lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
Log
Files
Refs
README
LICENSE

Commit: 8e31e9df7dc7e009f77e89645ba865ff4df6aba9
Parent: 5cd9ad9985341e9e491cc615d8f296831228463a
Author: Pablo <pablo-escobar@riseup.net>
Date: Sat, 17 Dec 2022 21:52:01 +0000

Capitalized named theorems and definitions

Diffstat

5 files changed, 95 insertions, 95 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/complete-reducibility.tex	16	8	8
Modified	sections/introduction.tex	28	14	14
Modified	sections/mathieu.tex	68	34	34
Modified	sections/semisimple-algebras.tex	60	30	30
Modified	sections/sl2-sl3.tex	18	9	9

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -185,7 +185,7 @@ clear things up.
   But \(V\) is indecomposable, so that either \(W = V\) or \(W = 0\). Since
   this holds for all \(W \subset V\), \(V\) is irreducible. For
   \(\textbf{(iii)} \implies \textbf{(iv)}\) it suffices to apply
-  theorem~\ref{thm:krull-schmidt}.
+  Theorem~\ref{thm:krull-schmidt}.
 
   Finally, for \(\textbf{(iv)} \implies \textbf{(i)}\), if we assume
   \(\textbf{(iv)}\) and let \(V\) be a representation of \(\mathfrak{g}\) with
@@ -213,7 +213,7 @@ clear things up.
 The advantage of working with irreducible representations as opposed to
 indecomposable ones is that they are generally much easier to find. The
 relationship between irreducible representations is also well understood. This
-is because of the following result, known as \emph{Schur's lemma}.
+is because of the following result, known as \emph{Schur's Lemma}.
 
 \begin{lemma}[Schur]
   Let \(V\) and \(W\) be irreducible representations of \(\mathfrak{g}\) and
@@ -544,7 +544,7 @@ implies\dots
     \end{tikzcd}
   \end{center}
 
-  By theorem~\ref{thm:ext-exacts-seqs} the sequence on the top is exact. Hence
+  By Theorem~\ref{thm:ext-exacts-seqs} the sequence on the top is exact. Hence
   so is the sequence on the bottom.
 \end{proof}
 
@@ -628,7 +628,7 @@ a representation}.
 \begin{proof}
   To see that \(C_V\) is central fix a basis \(\{X_i\}_i\) for \(\mathfrak{g}\)
   and denote by \(\{X^i\}_i\) its dual basis as in
-  definition~\ref{def:casimir-element}. Let \(X \in \mathfrak{g}\) and denote
+  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.
 
@@ -655,7 +655,7 @@ a representation}.
   for all representations \(W\) of \(\mathfrak{g}\): its action commutes with
   the action of any other element of \(\mathfrak{g}\).
 
-  In particular, it follows from Schur's lemma that if \(V\) is
+  In particular, it follows from Schur's Lemma that if \(V\) is
   finite-dimensional and irreducible then \(C_V\) acts on \(V\) as a scalar
   operator. To see that this scalar is nonzero we compute
   \[
@@ -679,7 +679,7 @@ establish\dots
 
 \begin{proof}
   We begin by the case where \(V\) is irreducible. Due to
-  theorem~\ref{thm:ext-1-classify-short-seqs}, it suffices to show that any
+  Theorem~\ref{thm:ext-1-classify-short-seqs}, it suffices to show that any
   exact sequence of the form
   \begin{equation}\label{eq:exact-seq-h1-vanishes}
     \begin{tikzcd}
@@ -829,7 +829,7 @@ We are now finally ready to prove\dots
     \end{tikzcd}
   \end{center}
   of vector spaces. But \(H^1(\mathfrak{g}, \operatorname{Hom}(U, W))\)
-  vanishes because of proposition~\ref{thm:first-cohomology-vanishes}. Hence we
+  vanishes because of Proposition~\ref{thm:first-cohomology-vanishes}. Hence we
   have an exact sequence
   \begin{center}
     \begin{tikzcd}
@@ -905,7 +905,7 @@ sequence
 This sequence always splits, which implies we can deduce information about the
 representations of \(\mathfrak{g}\) by studying those of its ``semisimple
 part'' \(\mfrac{\mathfrak{g}}{\mathfrak{rad}(\mathfrak{g})}\) -- see
-proposition~\ref{thm:quotients-by-rads}. In practice this translates to\dots
+Proposition~\ref{thm:quotients-by-rads}. In practice this translates to\dots
 
 \begin{theorem}\label{thm:semi-simple-part-decomposition}
   Every irreducible representation of \(\mathfrak{g}\) is the tensor product of

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -204,7 +204,7 @@ this last construction.
 \end{example}
 
 It is important to point out that the construction of the Lie algebra
-\(\mathfrak{g}\) of a Lie group \(G\) in example~\ref{ex:lie-alg-of-lie-grp} is
+\(\mathfrak{g}\) of a Lie group \(G\) in Example~\ref{ex:lie-alg-of-lie-grp} is
 functorial. Specifically, one can show the derivative \(d f_1 : \mathfrak{g}
 \cong T_1 G \to T_1 H \cong \mathfrak{h}\) of a smooth group homomorphism \(f :
 G \to H\) is a homomorphism of Lie algebras, and the chain rule implies \(d (f
@@ -225,11 +225,11 @@ invariants. Even more so\dots
 \end{theorem}
 
 This last theorem is a direct corollary of the so called \emph{first and third
-fundamental Lie theorems}. Lie's first theorem establishes that if \(G\) is a
+fundamental Lie Theorems}. Lie's first Theorem establishes that if \(G\) is a
 simply connected Lie group and \(H\) is a connected Lie group then the induced
 map \(\operatorname{Hom}(G, H) \to \operatorname{Hom}(\mathfrak{g},
 \mathfrak{h})\) is bijective, which implies the Lie functor is fully faithful.
-On the other hand, Lie's third theorem states that every finite-dimensional
+On the other hand, Lie's third Theorem states that every finite-dimensional
 real Lie algebra is the Lie algebra of a simply connected Lie group -- i.e. the
 Lie functor is essentially surjective.
 
