @@ between the category of Lie groups and smooth group homomorphisms and the
category of Lie algebras.
This goes to show Lie algebras are invariants of Lie groups. What's perhaps
-more surpring is the fact that, in certain contexts, Lie algebras are perfect
+more surprising is the fact that, in certain contexts, Lie algebras are perfect
invariants. Even more so\dots
\begin{theorem}[Lie]\label{thm:lie-theorems}
@@ -222,15 +222,15 @@ other hand, Lie's third theorem states every finite-dimensional real Lie
algebra is the Lie algebra of a simply connected Lie group -- i.e. the Lie
functor is essentially surjective.
-This goes to show that the ralationship between Lie groups and Lie algebras is
-deeper than the fact they share a name: in a very strong sence, studying simply
+This goes to show that the relationship between Lie groups and Lie algebras is
+deeper than the fact they share a name: in a very strong sense, studying simply
connected Lie groups is \emph{precisely} the same as studying
finite-dimensional Lie algebras. Such a vital connection between apparently
distant subjects is bound to produce interesting results. Indeed, the passage
from the algebraic and the geometric and vice-versa has proven itself a
fruitful one.
-This correspondance can be extended to the complex case too. In other words,
+This correspondence can be extended to the complex case too. In other words,
the Lie functor \(\mathbf{CLieGrp}_{\operatorname{simpl}} \to
\mathbb{C}\text{-}\mathbf{LieAlg}\) is also an equivalence of categories
between the category of simply connected complex Lie groups and the full
@@ -242,7 +242,7 @@ delicate in the algebraic case. For instance, given simply connected algebraic
homomorphism \(G \to H\).
In other words, the Lie functor \(K\text{-}\mathbf{Grp}_{\operatorname{simpl}}
-\to K\text{-}\mathbf{LieAlg}\) fails to be full. Furtheremore, there are
+\to K\text{-}\mathbf{LieAlg}\) fails to be full. Furthermore, there are
finite-dimension Lie algebras over \(K\) which are \emph{not} the Lie algebra
of an algebraic \(K\)-group, even if we allow for non-affine groups.
Nevertheless, Lie algebras are still powerful invariants of algebraic groups.
@@ -250,8 +250,8 @@ An interesting discussion of these delicacies can be found in sixth section of
\cite[ch.~II]{demazure-gabriel}.
All in all, there is a profound connection between groups and
-finite-dimensional Lie algebras troughtout multiple fields. While perhaps
-untituitive at first, the advantages of working with Lie algebras over their
+finite-dimensional Lie algebras throughout multiple fields. While perhaps
+unintuitive at first, the advantages of working with Lie algebras over their
group-theoretic counterparts are numerous. First, Lie algebras allow us to
avoid much of the delicacies of geometric objects such as real and complex Lie
groups. Even when working without additional geometric considerations, groups
@@ -260,7 +260,7 @@ On the other hand, Lie algebras are linear by nature, which makes them much
more flexible than groups.
Having thus hopefully established Lie algebras are interesting, we are now
-ready to dive deeper into them. We begin by analysing some of their most basic
+ready to dive deeper into them. We begin by analyzing some of their most basic
properties.
\section{Basic Structure of Lie Algebras}
@@ -281,7 +281,7 @@ is only natural to define\dots
\begin{note}
In the context of associative algebras, it's usual practice to distinguish
- between \emph{left ideals} and \emph{right ideals}. This is not neccessary
+ between \emph{left ideals} and \emph{right ideals}. This is not necessary
when dealing with Lie algebras, however, since any ``left ideal'' of a Lie
algebra is also a ``right ideal'': given \(\mathfrak{a} \normal
\mathfrak{g}\), \([Y, X] = - [X, Y] \in \mathfrak{a}\) for all \(X \in
@@ -292,7 +292,7 @@ is only natural to define\dots
Let \(f : \mathfrak{g} \to \mathfrak{h}\) be a homomorphism between Lie
algebras \(\mathfrak{g}\) and \(\mathfrak{h}\). Then \(\ker f \subset
\mathfrak{g}\) and \(\operatorname{im} f \subset \mathfrak{h}\) are
- subalgebras. Furtheremore, \(\ker f \normal \mathfrak{g}\).
+ subalgebras. Furthermore, \(\ker f \normal \mathfrak{g}\).
\end{example}
\begin{example}
@@ -320,7 +320,7 @@ There is also a natural analogue of quotients.
Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{a} \normal
\mathfrak{g}\), every homomorphism of Lie algebras \(f : \mathfrak{g} \to
\mathfrak{h}\) such that \(\mathfrak{a} \subset \ker f\) uniquely factors
- trought the projection \(\mathfrak{g} \to
+ through the projection \(\mathfrak{g} \to
\mfrac{\mathfrak{g}}{\mathfrak{a}}\).
\begin{center}
\begin{tikzcd}
@@ -423,7 +423,7 @@ Other interesting classes of Lie algebras are the so called \emph{simple} and
\end{definition}
\begin{example}
- Let \(G\) be a connected affinite algebraic \(K\)-group. Then \(G\) is
+ Let \(G\) be a connected affine algebraic \(K\)-group. Then \(G\) is
semisimple if, and only if \(\mathfrak{g}\) semisimple.
