diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -3,12 +3,12 @@
\epigraph{Nobody has ever bet enough on a winning horse.}{Some gambler}
% TODO: Update the 40 pages thing when we're done
-Having hopefully stablished in the previous chapter that Lia algebras are
-indeed usefull, we are now faced with the Herculian task of trying to
-understand them. We have seen that representations are a remarkbly effective
+Having hopefully established in the previous chapter that Lie algebras are
+indeed useful, we are now faced with the Herculean task of trying to
+understand them. We have seen that representations are a remarkably effective
way to derive information about groups -- and therefore algebras -- but the
question remains: how to we go about classifying the representations of a given
-Lie algebra? This is a question that have sparked an entire field of reasearch,
+Lie algebra? This is a question that have sparked an entire field of research,
and we cannot hope to provide a comprehensive answer the 40 pages we have left.
Nevertheless, we can work on particular cases.
@@ -17,8 +17,8 @@ Like any sane mathematician would do, we begin by studying a simpler case. The
restrictions we impose are twofold: restrictions on the algebras whose
representations we'll classify, and restrictions on the representations
themselves. First of all, we will work exclusively with finite-dimensional Lie
-algebras over an algebraicly closed field \(K\) of characteristic \(0\). This
-is a restriction we will cary throught these notes. Moreover, as indicated by
+algebras over an algebraically closed field \(K\) of characteristic \(0\). This
+is a restriction we will carry throughout these notes. Moreover, as indicated by
the title of this chapter, we will initially focus on the so called
\emph{semisimple} Lie algebras algebras -- we will later relax this restriction
a bit in the next chapter when we dive into \emph{reductive} Lie algebras.
@@ -65,8 +65,8 @@ A popular alternative to definition~\ref{thm:sesimple-algebra} is\dots
% TODO: Add gl_n(K) as an example of a reductive algebra
I suppose this last definition explains the nomenclature, but the reason why
-semisimple Lie algebras are interesting at all is still unclear. In particual,
-why are they simpler -- or perhaps \emph{semisimpler} -- to understnad than any
+semisimple Lie algebras are interesting at all is still unclear. In particular,
+why are they simpler -- or perhaps \emph{semisimpler} -- to understand than any
old Lie algebra? Well, the special thing about semisimple algebras is that the
relationship between their indecomposable representations
and their irreducible representations is much clearer -- at least in finite
@@ -108,7 +108,7 @@ is completely reducible, the equivalence between \textbf{(ii)} and \textbf{(v)}
implies a classification of the finite-dimensional irreducible representations
of \(\mathfrak{g}\) leads to a classification of \emph{all} finite-dimensional
representation of \(\mathfrak{g}\) -- it suffices to take direct sums of the
-already classifyed irreducible modules. This leads us to the third restriction
+already classified irreducible modules. This leads us to the third restriction
we will impose: for now, we will focus our attention exclusively on
finite-dimensional representations.
@@ -125,7 +125,7 @@ handy later on, is\dots
all \(Y \in \mathfrak{g}\) then \(X = 0\).
\end{proposition}
-\section{Comple Reducibility}
+\section{Complete Reducibility}
We are primarily interested in establishing\dots
@@ -138,25 +138,25 @@ Historically, this was first proved by Herman Weyl for \(K = \mathbb{C}\),
using his knowledge of unitary representations of compact groups. Namely, Weyl
showed that any finite-dimensional semisimple complex Lie algebra is
(isomorphic to) the complexification of the Lie algebra of a simply connected
-campact Lie group, so that the category of its finite-dimensional
+compact Lie group, so that the category of its finite-dimensional
representations is equivalent to that of the finite-dimensional smooth
representations of such compact group -- under which Mashcke's theorem for
compact groups applies. We refer the reader to (TODO: cite someone) for further
details.
-This proof, however, is heavely reliant on the geometric structure of
+This proof, however, is heavily reliant on the geometric structure of
\(\mathbb{C}\). In other words, there is no hope for generalizing this for some
arbitrary \(K\). Hopefully for us, there is a much simpler, completely
-algebraic proof which works for algebras over any algebraicly closed field of
+algebraic proof which works for algebras over any algebraically closed field of
characteristic zero. The algebraic proof included in here mainly based on that
-of \cite[ch. 6]{kirillov}, and uses some basic homological algebra. Admitdely,
-much of the homological algebra used in here could be conceiled from the reader
+of \cite[ch. 6]{kirillov}, and uses some basic homological algebra. Admittedly,
+much of the homological algebra used in here could be concealed from the reader
-- see \cite{humphreys} for instance -- which would make the exposition more
-accessible.