@@ -435,7 +435,7 @@ Other interesting classes of Lie algebras are the so called \emph{simple} and
   any ideal \(\mathfrak{a} \normal \mathfrak{sl}_2(K)\) must be stable under
   the operator \(\operatorname{ad}(h) : \mathfrak{sl}_2(K) \to
   \mathfrak{sl}_2(K)\) given by \(\operatorname{ad}(h) X = [h, X]\). But
-  example~\ref{ex:sl2-basis} implies \(\operatorname{ad}(h)\) is
+  Example~\ref{ex:sl2-basis} implies \(\operatorname{ad}(h)\) is
   diagonalizable, with eigenvalues \(0\) and \(\pm 2\). Hence \(\mathfrak{a}\)
   must be spanned by some of the eigenvectors \(e, f, h\) of
   \(\operatorname{ad}(h)\). If \(h \in \mathfrak{a}\), then \([e, h] = - 2 e
@@ -528,7 +528,7 @@ semisimple and reductive algebras by modding out by certain ideals, known as
   \(\mfrac{\mathfrak{g}}{\mathfrak{nil}(\mathfrak{g})}\) is reductive.
 \end{proposition}
 
-We have seen in example~\ref{ex:inclusion-alg-in-lie-alg} that we can pass from
+We have seen in Example~\ref{ex:inclusion-alg-in-lie-alg} that we can pass from
 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
@@ -570,7 +570,7 @@ subalgebra. In practice this means\dots
   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
+  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}
@@ -617,7 +617,7 @@ subalgebra. In practice this means\dots
 
 We should point out this construction is functorial. Indeed, if
 \(f : \mathfrak{g} \to \mathfrak{h}\) is a homomorphism of Lie algebras then
-proposition~\ref{thm:universal-env-uni-prop} implies there is a homomorphism of
+Proposition~\ref{thm:universal-env-uni-prop} implies there is a homomorphism of
 algebras \(\mathcal{U}(f) : \mathcal{U}(\mathfrak{g}) \to
 \mathcal{U}(\mathfrak{h})\) satisfying
 \begin{center}
@@ -635,13 +635,13 @@ It is important to note, however, that \(\mathcal{U} : K\text{-}\mathbf{LieAlg}
 \(K\text{-}\mathbf{Alg} \to K\text{-}\mathbf{LieAlg}\). For instance, if
 \(\mathfrak{g} = K\) is the \(1\)-dimensional Abelian Lie algebra then
 \(\mathcal{U}(\mathfrak{g}) \cong K[x]\), which is infinite-dimensional.
-Nevertheless, proposition~\ref{thm:universal-env-uni-prop} may be restated
+Nevertheless, Proposition~\ref{thm:universal-env-uni-prop} may be restated
 as\dots
 
 \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
+  Example~\ref{ex:inclusion-alg-in-lie-alg}, there is an adjunction
   \(\operatorname{Lie} \vdash \mathcal{U}\).
 \end{corollary}
 
@@ -659,7 +659,7 @@ we find\dots
   basis for \(\mathcal{U}(\mathfrak{g})\).
 \end{theorem}
 
-The Poincaré-Birkoff-Witt theorem is hugely important and will come up again
+The Poincaré-Birkoff-Witt Theorem is hugely important and will come up again
 and again throughout these notes. Among other things, it implies\dots
 
 \begin{corollary}
@@ -731,8 +731,8 @@ As one would expect, the same holds for complex Lie groups and algebraic groups
 too -- if we replace \(C^\infty(G)\) by \(\mathcal{O}(G)\) and \(K[G]\),
 respectively. This last proposition has profound implications. For example, it
 affords us an analytic proof of certain particular cases of the
-Poincaré-Birkoff-Witt theorem. Most surprising of all,
-proposition~\ref{thm:geometric-realization-of-uni-env} implies
+Poincaré-Birkoff-Witt Theorem. Most surprising of all,
+Proposition~\ref{thm:geometric-realization-of-uni-env} implies
 \(\mathcal{U}(\mathfrak{g})\)-modules are \emph{precisely} the same as modules
 over the ring of \(G\)-invariant differential operators -- i.e.
 \(\operatorname{Diff}(G)^G\)-modules. We can thus use
@@ -763,7 +763,7 @@ regarded as a \(K\)-vector space endowed with a ``linear action'' of
 \(\mathcal{U}(\mathfrak{g}) \to \operatorname{End}(V)\) 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
+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
@@ -916,7 +916,7 @@ define\dots
 
 \begin{example}\label{ex:sl2-polynomial-subrep}
   Let \(K[x, y]\) be the \(\mathfrak{sl}_2(K)\)-module as in
-  example~\ref{ex:sl2-polynomial-rep}. Since \(e\), \(f\) and \(h\) all
+  Example~\ref{ex:sl2-polynomial-rep}. Since \(e\), \(f\) and \(h\) all
   preserve the degree of monomials, the space \(K[x, y]^{(n)} = \bigoplus_{k +
   \ell = n} K x^k y^\ell\) of homogeneous polynomials of degree \(n\) is a
   finite-dimensional subrepresentation of \(K[x, y]\).

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -35,7 +35,7 @@ Indeed, our proof of the weight space decomposition in the finite-dimensional
 case relied heavily in the simultaneous diagonalization of commuting operators
 in a finite-dimensional space. Even if we restrict ourselves to irreducible
 modules, there is still a diverse spectrum of counterexamples to
-corollary~\ref{thm:finite-dim-is-weight-mod} in the infinite-dimensional
+Corollary~\ref{thm:finite-dim-is-weight-mod} in the infinite-dimensional
 setting. For instance, any representation \(V\) of \(\mathfrak{g}\) whose
 restriction to \(\mathfrak{h}\) is a free module satisfies \(V_\lambda = 0\)
 for all \(\lambda\) as in the previous example. These are called
@@ -66,9 +66,9 @@ to the case it holds. This brings us to the following definition.
 
