lie-algebras-and-their-representations

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -614,7 +614,7 @@ a representation}.
 \begin{proposition}
   The Casimir element \(C_V \in \mathcal{U}(\mathfrak{g})\) is central, so that
   \(C_V : W \to W\) is an intertwining operator for any \(\mathfrak{g}\)-module
-  \(W\). Furthermore, \(C_V\) acts in \(V\) as a nonzero scalar operator
+  \(W\). Furthermore, \(C_V\) acts on \(V\) as a nonzero scalar operator
   whenever \(V\) is a non-trivial finite-dimensional irreducible representation
   of \(\mathfrak{g}\).
 \end{proposition}
@@ -650,7 +650,7 @@ a representation}.
   the action of any other element of \(\mathfrak{g}\).
 
   In particular, it follows from Schur's lemma that if \(V\) is
-  finite-dimensional and irreducible then \(C_V\) acts in \(V\) as a scalar
+  finite-dimensional and irreducible then \(C_V\) acts on \(V\) as a scalar
   operator. To see that this scalar is nonzero we compute
   \[
     \operatorname{Tr}(C_V\!\restriction_V)
@@ -711,7 +711,7 @@ establish\dots
 
   Now suppose that \(V\) is non-trivial, so that \(C_V\) acts on \(V\) as
   \(\lambda \operatorname{Id}\) for some \(\lambda \ne 0\). Given an eigenvalue
-  \(\mu \in K\) of the action of \(C_V\) in \(W\), denote by \(W^\mu\) its
+  \(\mu \in K\) of the action of \(C_V\) on \(W\), denote by \(W^\mu\) its
   associated generalized eigenspace. We claim \(W^0\) is the image of the
   inclusion \(K \to W\). Since \(C_V\) acts as zero in \(K\), this image is
   clearly contained in \(W^0\). On the other hand, if \(w \in W\) is such that

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -130,7 +130,7 @@ this last construction.
   \(K\) with rational group operations -- and \(K[G]\) denote the ring of
   regular functions \(G \to K\). We call a derivation \(D : K[G] \to K[G]\)
   left invariant if \(D(g \cdot f) = g \cdot D f\) for all \(g \in G\) and \(f
-  \in K[G]\) -- where the action of \(G\) in \(K[G]\) is given by \((g \cdot
+  \in K[G]\) -- where the action of \(G\) on \(K[G]\) is given by \((g \cdot
   f)(h) = f(g^{-1} h)\). The commutator of left invariant derivations is
   invariant too, so the space \(\operatorname{Lie}(G) =
   \operatorname{Der}(G)^G\) of invariant derivations in \(K[G]\) has the
@@ -819,7 +819,7 @@ say\dots
 \begin{example}
   Given a Lie algebra \(\mathfrak{g}\), the algebra
   \(\mathcal{U}(\mathfrak{g})\) is a \(\mathfrak{g}\)-module, where the action
-  of \(\mathfrak{g}\) in \(\mathcal{U}(\mathfrak{g})\) is given by left
+  of \(\mathfrak{g}\) on \(\mathcal{U}(\mathfrak{g})\) is given by left
   multiplication. This is known as \emph{the regular representation of
   \(\mathfrak{g}\)}.
 \end{example}
@@ -845,7 +845,7 @@ representations.
   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 commutes with
-  the action of \(\mathfrak{g}\) in \(V\) and \(W\), in the sense that the
+  the action of \(\mathfrak{g}\) on \(V\) and \(W\), in the sense that the
   diagram
   \begin{center}
     \begin{tikzcd}