+accessible.
However, this does not change the fact the arguments used in this proof are
essentially homological in nature. Hence we consider it more productive to use
-the full force of the language of homological algebra, instead of buring the
+the full force of the language of homological algebra, instead of burring the
reader in a pile of unmotivated elementary arguments. Furthermore, the
homological algebra used in here is actually \emph{very basic}. In fact, all we
need to know is\dots
@@ -174,40 +174,40 @@ need to know is\dots
induces long exact sequences
\begin{center}
\begin{tikzcd}
- 0 \arrow{r} &
- \operatorname{Hom}_{\mathfrak{g}}(S, W) \arrow{r}{i \circ -} &
- \operatorname{Hom}_{\mathfrak{g}}(S, V) \arrow{r}{\pi \circ -} &
- \operatorname{Hom}_{\mathfrak{g}}(S, U) \arrow{r} &
+ 0 \arrow{r} &
+ \operatorname{Hom}_{\mathfrak{g}}(S, W) \arrow{r}{i \circ -} &
+ \operatorname{Hom}_{\mathfrak{g}}(S, V) \arrow{r}{\pi \circ -} &
+ \operatorname{Hom}_{\mathfrak{g}}(S, U) \arrow{r} &
\hphantom{0} \\
- \hphantom{0} \arrow{r} &
- \operatorname{Ext}^1(S, W) \arrow{r} &
- \operatorname{Ext}^1(S, V) \arrow{r} &
- \operatorname{Ext}^1(S, U) \arrow{r} &
+ \hphantom{0} \arrow{r} &
+ \operatorname{Ext}^1(S, W) \arrow{r} &
+ \operatorname{Ext}^1(S, V) \arrow{r} &
+ \operatorname{Ext}^1(S, U) \arrow{r} &
\hphantom{0} \\
- \hphantom{0} \arrow{r} &
- \operatorname{Ext}^2(S, W) \arrow{r} &
- \operatorname{Ext}^2(S, V) \arrow{r} &
- \operatorname{Ext}^2(S, U) \arrow{r} &
+ \hphantom{0} \arrow{r} &
+ \operatorname{Ext}^2(S, W) \arrow{r} &
+ \operatorname{Ext}^2(S, V) \arrow{r} &
+ \operatorname{Ext}^2(S, U) \arrow{r} &
\cdots
\end{tikzcd}
\end{center}
and
\begin{center}
\begin{tikzcd}
- 0 \arrow{r} &
- \operatorname{Hom}_{\mathfrak{g}}(U, S) \arrow{r}{- \circ \pi} &
- \operatorname{Hom}_{\mathfrak{g}}(V, S) \arrow{r}{- \circ i} &
- \operatorname{Hom}_{\mathfrak{g}}(W, S) \arrow{r} &
+ 0 \arrow{r} &
+ \operatorname{Hom}_{\mathfrak{g}}(U, S) \arrow{r}{- \circ \pi} &
+ \operatorname{Hom}_{\mathfrak{g}}(V, S) \arrow{r}{- \circ i} &
+ \operatorname{Hom}_{\mathfrak{g}}(W, S) \arrow{r} &
\hphantom{0} \\
- \hphantom{0} \arrow{r} &
- \operatorname{Ext}^1(U, S) \arrow{r} &
- \operatorname{Ext}^1(V, S) \arrow{r} &
- \operatorname{Ext}^1(W, S) \arrow{r} &
+ \hphantom{0} \arrow{r} &
+ \operatorname{Ext}^1(U, S) \arrow{r} &
+ \operatorname{Ext}^1(V, S) \arrow{r} &
+ \operatorname{Ext}^1(W, S) \arrow{r} &
\hphantom{0} \\
- \hphantom{0} \arrow{r} &
- \operatorname{Ext}^2(U, S) \arrow{r} &
- \operatorname{Ext}^2(V, S) \arrow{r} &
- \operatorname{Ext}^2(W, S) \arrow{r} &
+ \hphantom{0} \arrow{r} &
+ \operatorname{Ext}^2(U, S) \arrow{r} &
+ \operatorname{Ext}^2(V, S) \arrow{r} &
+ \operatorname{Ext}^2(W, S) \arrow{r} &
\cdots
\end{tikzcd}
\end{center}
@@ -216,7 +216,7 @@ need to know is\dots
% TODO: Make the correspondance more precise?