 \begin{example}\label{ex:submod-is-weight-mod}
   Proposition~\ref{thm:verma-is-weight-mod} and
-  proposition~\ref{thm:max-verma-submod-is-weight} imply that the Verma module
+  Proposition~\ref{thm:max-verma-submod-is-weight} imply that the Verma module
   \(M(\lambda)\) and its maximal subrepresentation are both weight modules. In
-  fact, the proof of proposition~\ref{thm:max-verma-submod-is-weight} is
+  fact, the proof of Proposition~\ref{thm:max-verma-submod-is-weight} is
   actually a proof of the fact that every subrepresentation \(W \subset V\) of
   a weight module \(V\) is a weight module, and \(W_\lambda = V_\lambda \cap
   W\) for all \(\lambda \in \mathfrak{h}^*\).
@@ -125,7 +125,7 @@ A particularly well behaved class of examples are the so called
   x^{-1}]_{2 k} = K x^k\) and \(K[x, x^{-1}]_\lambda = 0\) for any \(\lambda
   \notin 2 \mathbb{Z}\), so that \(K[x, x^{-1}] = \bigoplus_{k \in \mathbb{Z}}
   K x^k\) is a degree \(1\) admissible weight \(\mathfrak{sl}_2(K)\)-module. It
-  follows from the remark at the end of example~\ref{ex:submod-is-weight-mod}
+  follows from the remark at the end of Example~\ref{ex:submod-is-weight-mod}
   that any nonzero subrepresentation \(W \subset K[x, x^{-1}]\) must contain a
   monomial \(x^k\). But since the operators \(-\frac{\mathrm{d}}{\mathrm{d}x} +
   \frac{x^{-1}}{2}, x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{x}{2} : K[x,
@@ -170,7 +170,7 @@ isn't always the case. Nevertheless, in general we find\dots
   topology of $\mathfrak{h}^*$}.} in \(\mathfrak{h}^*\).
 \end{proposition}
 
-This proof was deemed too technical to be included in here, but see proposition
+This proof was deemed too technical to be included in here, but see Proposition
 3.5 of \cite{mathieu}. Again, there is plenty of examples of completely
 reducible modules which are \emph{not} weight modules. Nevertheless, weight
 modules constitute a large class of representations and understanding them can
@@ -237,8 +237,8 @@ construction very similar to that of Verma modules.
   The irreducible quotient \(L_{\mathfrak{p}}(V)\) is a weight module.
 \end{proposition}
 
-The proof of proposition~\ref{thm:generalized-verma-has-simple-quotient} is
-entirely analogous to that of proposition~\ref{thm:max-verma-submod-is-weight}.
+The proof of Proposition~\ref{thm:generalized-verma-has-simple-quotient} is
+entirely analogous to that of Proposition~\ref{thm:max-verma-submod-is-weight}.
 This leads us to the following definitions.
 
 \begin{definition}
@@ -302,7 +302,7 @@ relationship is well understood. Namely, Fernando himself established\dots
   independent of the choice of basis \(\Sigma\).
 \end{note}
 
-As a first consequence of Fernando's theorem, we provide two alternative
+As a first consequence of Fernando's Theorem, we provide two alternative
 characterizations of cuspidal modules.
 
 \begin{corollary}[Fernando]\label{thm:cuspidal-mod-equivs}
@@ -320,7 +320,7 @@ characterizations of cuspidal modules.
 \end{corollary}
 
 \begin{example}
-  As noted in example~\ref{ex:laurent-polynomial-mod}, the element \(f \in
+  As noted in Example~\ref{ex:laurent-polynomial-mod}, the element \(f \in
   \mathfrak{sl}_2(K)\) acts injectively in the space of Laurent polynomials.
   Hence \(K[x, x^{-1}]\) is a cuspidal representation of
   \(\mathfrak{sl}_2(K)\).
@@ -341,7 +341,7 @@ cuspidal representations? Specifically, given a cuspidal
 representations? To answer this question, we look back at the single example of
 a cuspidal representations we have encountered so far: the
 \(\mathfrak{sl}_2(K)\)-module \(K[x, x^{-1}]\) of Laurent polynomials -- i.e.
-example~\ref{ex:laurent-polynomial-mod}.
+Example~\ref{ex:laurent-polynomial-mod}.
 
 Our first observation is that \(\mathfrak{sl}_2(K)\) acts on \(K[x, x^{-1}]\)
 via differential operators. In other words, the action map
@@ -489,7 +489,7 @@ named \emph{coherent families}.
   1}, \sfrac{\mathrm{d}}{\mathrm{d}x} : \mathcal{M}(\lambda) \to
   \mathcal{M}(\lambda)\) are given by \(x^{\pm 1} x^\mu = x^{\mu \pm 1}\) and
   \(\sfrac{\mathrm{d}}{\mathrm{d}x} x^\mu = \mu x^{\mu - 1}\). It is easy to
-  check \(\mathcal{M}\) from example~\ref{ex:sl-laurent-family} is isomorphic
+  check \(\mathcal{M}\) from Example~\ref{ex:sl-laurent-family} is isomorphic
   to \(\mathcal{M}(\sfrac{1}{2})\) and \((\mathcal{M}(\sfrac{1}{2}))[0] \cong
   K[x, x^{-1}]\).
 \end{example}
@@ -506,7 +506,7 @@ named \emph{coherent families}.
 Our hope is that given an irreducible cuspidal representation \(V\), we can
 somehow fit \(V\) inside of a coherent \(\mathfrak{g}\)-family, such as in the
 case of \(K[x, x^{-1}]\) and \(\mathcal{M}\) from
-example~\ref{ex:sl-laurent-family}. This leads us to the following definition.
+Example~\ref{ex:sl-laurent-family}. This leads us to the following definition.
 
 \begin{definition}
   Given an admissible \(\mathfrak{g}\)-module \(V\) of degree \(d\), a
@@ -582,7 +582,7 @@ to a completely reducible coherent extension of \(V\).
   The uniqueness of \(\mathcal{M}^{\operatorname{ss}}\) should be clear:
   since \(\mathcal{M}^{\operatorname{ss}}\) is completely reducible, so is
   \(\mathcal{M}^{\operatorname{ss}}[\lambda]\). Hence by the Jordan-Hölder
-  theorem
+  Theorem
   \[
     \mathcal{M}^{\operatorname{ss}}[\lambda]
     \cong
@@ -653,8 +653,8 @@ to a completely reducible coherent extension of \(V\).
   In addition, there is no canonical choice of composition series.
 \end{note}
 
-The proof of lemma~\ref{thm:component-coh-family-has-finite-length} is
-extremely technical and may be found in \cite{mathieu} -- see lemma 3.3. As
+The proof of Lemma~\ref{thm:component-coh-family-has-finite-length} is
+extremely technical and may be found in \cite{mathieu} -- see Lemma 3.3. As
 promised, if \(\mathcal{M}\) is a coherent extension of \(V\) then so is
 \(\mathcal{M}^{\operatorname{ss}}\).
 