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -121,7 +121,7 @@ A particularly well behaved class of examples are the so called
 \end{definition}
 
 \begin{example}\label{ex:laurent-polynomial-mod}
-  There is a natural action of \(\mathfrak{sl}_2(K)\) in the space \(K[x,
+  There is a natural action of \(\mathfrak{sl}_2(K)\) on the space \(K[x,
   x^{-1}]\) of Laurent polynomials given by the formulas in
   (\ref{eq:laurent-polynomials-cusp-mod}). One can quickly verify \(K[x,
   x^{-1}]_{2 k} = K x^k\) and \(K[x, x^{-1}]_\lambda = 0\) for any \(\lambda
@@ -283,7 +283,7 @@ relationship is well understood. Namely, Fernando himself established\dots
   \mathfrak{p}^\sigma\) and \(V \cong \sigma W\) for some\footnote{Here
   $\mathfrak{p}^\sigma$ denotes the image of $\mathfrak{p}$ under the
   automorphism of $\sigma : \mathfrak{g} \to \mathfrak{g}$ given by the
-  canonical action of $\mathcal{W}$ in $\mathfrak{g}$ and $\sigma W$ is the
+  canonical action of $\mathcal{W}$ on $\mathfrak{g}$ and $\sigma W$ is the
   $\mathfrak{p}$-module given by composing the map $\mathfrak{p}' \to
   \mathfrak{gl}(W)$ with the restriction $\sigma\!\restriction_{\mathfrak{p}} :
   \mathfrak{p} \to \mathfrak{p}'$.} \(\sigma \in \mathcal{W}_V\), where
@@ -344,7 +344,7 @@ 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}.
 
-Our first observation is that \(\mathfrak{sl}_2(K)\) acts in \(K[x, x^{-1}]\)
+Our first observation is that \(\mathfrak{sl}_2(K)\) acts on \(K[x, x^{-1}]\)
 via differential operators. In other words, the action map
 \(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{End}(K[x, x^{-1}])\)
 factors through the inclusion of the algebra \(\operatorname{Diff}(K[x,
@@ -392,7 +392,7 @@ where the maps \(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{Diff}(K[x,
 x^{-1}])\) and \(\operatorname{Diff}(K[x, x^{1}]) \to \operatorname{End}(K[x,
 x^{-1}])\) are the ones from the previous diagram.
 
-Explicitly, we find that the action of \(\mathfrak{sl}_2(K)\) in
+Explicitly, we find that the action of \(\mathfrak{sl}_2(K)\) on
 \(\varphi_\lambda K[x, x^{-1}]\) is given by
 \begin{align*}
   p & \overset{e}{\mapsto}
@@ -692,8 +692,8 @@ coherent family are cuspidal representations?
   centralizer\footnote{This notation comes from the fact that the centralizer of
   $\mathfrak{h}$ in $\mathcal{U}(\mathfrak{g})$ coincides the weight space
   associated with $0 \in \mathfrak{h}^*$ in the adjoint action of
-  $\mathfrak{g}$ in $\mathcal{U}(\mathfrak{g})$ -- not to be confused with the
-  regular action of $\mathfrak{g}$ in $\mathcal{U}(\mathfrak{g})$.} of
+  $\mathfrak{g}$ on $\mathcal{U}(\mathfrak{g})$ -- not to be confused with the
+  regular action of $\mathfrak{g}$ on $\mathcal{U}(\mathfrak{g})$.} of
   \(\mathfrak{h}\) in \(\mathcal{U}(\mathfrak{g})\). Moreover, the multiplicity
   of a given irreducible representation \(W\) of \(\mathfrak{g}\) coincides
   with the multiplicity of \(W_\lambda\) in \(V_\lambda\) as a
@@ -887,14 +887,14 @@ 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
-verified that the action of \(f \in \mathfrak{sl}_2(K)\) in \(K[x, x^{-1}]\) is
+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
 nonzero vector in \(K[x, x^{-1}]_{2 k} = K x^k\) for any \(k \in \mathbb{Z}\)
 by translating between weight spaced using \(f\) and \(f^{-1}\) -- here
 \(f^{-1}\) denote the differential operator \((-
 \sfrac{\mathrm{d}}{\mathrm{d}x} + \sfrac{x^{-1}}{2})^{-1}\), which is the
-inverse of the action of \(f\) in \(K[x, x^{-1}]\).
+inverse of the action of \(f\) on \(K[x, x^{-1}]\).
 \begin{center}
   \begin{tikzcd}
     \cdots     \arrow[bend left=60]{r}{f^{-1}}
@@ -983,7 +983,7 @@ instance\dots
 \end{lemma}
 
 In our case, we are more interested in formally inverting the action of
-\(F_\alpha\) in \(V\) than in inverting \(F_\alpha\) itself. To that end, we
+\(F_\alpha\) on \(V\) than in inverting \(F_\alpha\) itself. To that end, we
 introduce one further construction, known as \emph{the localization of a
 module}.
 