\begin{theorem}\label{thm:ext-1-classify-short-seqs}
Given \(\mathfrak{g}\)-modules \(W\) and \(U\), there is a one-to-one
- correspondance between elements of \(\operatorname{Ext}^1(W, U)\) and
+ correspondence between elements of \(\operatorname{Ext}^1(W, U)\) and
isomorphism classes of short exact sequences
\begin{center}
\begin{tikzcd}
@@ -267,13 +267,13 @@ implies\dots
induces a long exact sequence
\begin{center}
\begin{tikzcd}
- 0 \arrow{r} &
- W^{\mathfrak{g}} \arrow{r}{i} &
- V^{\mathfrak{g}} \arrow{r}{\pi} &
- U^{\mathfrak{g}} \arrow{r} &
- H^1(\mathfrak{g}, W) \arrow{r} &
- H^1(\mathfrak{g}, V) \arrow{r} &
- H^1(\mathfrak{g}, U) \arrow{r} &
+ 0 \arrow{r} &
+ W^{\mathfrak{g}} \arrow{r}{i} &
+ V^{\mathfrak{g}} \arrow{r}{\pi} &
+ U^{\mathfrak{g}} \arrow{r} &
+ H^1(\mathfrak{g}, W) \arrow{r} &
+ H^1(\mathfrak{g}, V) \arrow{r} &
+ H^1(\mathfrak{g}, U) \arrow{r} &
\cdots
\end{tikzcd}
\end{center}
@@ -283,19 +283,19 @@ implies\dots
We have an isomorphism of sequences
\begin{center}
\begin{tikzcd}
- 0 \arrow{r} &
- \operatorname{Hom}_{\mathfrak{g}}(K, W)
- \arrow{r}{i \circ -} \arrow{d} &
- \operatorname{Hom}_{\mathfrak{g}}(K, V)
- \arrow{r}{\pi \circ -} \arrow{d} &
- \operatorname{Hom}_{\mathfrak{g}}(K, U) \arrow{r} \arrow{d} &
- H^1(\mathfrak{g}, W) \arrow{r} \arrow[Rightarrow, no head]{d} &
+ 0 \arrow{r} &
+ \operatorname{Hom}_{\mathfrak{g}}(K, W)
+ \arrow{r}{i \circ -} \arrow{d} &
+ \operatorname{Hom}_{\mathfrak{g}}(K, V)
+ \arrow{r}{\pi \circ -} \arrow{d} &
+ \operatorname{Hom}_{\mathfrak{g}}(K, U) \arrow{r} \arrow{d} &
+ H^1(\mathfrak{g}, W) \arrow{r} \arrow[Rightarrow, no head]{d} &
\cdots \\
- 0 \arrow{r} &
- W^{\mathfrak{g}} \arrow[swap]{r}{i} &
- V^{\mathfrak{g}} \arrow[swap]{r}{\pi} &
- U^{\mathfrak{g}} \arrow{r} &
- H^1(\mathfrak{g}, W) \arrow{r} &
+ 0 \arrow{r} &
+ W^{\mathfrak{g}} \arrow[swap]{r}{i} &
+ V^{\mathfrak{g}} \arrow[swap]{r}{\pi} &
+ U^{\mathfrak{g}} \arrow{r} &
+ H^1(\mathfrak{g}, W) \arrow{r} &
\cdots
\end{tikzcd}
\end{center}
@@ -310,11 +310,11 @@ trying to control obstructions of some kind. In our case, the bifunctor
\(H^1(\mathfrak{g}, \operatorname{Hom}(-, -)) :
\mathfrak{g}\text{-}\mathbf{Mod} \times \mathfrak{g}\text{-}\mathbf{Mod} \to
\mathbf{Ab}\) classifies obstructions to complete reducibility.