@@ -718,10 +718,10 @@ submodules of a \emph{nice} coherent family are cuspidal representations?
 
 \begin{proof}
   The fact that \strong{(i)} and \strong{(iii)} are equivalent follows directly
-  from corollary~\ref{thm:cuspidal-mod-equivs}. Likewise, it is clear from the
+  from Corollary~\ref{thm:cuspidal-mod-equivs}. Likewise, it is clear from the
   corollary that \strong{(iii)} implies \strong{(ii)}. All it is left is to show
   \strong{(ii)} implies \strong{(iii)}\footnote{This isn't already clear from
-  corollary~\ref{thm:cuspidal-mod-equivs} because, at first glance,
+  Corollary~\ref{thm:cuspidal-mod-equivs} because, at first glance,
   $\mathcal{M}[\lambda]$ may not be irreducible for some $\lambda$ satisfying
   \strong{(ii)}. We will show this is never the case.}.
 
@@ -729,7 +729,7 @@ submodules of a \emph{nice} coherent family are cuspidal representations?
   \(\mathcal{M}[\lambda]\), for all \(\alpha \in \Delta\). Since
   \(\mathcal{M}[\lambda]\) has finite length, \(\mathcal{M}[\lambda]\) contains
   an infinite-dimensional irreducible \(\mathfrak{g}\)-submodule \(V\).
-  Moreover, again by corollary~\ref{thm:cuspidal-mod-equivs} we conclude \(V\)
+  Moreover, again by Corollary~\ref{thm:cuspidal-mod-equivs} we conclude \(V\)
   is a cuspidal representation, and its degree is bounded by \(d\). We want to
   show \(\mathcal{M}[\lambda] = V\).
 
@@ -744,15 +744,15 @@ submodules of a \emph{nice} coherent family are cuspidal representations?
 
   In particular, \(V_\mu \ne 0\), so \(V_\mu = \mathcal{M}_\mu\). Now given any
   irreducible \(\mathfrak{g}\)-module \(W\), it follows from
-  proposition~\ref{thm:centralizer-multiplicity} that the multiplicity of \(W\)
+  Proposition~\ref{thm:centralizer-multiplicity} that the multiplicity of \(W\)
   in \(\mathcal{M}[\lambda]\) is the same as the multiplicity \(W_\mu\) in
   \(\mathcal{M}_\mu\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module -- which is,
   of course, \(1\) if \(W \cong V\) and \(0\) otherwise. Hence
   \(\mathcal{M}[\lambda] = V\) and \(\mathcal{M}[\lambda]\) is cuspidal.
 \end{proof}
 
-Once more, the proof of proposition~\ref{thm:centralizer-multiplicity} wasn't
-deemed informative enough to be included in here, but see the proof of lemma
+Once more, the proof of Proposition~\ref{thm:centralizer-multiplicity} wasn't
+deemed informative enough to be included in here, but see the proof of Lemma
 2.3 of \cite{mathieu}. To finish the proof, we now show\dots
 
 \begin{lemma}
@@ -815,7 +815,7 @@ deemed informative enough to be included in here, but see the proof of lemma
   that \(W = 0\) or \(W = \mathcal{M}_\lambda\).
 
   On the other hand, if \(\mathcal{M}_\lambda\) is simple then by Burnside's
-  theorem on matrix algebras the map \(\mathcal{U}(\mathfrak{g})_0 \to
+  Theorem on matrix algebras the map \(\mathcal{U}(\mathfrak{g})_0 \to
   \operatorname{End}(\mathcal{M}_\lambda)\) is surjective. Hence the
   commutativity of the previously drawn diagram, as well as the fact that
   \(\operatorname{rank}(\mathcal{U}(\mathfrak{g})_0 \to
@@ -892,7 +892,7 @@ Our first task is constructing \(\mathcal{M}[\lambda]\). The issue here is that
 find \(V \subsetneq \mathcal{M}[\lambda]\). In fact, we may find
 \(\operatorname{supp} V \subsetneq \lambda + Q\).
 
-This wasn't an issue an example~\ref{ex:laurent-polynomial-mod} because we
+This wasn't an issue an Example~\ref{ex:laurent-polynomial-mod} because we
 verified that the action of \(f \in \mathfrak{sl}_2(K)\) on \(K[x, x^{-1}]\) is
 injective. Since all weight spaces of \(K[x, x^{-1}]\) are \(1\)-dimensional,
 this implies the action of \(f\) is actually bijective, so we can obtain a
@@ -1046,18 +1046,18 @@ well-behaved. For example, we can show\dots
 \end{lemma}
 
 \begin{note}
-  The basis \(\Sigma\) in lemma~\ref{thm:nice-basis-for-inversion} may very
+  The basis \(\Sigma\) in Lemma~\ref{thm:nice-basis-for-inversion} may very
   well depend on the representation \(V\)! This is another obstruction to the
   functoriality of our constructions.
 \end{note}
 
-The proof of the previous lemma is quite technical and was deemed too tedious
-to be included in here. See lemma 4.4 of \cite{mathieu} for a full proof. Since
+The proof of the previous Lemma is quite technical and was deemed too tedious
+to be included in here. See Lemma 4.4 of \cite{mathieu} for a full proof. Since
 \(F_\alpha\) is locally \(\operatorname{ad}\)-nilpotent for all \(\alpha \in
 \Delta\), we can see\dots
 
 \begin{corollary}
-  Let \(\Sigma\) be as in lemma~\ref{thm:nice-basis-for-inversion} and
+  Let \(\Sigma\) be as in Lemma~\ref{thm:nice-basis-for-inversion} and
   \((F_\beta)_{\beta \in \Sigma} \subset \mathcal{U}(\mathfrak{g})\) be the
   multiplicative subset generated by the \(F_\beta\)'s. The \(K\)-algebra
   \(\Sigma^{-1} \mathcal{U}(\mathfrak{g}) = (F_\beta)_{\beta \in \Sigma}^{-1}
@@ -1067,7 +1067,7 @@ to be included in here. See lemma 4.4 of \cite{mathieu} for a full proof. Since
 \end{corollary}
 
 From now on let \(\Sigma\) be some fixed basis for \(\Delta\) satisfying the
-hypothesis of lemma~\ref{thm:nice-basis-for-inversion}. We now show that
+hypothesis of Lemma~\ref{thm:nice-basis-for-inversion}. We now show that
 \(\Sigma^{-1} V\) is a weight module whose support is an entire \(Q\)-coset.
 
 \begin{proposition}\label{thm:irr-admissible-is-contained-in-nice-mod}
@@ -1141,7 +1141,7 @@ extension of the \(\mathfrak{sl}_2(K)\)-module \(K[x, x^{-1}]\). Specifically,
 the idea is that if twist \(\Sigma^{-1} V\) by an automorphism which shifts its
 support by some \(\lambda \in \mathfrak{h}^*\), we can construct a coherent
 family by summing these modules over \(\lambda\) as in
-example~\ref{ex:sl-laurent-family}.
+Example~\ref{ex:sl-laurent-family}.
 