\end{example}
@@ -458,7 +458,7 @@ A slight generalization is\dots
As suggested by their names, simple and semisimple algebras are quite well
behaved when compared with the general case. To a lesser degree, reductive
-algebras are also unusualy well behaved. In the next chapter we will explore
+algebras are also unusually well behaved. In the next chapter we will explore
the question of why this is the case, but for now we note that we can get
semisimple and reductive algebras by modding out by certain ideals, known as
\emph{radicals}.
@@ -501,7 +501,7 @@ associative algebras to Lie algebras using the functor \(\operatorname{Lie} :
K\text{-}\mathbf{Alg} \to K\text{-}\mathbf{LieAlg}\) that takes an algebra
\(A\) to the Lie algebra \(A\) with brackets given by commutators. We can also
go the other direction by embedding a Lie algebra \(\mathfrak{g}\) in an
-associative algebra, kwown as \emph{the universal enveloping algebra of
+associative algebra, known as \emph{the universal enveloping algebra of
\(\mathfrak{g}\)}.
\begin{definition}
@@ -616,9 +616,9 @@ as\dots
The structure of \(\mathcal{U}(\mathfrak{g})\) can often be described in terms
of the structure of \(\mathfrak{g}\). For instance, \(\mathfrak{g}\) is Abelian
-if, and only if \(\mathcal{U}(\mathfrak{g})\) is cummutative, in which case any
+if, and only if \(\mathcal{U}(\mathfrak{g})\) is commutative, in which case any
basis \(\{X_i\}_i\) for \(\mathfrak{g}\) induces an isomorphism
-\(\mathcal{U}(\mathfrak{g}) \cong K[X_1, \ldots, X_i, \ldots]\). More generaly,
+\(\mathcal{U}(\mathfrak{g}) \cong K[X_1, \ldots, X_i, \ldots]\). More generally,
we find\dots
\begin{theorem}[Poincaré-Birkoff-Witt]
@@ -643,7 +643,7 @@ and again throughout these notes. Among other things, it implies\dots
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 intemately related with the algebra
+Lie group \(G\) is in fact intimately related with the algebra
\(\operatorname{Diff}(G)\) of differential operators \(C^\infty(G) \to
C^\infty(G)\) -- as defined in Coutinho's \citetitle{coutinho}
\cite[ch.~3]{coutinho}, for example. Algebras of differential operators and
@@ -663,7 +663,7 @@ remarkable progress in the past century. Specifically, we find\dots
\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
+ \(G\)-invariant differential 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
@@ -673,8 +673,8 @@ remarkable progress in the past century. Specifically, we find\dots
Since \(f\) is a homomorphism of Lie algebras, it can be extended to an
algebra homomorphism \(\bar f : \mathcal{U}(\mathfrak{g}) \to
- \operatorname{Diff}(G)^G\). We claim \(g\) is an isomorphim. To see that
- \(g\) is injective, it suffices to notice
+ \operatorname{Diff}(G)^G\). We claim \(\bar f\) is an isomorphism. To see
+ that \(\bar f\) is injective, it suffices to notice
\[
\bar f(X_1 \cdots X_n)
= \bar f(X_1) \cdots \bar f(X_n)
@@ -702,7 +702,7 @@ over the ring of \(G\)-invariant differential operators -- i.e.
\(G\).
Proposition~\ref{thm:geometric-realization-of-uni-env} is in fact only the
-beggining of a profound connection between the theory of \(D\)-modules and and
+beginning of a profound connection between the theory of \(D\)-modules and and
the so called \emph{representations} of Lie algebras. These will be the focus
of our next section.
@@ -769,8 +769,8 @@ representations too.
\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
+ intertwiner} or \emph{a homomorphism of representations} if it commutes with
+ the action of \(\mathfrak{g}\) in \(V\) and \(W\), in the sense that the
diagram
\begin{center}
\begin{tikzcd}
@@ -784,10 +784,10 @@ representations too.
The collection of representations of a fixed Lie algebra \(\mathfrak{g}\) thus
forms a category, which we call \(\mathfrak{g}\text{-}\mathbf{Mod}\). As
-promised, representations of \(\mathfrak{g}\) are intemately related to
+promised, representations of \(\mathfrak{g}\) are intimately related to
\(\mathcal{U}(\mathfrak{g})\)-modules. In fact, given a \(K\)-vector space
\(V\) proposition~\ref{thm:universal-env-uni-prop} implies there is a
-one-to-one correspondance between homomorphisms of Lie algebras \(\mathfrak{g}
+one-to-one correspondence between homomorphisms of Lie algebras \(\mathfrak{g}
\to \mathfrak{gl}(V)\) and homomorphisms of algebras
\(\mathcal{U}(\mathfrak{g}) \to \operatorname{End}(V)\) -- which takes a
homomorphism \(f : \mathfrak{g} \to \mathfrak{gl}(V)\) to its extension
@@ -798,7 +798,7 @@ then follows\dots
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.
+ finite-dimensional representations to finitely generated modules.
\end{proposition}
\begin{note}
@@ -817,7 +817,7 @@ dedicated to understanding a Lie algebra \(\mathfrak{g}\) via its
simple one: classifying all representations of a given Lie algebra up to
isomorphism. However, understanding the relationship between representations is
also of huge importance. In other words, to understand the whole of
-\(\mathfrak{g}\text{-}\mathbf{Mod}\) we need to study the collective behaviour
+\(\mathfrak{g}\text{-}\mathbf{Mod}\) we need to study the collective behavior
of representations -- as opposed to individual examples.
To that end, we define\dots