@@ -1022,7 +1022,7 @@ in\dots
 \end{lemma}
 
 Again, in our case we are interested in inverting the actions of the
-\(F_\alpha\) in \(V\). However, for us to be able to translate between all
+\(F_\alpha\) on \(V\). However, for us to be able to translate between all
 weight spaces associated with elements of \(\lambda + Q\), \(\lambda \in
 \operatorname{supp} V\), we only need to invert the \(F_\alpha\)'s for
 \(\alpha\) in some subset of \(\Delta\) which spans all of \(Q = \mathbb{Z}
@@ -1105,7 +1105,7 @@ that \(\Sigma^{-1} V\) contains \(V\) and that its support is an entire
   module and every element of \(\Sigma^{-1} V\) has the form \(s^{-1} v =
   s^{-1} \otimes v\) for \(s \in (F_\beta)_{\beta \in \Sigma}\) and \(v \in
   V\), we can see that \(\Sigma^{-1} V = \bigoplus_\lambda \Sigma^{-1}
-  V_\lambda\). Furthermore, since the action of each \(F_\beta\) in
+  V_\lambda\). Furthermore, since the action of each \(F_\beta\) on
   \(\Sigma^{-1} V\) is bijective and \(\Sigma\) is a basis for \(Q\) we obtain
   \(\operatorname{supp} \Sigma^{-1} V = Q + \operatorname{supp} V\).
 

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -159,7 +159,7 @@ words\dots
 
 \begin{proposition}\label{thm:preservation-jordan-form}
   Let \(V\) be a finite-dimensional representation of \(\mathfrak{g}\) and \(X
-  \in \mathfrak{g}\). Denote by \(X\!\restriction_V\) the action of \(X\) in
+  \in \mathfrak{g}\). Denote by \(X\!\restriction_V\) the action of \(X\) on
   \(V\). Then \(X_s\!\restriction_V = (X\!\restriction)_s\) and
   \(X_n\!\restriction_V = (X\!\restriction)_n\).
 \end{proposition}
@@ -212,7 +212,7 @@ 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
-\(V \to V\) -- which corresponds to the action of \(1 \in \mathfrak{g}\) in
+\(V \to V\) -- which corresponds to the action of \(1 \in \mathfrak{g}\) on
 \(V\). In particular, if we choose an operator \(V \to V\) which is \emph{not}
 diagonalizable we find \(V \ne \bigoplus_{\lambda \in \mathfrak{h}^*}
 V_\lambda\).
@@ -300,7 +300,7 @@ be a general fact, which is a consequence of the non-degeneracy of the
 restriction of the Killing form to the Cartan subalgebra.
 
 \begin{proposition}\label{thm:weights-symmetric-span}
-  The eigenvalues \(\alpha\) of the adjoint action of \(\mathfrak{h}\) in
+  The eigenvalues \(\alpha\) of the adjoint action of \(\mathfrak{h}\) on
   \(\mathfrak{g}\) are symmetrical about the origin -- i.e. \(- \alpha\) is
   also an eigenvalue -- and they span all of \(\mathfrak{h}^*\).
 \end{proposition}
@@ -352,7 +352,7 @@ Furthermore, as in the case of \(\mathfrak{sl}_2(K)\) and
 The proof of the first statement of
 proposition~\ref{thm:weights-symmetric-span} highlights something interesting:
 if we fix some some eigenvalue \(\alpha\) of the adjoint action of
-\(\mathfrak{h}\) in \(\mathfrak{g}\) and a eigenvector \(X \in
+\(\mathfrak{h}\) 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
 \[
@@ -526,7 +526,7 @@ have that \(\lambda + \alpha\) is a weight with \(\lambda \prec \lambda +
   = \bigoplus_{k \in \mathbb{N}} V_{\lambda - k \alpha}
 \]
 and \(\lambda(H_\alpha)\) is the right-most eigenvalue of the action of \(h\)
-in the \(\mathfrak{sl}_2(K)\)-module \(\bigoplus_k V_{\lambda - k \alpha}\).
+on the \(\mathfrak{sl}_2(K)\)-module \(\bigoplus_k V_{\lambda - k \alpha}\).
 