 For \(K[x, x^{-1}]\) this was achieved by twisting the
 \(\operatorname{Diff}(K[x, x^{-1}])\)-module \(K[x, x^{-1}]\) by the
@@ -1327,7 +1327,7 @@ It should now be obvious\dots
       (\theta_\lambda(u)\!\restriction_{\Sigma^{-1} V_{\mu - \lambda}})
   \]
   is polynomial in \(\mu\) because of the second item of
-  proposition~\ref{thm:nice-automorphisms-exist}.
+  Proposition~\ref{thm:nice-automorphisms-exist}.
 \end{proof}
 
 Lo and behold\dots
@@ -1346,11 +1346,11 @@ Lo and behold\dots
   \(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\).
 
   To see that \(\operatorname{Ext}(V)\) is simple, recall from
-  corollary~\ref{thm:admissible-is-submod-of-extension} that \(V\) is a
+  Corollary~\ref{thm:admissible-is-submod-of-extension} that \(V\) is a
   subrepresentation of \(\operatorname{Ext}(V)\). Since the degree of \(V\) is
   the same as the degree of \(\operatorname{Ext}(V)\), some of its weight
   spaces have maximal dimension inside of \(\operatorname{Ext}(V)\). In
-  particular, it follows from proposition~\ref{thm:centralizer-multiplicity}
+  particular, it follows from Proposition~\ref{thm:centralizer-multiplicity}
   that \(\operatorname{Ext}(V)_\lambda = V_\lambda\) is a simple
   \(\mathcal{U}(\mathfrak{g})_0\)-module for some \(\lambda \in
   \operatorname{supp} V\).
@@ -1445,7 +1445,7 @@ irreducible representations of an arbitrary reductive algebra it suffices to
 classify those of its simple components. To classify these representations we
 can apply Fernando's results and reduce the problem to constructing the
 cuspidal representation of the simple Lie algebras. But by
-proposition~\ref{thm:only-sl-n-sp-have-cuspidal} only \(\mathfrak{sl}_n(K)\)
+Proposition~\ref{thm:only-sl-n-sp-have-cuspidal} only \(\mathfrak{sl}_n(K)\)
 and \(\mathfrak{sp}_{2 n}(K)\) admit cuspidal representation, so it suffices to
 consider these two cases.
 
@@ -1459,7 +1459,7 @@ complicated on its own, the geometric construction is more concrete than its
 combinatorial counterpart.
 
 This construction also brings us full circle to the beginning of these notes,
-where we saw in proposition~\ref{thm:geometric-realization-of-uni-env} that
+where we saw in Proposition~\ref{thm:geometric-realization-of-uni-env} that
 \(\mathfrak{g}\)-modules may be understood as geometric objects. In fact,
 throughout the previous four chapters we have seen a tremendous number of
 geometrically motivated examples, which further emphasizes the connection

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -73,7 +73,7 @@ Hence we are generally interested in maximal Abelian subalgebras \(\mathfrak{h}
   such that \(\operatorname{ad}(H)\) is a diagonal operator for each \(H \in
   \mathfrak{h}_i\), the subalgebra \(\bigcup_i \mathfrak{h}_i \subset
   \mathfrak{g}\) is Abelian, and its elements also act diagonally in
-  \(\mathfrak{g}\). It then follows from Zorn's lemma that there exists a
+  \(\mathfrak{g}\). It then follows from Zorn's Lemma that there exists a
   subalgebra \(\mathfrak{h}\) which is maximal with respect to both these
   properties, also known as a Cartan subalgebra.
 \end{proof}
@@ -214,9 +214,9 @@ implies\dots
 
 We should point out that this last proof only works for semisimple Lie
 algebras. This is because we rely heavily on
-proposition~\ref{thm:preservation-jordan-form}, as well in the fact that
+Proposition~\ref{thm:preservation-jordan-form}, as well in the fact that
 semisimple Lie algebras are centerless. In fact,
-corollary~\ref{thm:finite-dim-is-weight-mod} fails even for reductive Lie
+Corollary~\ref{thm:finite-dim-is-weight-mod} fails even for reductive Lie
 algebras. For a counterexample, consider the algebra \(\mathfrak{g} = K\): the
 Cartan subalgebra of \(\mathfrak{g}\) is \(\mathfrak{g}\) itself, and a
 \(\mathfrak{g}\)-module is simply a vector space \(V\) endowed with an operator
@@ -225,14 +225,14 @@ Cartan subalgebra of \(\mathfrak{g}\) is \(\mathfrak{g}\) itself, and a
 diagonalizable we find \(V \ne 0 = \bigoplus_{\lambda \in \mathfrak{h}^*}
 V_\lambda\).
 
-However, corollary~\ref{thm:finite-dim-is-weight-mod} does work for reductive
+However, Corollary~\ref{thm:finite-dim-is-weight-mod} does work for reductive
 \(\mathfrak{g}\) if we assume that the representation in question is
 irreducible, since central elements of \(\mathfrak{g}\) act on irreducible
 representations as scalar operators. The hypothesis of finite-dimensionality is
 also of huge importance. In the next chapter we will encounter
 infinite-dimensional \(\mathfrak{g}\)-modules for which the eigenspace
 decomposition \(V = \bigoplus_{\lambda \in \mathfrak{h}^*} V_\lambda\) fails.
-As a first consequence of corollary~\ref{thm:finite-dim-is-weight-mod} we
+As a first consequence of Corollary~\ref{thm:finite-dim-is-weight-mod} we
 show\dots
 
 \begin{corollary}
@@ -358,7 +358,7 @@ Furthermore, as in the case of \(\mathfrak{sl}_2(K)\) and
 \end{proposition}
 