-Explicitely\dots
+Explicitly\dots
\begin{theorem}
Given \(\mathfrak{g}\)-modules \(W\) and \(U\), there is a one-to-one
- correspondance between elements of \(H^1(\mathfrak{g}, \operatorname{Hom}(W,
+ correspondence between elements of \(H^1(\mathfrak{g}, \operatorname{Hom}(W,
U))\) and isomorphism classes of short exact sequences
\begin{center}
\begin{tikzcd}
@@ -323,26 +323,26 @@ Explicitely\dots
\end{center}
\end{theorem}
-For the readers already familiar with homological algebra: this correspondance
+For the readers already familiar with homological algebra: this correspondence
can computed very concretely by considering the canonical acyclic resolution
\begin{center}
\begin{tikzcd}
0 \rar &
- K \rar &
- \mathfrak{g} \rar &
- \wedge^2 \mathfrak{g} \rar &
- \wedge^3 \mathfrak{g} \rar &
+ K \rar &
+ \mathfrak{g} \rar &
+ \wedge^2 \mathfrak{g} \rar &
+ \wedge^3 \mathfrak{g} \rar &
\cdots
\end{tikzcd}
\end{center}
of the trivial representation \(K\), which provides an explicit construction of
the cohomology groups -- see \cite[sec. 9]{lie-groups-serganova-student} for
-further details. We will use the previous result implicitely in our proof, but
+further details. We will use the previous result implicitly in our proof, but
we will not prove it in its full force. Namely, we will show that
\(H^1(\mathfrak{g}, V) = 0\) for all finite-dimensional \(V\), and that the
fact that \(H^1(\mathfrak{g}, \operatorname{Hom}(W, U)) = 0\) for all
finite-dimensional \(W\) and \(U\) implies complete reducibility. To that end,
-we introduce a distinguised element of \(\mathcal{U}(\mathfrak{g})\), known as
+we introduce a distinguished element of \(\mathcal{U}(\mathfrak{g})\), known as
\emph{the Casimir element}.
\begin{definition}\label{def:casimir-element}
@@ -356,7 +356,7 @@ we introduce a distinguised element of \(\mathcal{U}(\mathfrak{g})\), known as
\end{definition}
\begin{lemma}
- The definition of \(C\) is independant of the choice of basis \(\{X_i\}_i\).
+ The definition of \(C\) is independent of the choice of basis \(\{X_i\}_i\).
\end{lemma}
\begin{proof}
@@ -371,7 +371,7 @@ we introduce a distinguised element of \(\mathcal{U}(\mathfrak{g})\), known as
\begin{proposition}
The Casimir element \(C \in \mathcal{U}(\mathfrak{g})\) is central, so that
- \(C : V \to V\) is an intertnining operator for any \(\mathfrak{g}\)-module
+ \(C : V \to V\) is an intertwining operator for any \(\mathfrak{g}\)-module
\(V\). Furthermore, \(C\) acts as a non-zero scalar operator whenever \(V\)
is a non-trivial finite-dimensional irreducible representation of
\(\mathfrak{g}\).
@@ -382,7 +382,7 @@ we introduce a distinguised element of \(\mathcal{U}(\mathfrak{g})\), known as
and denote by \(\{X^i\}_i\) its dual basis as in
definition~\ref{def:casimir-element}. Let \(X \in \mathfrak{g}\) and denote
by \(\lambda_{i j}, \mu_{i j} \in K\) the coefficients of \(X_j\) and \(X^j\)
- in \([X, X_i]\) and \([X, X^i]\), respectively.
+ in \([X, X_i]\) and \([X, X^i]\), respectively.
% TODO: Comment on the invariance of the Killing form beforehand
The invariance of \(B\) implies
@@ -449,13 +449,13 @@ As promised, the Casimir element can be used to establish\dots
particular, \(X w = 0\) for all \(X \in \mathfrak{g}\). Since \(w\) lies
outside the image of the inclusion \(K \to W\), \(\pi(w) \ne 0\) -- which is
to say, \(w \notin \ker \pi\). This implies the map \(K \to W\) that takes
- \(1\) to \(\sfrac{w}{\pi(w)}\) is a spliting of
+ \(1\) to \(\sfrac{w}{\pi(w)}\) is a splitting of
(\ref{eq:trivial-extrems-exact-seq}).