 This has a number of important consequences. For instance\dots
 
@@ -542,7 +542,7 @@ This has a number of important consequences. For instance\dots
   Notice that any \(\mu \in P\) in the line joining \(\lambda\) and
   \(\sigma_\alpha(\lambda)\) has the form \(\mu = \lambda - k \alpha\) for some
   \(k\), so that \(V_\mu\) corresponds the eigenspace associated with the
-  eigenvalue \(\lambda(H_\alpha) - 2k\) of the action of \(h\) in \(\bigoplus_k
+  eigenvalue \(\lambda(H_\alpha) - 2k\) of the action of \(h\) on \(\bigoplus_k
   V_{\lambda - k \alpha}\). If \(\mu\) lies between \(\lambda\) and
   \(\sigma_\alpha(\lambda)\) then \(k\) lies between \(0\) and
   \(\lambda(H_\alpha)\), in which case \(V_\mu \neq 0\) and therefore \(\mu\)
@@ -579,7 +579,7 @@ class of arguments leads us to the conclusion\dots
 
 Aside from showing up the previous theorem, the Weyl group will also play an
 important role in chapter~\ref{ch:mathieu} by virtue of the existence of a
-canonical action of \(\mathcal{W}\) in \(\mathfrak{h}\). By its very nature
+canonical action of \(\mathcal{W}\) on \(\mathfrak{h}\). By its very nature
 \(\mathcal{W}\) acts in \(\mathfrak{h}^*\). If we conjugate the action
 \(\sigma\!\restriction_{\mathfrak{h}^*} : \mathfrak{h}^* \isoto
 \mathfrak{h}^*\) of some \(\sigma \in \mathcal{W}\) by the isomorphism
@@ -601,14 +601,14 @@ prove -- but see \cite[sec.~14.3]{humphreys}.
   e^{\operatorname{ad}(E_\alpha)} e^{- \operatorname{ad}(F_\alpha)}
   e^{\operatorname{ad}(E_\alpha)} : \mathfrak{g} \isoto \mathfrak{g}\). Then
   \(\tilde\sigma_\alpha\) is an automorphism of Lie algebras, and this defines
-  an action of \(\mathcal{W}\) in \(\mathfrak{g}\) which is compatible with the
-  canonical action of \(\mathcal{W}\) in \(\mathfrak{h}\) -- i.e.
+  an action of \(\mathcal{W}\) on \(\mathfrak{g}\) which is compatible with the
+  canonical action of \(\mathcal{W}\) on \(\mathfrak{h}\) -- i.e.
   \(\tilde\sigma\!\restriction_{\mathfrak{h}} =
   \sigma\!\restriction_{\mathfrak{h}}\) for all \(\sigma \in \mathcal{W}\).
 \end{proposition}
 