 The proof of the first statement of
-proposition~\ref{thm:weights-symmetric-span} highlights something interesting:
+Proposition~\ref{thm:weights-symmetric-span} highlights something interesting:
 if we fix some eigenvalue \(\alpha\) of the adjoint action of \(\mathfrak{h}\)
 on \(\mathfrak{g}\) and a eigenvector \(X \in \mathfrak{g}_\alpha\), then for
 each \(H \in \mathfrak{h}\) and \(v \in V_\lambda\) we find
@@ -400,7 +400,7 @@ as in the case of \(\mathfrak{sl}_3(K)\) we show\dots
   and the weight spaces in this string match the eigenspaces of \(h\).
 \end{corollary}
 
-The proof of proposition~\ref{thm:distinguished-subalgebra} is very technical
+The proof of Proposition~\ref{thm:distinguished-subalgebra} is very technical
 in nature and we won't include it here, but the idea behind it is simple:
 recall that \(\mathfrak{g}_\alpha\) and \(\mathfrak{g}_{- \alpha}\) are both
 \(1\)-dimensional, so that \(\dim [\mathfrak{g}_\alpha, \mathfrak{g}_{- \alpha}]\)
@@ -417,7 +417,7 @@ satisfies
 
 The elements \(E_\alpha, F_\alpha \in \mathfrak{g}\) are not uniquely
 determined by this condition, but \(H_\alpha\) is. As promised, the second
-statement of corollary~\ref{thm:distinguished-subalg-rep} imposes strong
+statement of Corollary~\ref{thm:distinguished-subalg-rep} imposes strong
 restrictions on the weights of \(V\). Namely, if \(\lambda\) is a weight,
 \(\lambda(H_\alpha)\) is an eigenvalue of \(h\) on some representation of
 \(\mathfrak{sl}_2(K)\), so it must be an integer. In other words\dots
@@ -435,8 +435,8 @@ restrictions on the weights of \(V\). Namely, if \(\lambda\) is a weight,
 \end{proposition}
 
 Proposition~\ref{thm:weights-fit-in-weight-lattice} is clearly analogous to
-corollary~\ref{thm:sl3-weights-fit-in-weight-lattice}. In fact, the weight
-lattice of \(\mathfrak{sl}_3(K)\) -- as in definition~\ref{def:weight-lattice}
+Corollary~\ref{thm:sl3-weights-fit-in-weight-lattice}. In fact, the weight
+lattice of \(\mathfrak{sl}_3(K)\) -- as in Definition~\ref{def:weight-lattice}
 -- is precisely \(\mathbb{Z} \langle \alpha_1, \alpha_2, \alpha_3 \rangle\). To
 proceed further, we would like to take \emph{the highest weight of \(V\)} as in
 section~\ref{sec:sl3-reps}, but the meaning of \emph{highest} is again unclear
@@ -500,7 +500,7 @@ for \(\Delta\)?
   There is a basis \(\Sigma\) for \(\Delta\).
 \end{proposition}
 
-The intuition behind the proof of this proposition is similar to our original
+The intuition behind the proof of this Proposition is similar to our original
 idea of fixing a direction in \(\mathfrak{h}^*\) in the case of
 \(\mathfrak{sl}_3(K)\). Namely, one can show that \(B(\alpha, \beta) \in
 \mathbb{Z}\) for all \(\alpha, \beta \in \Delta\), so that the Killing form
@@ -519,7 +519,7 @@ Fix some basis \(\Sigma\) for \(\Delta\), with corresponding decomposition
 \(V\). We call \(\lambda\) \emph{the highest weight of \(V\)}, and we call any
 nonzero \(v \in V_\lambda\) \emph{a highest weight vector}. The strategy then
 is to describe all weight spaces of \(V\) in terms of \(\lambda\) and \(v\), as
-in theorem~\ref{thm:sl3-irr-weights-class}. Unsurprisingly we do so by
+in Theorem~\ref{thm:sl3-irr-weights-class}. Unsurprisingly we do so by
 reproducing the proof of the case of \(\mathfrak{sl}_3(K)\).
 
 First, we note that any highest weight vector \(v \in V_\lambda\) is
@@ -615,7 +615,7 @@ translates into the following results, which we do not prove -- but see
 
 \begin{note}
   Notice that the action of \(\mathcal{W}\) on \(\mathfrak{g}\) from
-  proposition~\ref{thm:weyl-group-action} is not canonical, since it depends on
+  Proposition~\ref{thm:weyl-group-action} is not canonical, since it depends on
   the choice of \(E_\alpha\) and \(F_\alpha\). Nevertheless, \(\mathfrak{h}\)
   is stable under the action of \(\mathcal{W}\) -- i.e. \(\mathcal{W} \cdot
   \mathfrak{h} \subset \mathfrak{h}\) -- and the restriction of this action to
@@ -629,8 +629,8 @@ century to classify all finite-dimensional simple complex Lie algebras in terms
 of Dynking diagrams. This and much more can be found in \cite[III]{humphreys}
 and \cite[ch.~21]{fulton-harris}. Having found all of the weights of \(V\), the
 only thing we are missing for a complete classification is an existence and
-uniqueness theorem analogous to theorem~\ref{thm:sl2-exist-unique} and
-theorem~\ref{thm:sl3-existence-uniqueness}. This will be the focus of the next
+uniqueness theorem analogous to Theorem~\ref{thm:sl2-exist-unique} and
+Theorem~\ref{thm:sl3-existence-uniqueness}. This will be the focus of the next
 section.
 
 \section{Verma Modules \& the Highest Weight Theorem}
@@ -655,7 +655,7 @@ words\dots
   highest weight is \(\lambda\).
 \end{theorem}
 
-This is known as \emph{the highest weight theorem}, and its proof is the focus
+This is known as \emph{the Highest Weight Theorem}, and its proof is the focus
 of this section. The ``uniqueness'' part of the theorem follows at once from
 the argument used for \(\mathfrak{sl}_3(K)\). Namely\dots
 
@@ -666,8 +666,8 @@ the argument used for \(\mathfrak{sl}_3(K)\). Namely\dots
   irreducible.
 \end{proposition}
 