Now suppose that \(V\) is non-trivial, so that \(C\) acts on \(V\) as
\(\lambda \operatorname{Id}\) for some \(\lambda \ne 0\). Given an eigenvalue
\(\mu \in K\) of the action of \(C\) in \(W\), denote by \(W^\mu\) its
- associated generalized eigespace. We claim \(W^0\) is the image of the
+ associated generalized eigenspace. We claim \(W^0\) is the image of the
inclusion \(K \to W\). Since \(C\) acts as zero in \(K\), this image is
clearly contained in \(W^0\). On the other hand, if \(w \in W\) is such that
\(C^n w = 0\) then
@@ -466,11 +466,11 @@ As promised, the Casimir element can be used to establish\dots
= 0,
\]
so that \(w \in \ker \pi\) -- because \(\lambda^n \ne 0\). The exactness of
- (\ref{eq:exact-seq-h1-vanishes}) then implies the desired conclusion.
+ (\ref{eq:exact-seq-h1-vanishes}) then implies the desired conclusion.
We furthermore claim that the only eigenvalues of \(C\) in \(W\) are \(0\)
and \(\lambda\). Indeed, if \(\mu \ne 0\) is eigenvalue and \(w\) is an
- associated eigenvector, then
+ associated eigenvector, then
\[
\mu \pi(w) = \pi(C w) = C \pi(w) = \lambda \pi(w)
\]
@@ -506,7 +506,7 @@ As promised, the Casimir element can be used to establish\dots
Since \(0 < \dim W, \dim \sfrac{V}{W} < \dim V\) it follows
\(H^1(\mathfrak{g}, W) = H^1(\mathfrak{g}, \sfrac{V}{W}) = 0\). The exactness
- of
+ of
\begin{center}
\begin{tikzcd}
0 \arrow{r} &
@@ -519,7 +519,7 @@ As promised, the Casimir element can be used to establish\dots
done.
\end{proof}
-We are now finally ready to prove comple reducibility once and for all.
+We are now finally ready to prove complete reducibility once and for all.
\begin{proof}[Proof of theorem~\ref{thm:complete-reducibility-holds-for-ss}]
Let
@@ -535,9 +535,9 @@ We are now finally ready to prove comple reducibility once and for all.
We have an exact sequence
\begin{center}
\begin{tikzcd}
- 0 \arrow{r} &
- \operatorname{Hom}(U, W) \arrow{r} &
- \operatorname{Hom}(U, V) \arrow{r}{\pi \circ -} &
+ 0 \arrow{r} &
+ \operatorname{Hom}(U, W) \arrow{r} &
+ \operatorname{Hom}(U, V) \arrow{r}{\pi \circ -} &
\operatorname{Hom}(U, U) \arrow{r} & 0
\end{tikzcd}
\end{center}
@@ -547,15 +547,15 @@ We are now finally ready to prove comple reducibility once and for all.
exact sequence
\begin{center}
\begin{tikzcd}
- 0 \arrow{r} &
- \operatorname{Hom}(U, W)^{\mathfrak{g}} \arrow{r} &
- \operatorname{Hom}(U, V)^{\mathfrak{g}} \arrow{r}{\pi \circ -} &
+ 0 \arrow{r} &
+ \operatorname{Hom}(U, W)^{\mathfrak{g}} \arrow{r} &
+ \operatorname{Hom}(U, V)^{\mathfrak{g}} \arrow{r}{\pi \circ -} &
\operatorname{Hom}(U, U)^{\mathfrak{g}} \arrow{r} &
\hphantom{0}
\\
\hphantom{0} \arrow{r} &
- H^1(\mathfrak{g}, \operatorname{Hom}(U, W)) \arrow{r} &
- H^1(\mathfrak{g}, \operatorname{Hom}(U, V)) \arrow{r} &
+ H^1(\mathfrak{g}, \operatorname{Hom}(U, W)) \arrow{r} &
+ H^1(\mathfrak{g}, \operatorname{Hom}(U, V)) \arrow{r} &
H^1(\mathfrak{g}, \operatorname{Hom}(U, U)) \arrow{r} &
\cdots
\end{tikzcd}
@@ -565,9 +565,9 @@ We are now finally ready to prove comple reducibility once and for all.