 \begin{note}
-  Notice that the action of \(\mathcal{W}\) in \(\mathfrak{g}\) from
+  Notice that the action of \(\mathcal{W}\) on \(\mathfrak{g}\) from
   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
@@ -686,7 +686,7 @@ known as \emph{Verma modules}.
 \begin{definition}\label{def:verma}
   The \(\mathfrak{g}\)-module \(M(\lambda) =
   \operatorname{Ind}_{\mathfrak{b}}^{\mathfrak{g}} K v^+\), where the action of
-  \(\mathfrak{b}\) in \(K v^+\) is given by \(H v^+ = \lambda(H) \cdot v^+\)
+  \(\mathfrak{b}\) on \(K v^+\) is given by \(H v^+ = \lambda(H) \cdot v^+\)
   for all \(H \in \mathfrak{h}\) and \(X v^+ = 0\) for \(X \in
   \mathfrak{g}_{\alpha}\), \(\alpha \in \Delta^+\), is called \emph{the Verma
   module of weight \(\lambda\)}
@@ -783,7 +783,7 @@ Moreover, we find\dots
   If \(\mathfrak{g} = \mathfrak{sl}_2(K)\), then we can take \(\mathfrak{h} = K
   h\) and \(\mathfrak{b} = K e \oplus K h\). If \(\lambda \in
   \mathfrak{h}^*\) is the map \(h \mapsto 2\) then \(M(\lambda) =
-  \bigoplus_{k \ge 0} K f^k v^+\), and the action of \(\mathfrak{sl}_2(K)\) in
+  \bigoplus_{k \ge 0} K f^k v^+\), and the action of \(\mathfrak{sl}_2(K)\) on
   \(M(\lambda)\) is given by
   \begin{align*}
     f^{k + 1} v^+ & \overset{e}{\mapsto} (2 - k (k - 1)) f^k v^+ &
@@ -897,7 +897,7 @@ semisimple \(\mathfrak{g}\). Namely\dots
 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 \(\sfrac{M(\lambda)}{N(\lambda)}\) is stable under the natural
-action of the Weyl group \(\mathcal{W}\) in \(\mathfrak{h}^*\). One can then
+action of the Weyl group \(\mathcal{W}\) on \(\mathfrak{h}^*\). One can then
 show that the every weight of \(V\) is conjugate to a single dominant integral
 weight of \(\sfrac{M(\lambda)}{N(\lambda)}\), and that the set of dominant
 integral weights of such irreducible quotient is finite. Since \(W\) is
@@ -937,7 +937,7 @@ non-dominant \(\lambda \in P\). While \(\lambda\) is always a maximal weight of
 \(M(\lambda)\), one can show show that if \(\lambda \in P\) is not dominant
 then \(N(\lambda) = 0\) and \(M(\lambda)\) is irreducible. For instance, if
 \(\mathfrak{g} = \mathfrak{sl}_2(K)\) and \(\lambda = -2\) then the action of
-\(\mathfrak{g}\) in \(M(\lambda)\) is given by
+\(\mathfrak{g}\) on \(M(\lambda)\) is given by
 \begin{center}
   \begin{tikzcd}
     \cdots \arrow[bend left=60]{r}{-20}

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -25,7 +25,7 @@ form a basis of \(\mathfrak{sl}_2(K)\) and satisfy
 \end{align*}
 
 Let \(V\) be a finite-dimensional irreducible \(\mathfrak{sl}_2(K)\)-module. We
-now turn our attention to the action of \(h\) in \(V\), in particular, to the
+now turn our attention to the action of \(h\) on \(V\), in particular, to the
 eigenspace decomposition
 \[
   V = \bigoplus_{\lambda} V_\lambda
@@ -131,7 +131,7 @@ V_\lambda\) and consider the set \(\{v, f v, f^2, v, \ldots\}\).
 
 Proposition~\ref{thm:basis-of-irr-rep} may seem unrelated to our problem at first,
 but its significance lies in the fact that we have just provided a complete
-description of the action of \(\mathfrak{sl}_2(K)\) in \(V\). In other
+description of the action of \(\mathfrak{sl}_2(K)\) on \(V\). In other
 words\dots
 
 \begin{corollary}
@@ -194,7 +194,7 @@ Other important consequences of proposition~\ref{thm:basis-of-irr-rep} are\dots
 \end{proof}
 
 Corollary~\ref{thm:sl2-find-weights} can be used to find the eigenvalues of the
-action of \(h\) in an arbitrary finite-dimensional
+action of \(h\) on an arbitrary finite-dimensional
 \(\mathfrak{sl}_2(K)\)-module. Namely, if \(V\) and \(W\) are representations
 of \(\mathfrak{sl}_2(K)\), \(v \in V_\lambda\) and \(w \in W_\lambda\) then by
 computing
@@ -374,7 +374,7 @@ integral linear combinations of the eigenvalues of the adjoint action of
   \operatorname{ad}(h) f & = -2 f &
   \operatorname{ad}(h) h & = 0,
 \end{align*}
-and the eigenvalues of the action of \(h\) in an irreducible
+and the eigenvalues of the action of \(h\) on an irreducible
 \(\mathfrak{sl}_2(K)\)-modules differ from one another by multiples of \(\pm
 2\).
 