-The proof of proposition~\ref{thm:irr-subrep-generated-by-vec} is very similar
-in spirit to that of proposition~\ref{thm:sl3-positive-roots-span-all-irr-rep}:
+The proof of Proposition~\ref{thm:irr-subrep-generated-by-vec} is very similar
+in spirit to that of Proposition~\ref{thm:sl3-positive-roots-span-all-irr-rep}:
 we use the commutator relations of \(\mathfrak{g}\) to inductively show that
 the subspace spanned by the images of a highest weight vector under successive
 applications of negative root vectors is invariant under the action of
@@ -684,7 +684,7 @@ Of course, what we are really interested in is\dots
   Let \(v \in V\) and \(w \in W\) be highest weight vectors and \(U =
   \mathcal{U}(\mathfrak{g}) \cdot v + w \subset V \oplus W\). It is clear that
   \(v + w\) is a highest weight vector of \(V \oplus W\). Hence by
-  proposition~\ref{thm:irr-subrep-generated-by-vec} \(U\) is irreducible. The
+  Proposition~\ref{thm:irr-subrep-generated-by-vec} \(U\) is irreducible. The
   projections \(\pi_1 : U \to V\) and \(\pi_2 : U \to W\) are thus nonzero
   intertwiners between irreducible representations of \(\mathfrak{g}\) and are
   therefore isomorphisms. Hence \(V \cong U \cong W\).
@@ -712,7 +712,7 @@ Verma modules are \emph{highly infinite-dimensional}. Indeed, the dimension of
 in \(\mathcal{U}(\mathfrak{g})\), which is always infinite. Nevertheless,
 \(M(\lambda)\) turns out to be quite well behaved. For instance, by
 construction \(M(\lambda) = \mathcal{U}(\mathfrak{g}) \cdot v^+\) -- where
-\(v^+ = 1 \otimes v^+ \in M(\lambda)\) is as in definition~\ref{def:verma}.
+\(v^+ = 1 \otimes v^+ \in M(\lambda)\) is as in Definition~\ref{def:verma}.
 Moreover, we find\dots
 
 \begin{proposition}\label{thm:verma-is-weight-mod}
@@ -723,14 +723,14 @@ Moreover, we find\dots
   holds. Furthermore, \(\dim M(\lambda)_\mu < \infty\) for all \(\mu \in
   \mathfrak{h}^*\) and \(\dim M(\lambda) = 1\). Finally, \(\lambda\) is the
   highest weight of \(M(\lambda)\), with highest weight vector given by \(v^+ =
-  1 \otimes v^+ \in M(\lambda)\) as in definition~\ref{def:verma}.
+  1 \otimes v^+ \in M(\lambda)\) as in Definition~\ref{def:verma}.
 \end{proposition}
 
 \begin{proof}
-  The Poincaré-Birkhoff-Witt theorem implies that \(M(\lambda)\) is spanned by
+  The Poincaré-Birkhoff-Witt Theorem implies that \(M(\lambda)\) is spanned by
   the vectors \(F_{\alpha_1} F_{\alpha_2} \cdots F_{\alpha_n} v^+\) for
   \(\alpha_i \in \Delta^-\) and \(F_{\alpha_i} \in \mathfrak{g}_{\alpha_i}\) as
-  in the proof of proposition~\ref{thm:distinguished-subalgebra}. But
+  in the proof of Proposition~\ref{thm:distinguished-subalgebra}. But
   \[
     \begin{split}
       H F_{\alpha_1} F_{\alpha_2} \cdots F_{\alpha_n} v^+
@@ -765,7 +765,7 @@ Moreover, we find\dots
   This already gives us that the weights of \(M(\lambda)\) are bounded by
   \(\lambda\). To see that \(\lambda\) is indeed a weight, we show that
   \(v^+\) is nonzero weight vector. Clearly \(v^+ \in V_\lambda\). The
-  Poincaré-Birkhoff-Witt theorem implies \(\mathcal{U}(\mathfrak{g})\) is a
+  Poincaré-Birkhoff-Witt Theorem implies \(\mathcal{U}(\mathfrak{g})\) is a
   free \(\mathcal{U}(\mathfrak{b})\)-module, so that
   \[
     M(\lambda)
@@ -826,7 +826,7 @@ Moreover, we find\dots
 
 What is interesting to us about all this is that we have just constructed a
 \(\mathfrak{g}\)-module whose highest weight is \(\lambda\). This is not a
-proof of theorem~\ref{thm:dominant-weight-theo}, however, since \(M(\lambda)\)
+proof of Theorem~\ref{thm:dominant-weight-theo}, however, since \(M(\lambda)\)
 is neither irreducible nor finite-dimensional. Nevertheless, we can use
 \(M(\lambda)\) to construct an irreducible representation of \(\mathfrak{g}\)
 whose highest weight is \(\lambda\).
@@ -840,7 +840,7 @@ whose highest weight is \(\lambda\).
 
 \begin{proof}
   Let \(V \subset M(\lambda)\) be a subrepresentation and take any nonzero \(v
-  \in V\). Because of proposition~\ref{thm:verma-is-weight-mod}, we know there
+  \in V\). Because of Proposition~\ref{thm:verma-is-weight-mod}, we know there
   are \(\mu_1, \ldots, \mu_n \in \mathfrak{h}^*\) and nonzero \(v_i \in
   M(\lambda)_{\mu_i}\) such that \(v = v_1 + \cdots + v_n\). We want to show
   \(v_i \in V\) for all \(i\).
@@ -892,7 +892,7 @@ whose highest weight is \(\lambda\).
 
 \begin{example}\label{ex:sl2-verma-quotient}
   If \(\mathfrak{g} = \mathfrak{sl}_2(K)\) and \(\lambda : h \mapsto 2\), we
-  can see from example~\ref{ex:sl2-verma} that \(N(\lambda) = \bigoplus_{k \ge
+  can see from Example~\ref{ex:sl2-verma} that \(N(\lambda) = \bigoplus_{k \ge
   3} K f^k v^+\), so that \(L(\lambda)\) is the \(3\)-dimensional irreducible
   representation of \(\mathfrak{sl}_2(K)\) -- i.e. the finite-dimensional
   irreducible representation with highest weight \(\lambda\) constructed in
@@ -902,7 +902,7 @@ whose highest weight is \(\lambda\).
 This last example is particularly interesting to us, since it indicates that
 the finite-dimensional irreducible representations of \(\mathfrak{sl}_2(K)\) as
 quotients of Verma modules. This is because the quotient
-\(\sfrac{M(\lambda)}{N(\lambda)}\) in example~\ref{ex:sl2-verma-quotient}
+\(\sfrac{M(\lambda)}{N(\lambda)}\) in Example~\ref{ex:sl2-verma-quotient}
 is finite-dimensional. As it turns out, this is not a coincidence.
 
 \begin{proposition}\label{thm:verma-is-finite-dim}
@@ -910,7 +910,7 @@ is finite-dimensional. As it turns out, this is not a coincidence.
   the unique irreducible quotient of \(M(\lambda)\) is finite-dimensional.
 \end{proposition}
 