have an exact sequence
\begin{center}
\begin{tikzcd}
- 0 \arrow{r} &
- \operatorname{Hom}(U, W)^{\mathfrak{g}} \arrow{r} &
- \operatorname{Hom}(U, V)^{\mathfrak{g}} \arrow{r}{\pi \circ -} &
+ 0 \arrow{r} &
+ \operatorname{Hom}(U, W)^{\mathfrak{g}} \arrow{r} &
+ \operatorname{Hom}(U, V)^{\mathfrak{g}} \arrow{r}{\pi \circ -} &
\operatorname{Hom}(U, U)^{\mathfrak{g}} \arrow{r} &
0
\end{tikzcd}
@@ -587,9 +587,9 @@ We are now finally ready to prove comple reducibility once and for all.
for all \(\mathfrak{g}\)-module \(S\). We thus have a short exact sequence
\begin{center}
\begin{tikzcd}
- 0 \arrow{r} &
- \operatorname{Hom}_{\mathfrak{g}}(U, W) \arrow{r} &
- \operatorname{Hom}_{\mathfrak{g}}(U, V) \arrow{r}{\pi \circ -} &
+ 0 \arrow{r} &
+ \operatorname{Hom}_{\mathfrak{g}}(U, W) \arrow{r} &
+ \operatorname{Hom}_{\mathfrak{g}}(U, V) \arrow{r}{\pi \circ -} &
\operatorname{Hom}_{\mathfrak{g}}(U, U) \arrow{r} &
0
\end{tikzcd}
@@ -599,10 +599,10 @@ We are now finally ready to prove comple reducibility once and for all.
\circ T : U \to U\) is the identity operator. In other words
\begin{center}
\begin{tikzcd}
- 0 \arrow{r} &
- W \arrow{r} &
- V \arrow{r}{\pi} &
- U \arrow{r} \arrow[bend left]{l}{T} &
+ 0 \arrow{r} &
+ W \arrow{r} &
+ V \arrow{r}{\pi} &
+ U \arrow{r} \arrow[bend left]{l}{T} &
0
\end{tikzcd}
\end{center}
@@ -611,16 +611,16 @@ We are now finally ready to prove comple reducibility once and for all.
% TODO: Define what the semisimple form of a complex Lie algebra is in the
% introduction
-We should point out that this last results are just the beggining of a well
+We should point out that this last results are just the beginning of a well
developed cohomology theory. For example, a similar argument using the Casimir
element can be used to show that \(H^i(\mathfrak{g}, V) = 0\) for all
non-trivial finite-dimensional irreducible \(V\), \(i > 0\). For \(K =
\mathbb{C}\), the Lie algebra cohomology groups of an algebra \(\mathfrak{g}\)
-are intemately related with the topological cohomologies -- i.e. singular
+are intimately related with the topological cohomologies -- i.e. singular
cohomology, de Rham cohomology, etc. -- of its simply connected form. We refer
the reader to \cite{cohomologies-lie} for further details.
-Complete reducibility can be generalized to a certain extent for aribitrary --
+Complete reducibility can be generalized to a certain extent for arbitrary --
not necessarily semisimple -- \(\mathfrak{g}\) by considering the exact
sequence
\begin{center}
@@ -1738,14 +1738,14 @@ Hence we are generally interested in maximal Abelian subalgebras \(\mathfrak{h}
Cartan subalgebra of \(\mathfrak{g}\)} if it is Abelian,
\(\operatorname{ad}(H)\) is a diagonal operator for each \(H \in
\mathfrak{h}\) and if \(\mathfrak{h}\) is maximal with respect to the former
- two properties\footnote{More generaly, a Cartan subalgebra of an arbitrary
- Lie algebra \(\mathfrak{g}\) -- not necessarily semisimple -- is definided as
+ two properties\footnote{More generally, a Cartan subalgebra of an arbitrary
+ Lie algebra \(\mathfrak{g}\) -- not necessarily semisimple -- is defined as
a self-normalizing nilpotent subalgebra. This definition turns out to be
equivalent to our characterization whenever \(\mathfrak{g}\) is reductive.}.
\end{definition}
\begin{proposition}
- There exisits a Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{g}\).
+ There exists a Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{g}\).
\end{proposition}
\begin{proof}