@@ -423,7 +423,7 @@ Visually we may draw
   \end{tikzpicture}
 \end{figure}
 
-If we denote the eigenspace of the adjoint action of \(\mathfrak{h}\) in
+If we denote the eigenspace of the adjoint action of \(\mathfrak{h}\) on
 \(\mathfrak{sl}_3(K)\) associated to \(\alpha\) by
 \(\mathfrak{sl}_3(K)_\alpha\) and fix some \(X \in \mathfrak{sl}_3(K)_\alpha\),
 \(H \in \mathfrak{h}\) and \(v \in V_\lambda\) then
@@ -464,10 +464,10 @@ This is again entirely analogous to the situation we observed in
 \(\mathfrak{sl}_2(K)\). In fact, we may once more conclude\dots
 
 \begin{theorem}\label{thm:sl3-weights-congruent-mod-root}
-  The eigenvalues of the action of \(\mathfrak{h}\) in an irreducible
+  The eigenvalues of the action of \(\mathfrak{h}\) on an irreducible
   \(\mathfrak{sl}_3(K)\)-representation \(V\) differ from one another by
   integral linear combinations of the eigenvalues \(\alpha_i - \alpha_j\) of
-  adjoint action of \(\mathfrak{h}\) in \(\mathfrak{sl}_3(K)\).
+  adjoint action of \(\mathfrak{h}\) on \(\mathfrak{sl}_3(K)\).
 \end{theorem}
 
 \begin{proof}
@@ -481,12 +481,12 @@ This is again entirely analogous to the situation we observed in
 \end{proof}
 
 To avoid confusion we better introduce some notation to differentiate between
-eigenvalues of the action of \(\mathfrak{h}\) in \(V\) and eigenvalues of the
+eigenvalues of the action of \(\mathfrak{h}\) on \(V\) and eigenvalues of the
 adjoint action of \(\mathfrak{h}\).
 
 \begin{definition}
   Given a representation \(V\) of \(\mathfrak{sl}_3(K)\), we'll call the
-  nonzero eigenvalues of the action of \(\mathfrak{h}\) in \(V\) \emph{weights
+  nonzero eigenvalues of the action of \(\mathfrak{h}\) on \(V\) \emph{weights
   of \(V\)}. As you might have guessed, we'll correspondingly refer to
   eigenvectors and eigenspaces of a given weight by \emph{weight vectors} and
   \emph{weight spaces}.
@@ -817,7 +817,7 @@ Notice that \(\lambda([E_{1 2}, E_{2 1}]) \in \mathbb{Z}\) is the right-most
 eigenvalue of the \(\mathfrak{sl}_2(K)\)-module \(\bigoplus_{k \in \mathbb{Z}}
 V_{\lambda + k (\alpha_i - \alpha_j)}\). In particular, \(\lambda([E_{1 2},
 E_{2 1}]\) must be positive. In addition, since the eigenspace of the
-eigenvalue \(\lambda([E_{1 2}, E_{2 1}]) - 2k\) of the action of \(h\) in
+eigenvalue \(\lambda([E_{1 2}, E_{2 1}]) - 2k\) of the action of \(h\) on
 \(\bigoplus_{k \in \mathbb{N}} V_{\lambda + k (\alpha_1 - \alpha_2)}\) is
 \(V_{\lambda + k (\alpha_1 - \alpha_2)}\), the weights of \(V\) appearing the
 string \(\lambda, \lambda + (\alpha_1 - \alpha_2), \ldots, \lambda + k