-The proof of proposition~\ref{thm:verma-is-finite-dim} is very technical and we
+The proof of Proposition~\ref{thm:verma-is-finite-dim} is very technical and we
 won't include it here, but the idea behind it is to show that the set of
 weights of \(L(\lambda)\) is stable under the natural action of the Weyl group
 \(\mathcal{W}\) on \(\mathfrak{h}^*\). One can then show that the every weight
@@ -949,7 +949,7 @@ are really interested in is\dots
   weight \(\mu\) of \(M(\lambda)\) with \(\mu \succ \lambda\).
 \end{proof}
 
-We should point out that proposition~\ref{thm:verma-is-finite-dim} fails for
+We should point out that Proposition~\ref{thm:verma-is-finite-dim} fails for
 non-dominant \(\lambda \in P\). While \(\lambda\) is always a maximal weight of
 \(M(\lambda)\), one can show that if \(\lambda \in P\) is not dominant then
 \(N(\lambda) = 0\) and \(M(\lambda)\) is irreducible. For instance, if

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -134,7 +134,7 @@ V_\lambda\) and consider the set \(\{v, f v, f^2 v, \ldots\}\).
   which which is the first formula of (\ref{eq:irr-rep-of-sl2}).
 \end{proof}
 
-The significance of proposition~\ref{thm:basis-of-irr-rep} should be
+The significance of Proposition~\ref{thm:basis-of-irr-rep} should be
 self-evident: we have just provided a complete description of the action of
 \(\mathfrak{sl}_2(K)\) on \(V\). In particular, this goes to show\dots
 
@@ -216,7 +216,7 @@ Surprisingly, we have already encountered such a \(V\).
 \begin{proof}
   Let \(V = K[x, y]^{(n)}\) be the \(\mathfrak{sl}_2(K)\)-module of homogeneous
   polynomials of degree \(n\) in two variables, as in
-  example~\ref{ex:sl2-polynomial-subrep}. A simple calculation shows \(V_{n - 2
+  Example~\ref{ex:sl2-polynomial-subrep}. A simple calculation shows \(V_{n - 2
   k} = K x^{n - k} y^k\) for \(k = 0, \ldots, n\) and \(V_\lambda = 0\)
   otherwise. In particular, the right-most eigenvalue of \(V\) is \(n\).
   Alternatively, one can readily check that if \(K^2\) is the natural
@@ -286,7 +286,7 @@ chapter, but for now we note that perhaps the most fundamental property of
 annihilated by \(e\)} -- that being the generator of the right-most eigenspace
 of \(h\). This was instrumental to our explicit description of the irreducible
 representations of \(\mathfrak{sl}_2(K)\) culminating in
-theorem~\ref{thm:sl2-exist-unique}.
+Theorem~\ref{thm:sl2-exist-unique}.
 
 Our first task is to find some analogue of \(h\) in \(\mathfrak{sl}_3(K)\), but
 it is still unclear what exactly we are looking for. We could say we are looking
@@ -485,7 +485,7 @@ arguments from the previous section in the context of \(\mathfrak{sl}_3(K)\).
 However, it is more profitable to use our knowledge of the representations of
 \(\mathfrak{sl}_2(K)\) instead. Notice that the canonical inclusion
 \(\mathfrak{gl}_2(K) \to \mathfrak{gl}_3(K)\) -- as described in
-example~\ref{ex:gln-inclusions} -- restricts to an injective homomorphism
+Example~\ref{ex:gln-inclusions} -- restricts to an injective homomorphism
 \(\mathfrak{sl}_2(K) \to \mathfrak{sl}_3(K)\). In other words,
 \(\mathfrak{sl}_2(K)\) is isomorphic to the image \(\mathfrak{s}_{1 2} = K
 \langle E_{1 2}, E_{2 1}, [E_{1 2}, E_{2 1}] \rangle \subset
@@ -952,7 +952,7 @@ This final picture is known as \emph{the weight diagram of \(V\)}. Finally\dots
 
 Having found all of the weights of \(V\), the only thing we are missing is an
 existence and uniqueness theorem analogous to
-theorem~\ref{thm:sl2-exist-unique}. In other words, our next goal is
+Theorem~\ref{thm:sl2-exist-unique}. In other words, our next goal is
 establishing\dots
 
 \begin{theorem}\label{thm:sl3-existence-uniqueness}
@@ -963,7 +963,7 @@ establishing\dots
 
 To proceed further we once again refer to the approach we employed in the case
 of \(\mathfrak{sl}_2(K)\): next we showed in
-proposition~\ref{thm:basis-of-irr-rep} that any irreducible representation of
+Proposition~\ref{thm:basis-of-irr-rep} that any irreducible representation of
 \(\mathfrak{sl}_2(K)\) is spanned by the images of its highest weight vector
 under \(f\). A more abstract way of putting it is to say that an irreducible
 representation \(V\) of \(\mathfrak{sl}_2(K)\) is spanned by the images of its
@@ -1102,7 +1102,7 @@ The same argument also goes to show\dots
 This is very interesting to us since it implies that finding \emph{any}
 finite-dimensional representation whose highest weight is \(n \alpha_1 - m
 \alpha_2\) is enough for establishing the ``existence'' part of
-theorem~\ref{thm:sl3-existence-uniqueness}. Moreover, constructing such
+Theorem~\ref{thm:sl3-existence-uniqueness}. Moreover, constructing such
 representation turns out to be quite simple.
 
 \begin{proof}[Proof of existence]
@@ -1173,13 +1173,13 @@ representation turns out to be quite simple.
   highest weight vector \(e_1^n \otimes f_3^m\).
 \end{proof}
 
-The ``uniqueness'' part of theorem~\ref{thm:sl3-existence-uniqueness} is even
+The ``uniqueness'' part of Theorem~\ref{thm:sl3-existence-uniqueness} is even
 simpler than that.
 
 \begin{proof}[Proof of uniqueness]
   Let \(V\) and \(W\) be two irreducible representations of
   \(\mathfrak{sl}_3(K)\) with highest weight \(\lambda\). By
-  theorem~\ref{thm:sl3-irr-weights-class}, the weights of \(V\) are precisely
+  Theorem~\ref{thm:sl3-irr-weights-class}, the weights of \(V\) are precisely
   the same as those of \(W\).
 
   Now by computing