lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -67,7 +67,7 @@ A popular alternative to definition~\ref{thm:sesimple-algebra} is\dots
   \mathfrak{a}\) then \([f, e] = - h \in \mathfrak{a}\), so again
   \(\mathfrak{a} = \mathfrak{sl}_2(K)\). Similarly, if \(f \in \mathfrak{a}\)
   then \([e, f] = h \in \mathfrak{a}\) and \(\mathfrak{a} =
-  \mathfrak{sl}_2(K)\). More generaly, the Lie algebra \(\mathfrak{sl}_n(K)\)
+  \mathfrak{sl}_2(K)\). More generally, the Lie algebra \(\mathfrak{sl}_n(K)\)
   is simple for each \(n > 0\) -- see the section of \cite[ch. 6]{kirillov} on
   invariant bilinear forms and the semisimplicity of classical Lie algebras.
 \end{example}
@@ -1195,7 +1195,7 @@ direction. For instance, let's say we fix the direction
 and let \(\lambda\) be the weight lying the furthest in this direction.
 
 Its easy to see what we mean intuitively by looking at the previous picture,
-but its precise meaning is still alusive. Formally this means we'll choose a
+but its precise meaning is still allusive. Formally this means we'll choose a
 linear functional \(f : \mathfrak{h}^* \to \QQ\) and pick the weight that
 maximizes \(f\). To avoid any ambiguity we should choose the direction of a
 line irrational with respect to the root lattice \(Q\). For instance if we
@@ -1876,7 +1876,7 @@ As promised, this implies\dots
   \mathfrak{h} = 0\) -- i.e. \(\mathfrak{h} \subset \mathfrak{g}_0\). On the
   other hand, since \(\mathfrak{h}\) is self-normalizing, if \([X, H] = 0 \in
   \mathfrak{h}\) for all \(H \in \mathfrak{h}\) then \(X \in \mathfrak{h}\) --
-  i.e. \(\mathfrak{g}_0 \subset \mathfrak{h}\). So the eigespace decomposition
+  i.e. \(\mathfrak{g}_0 \subset \mathfrak{h}\). So the eigenspace decomposition
   becomes
   \[
     \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_\alpha \mathfrak{g}_\alpha
@@ -1894,7 +1894,7 @@ As promised, this implies\dots
     = 0
   \]
 
-  Hence the non-degeneracy of \(B\) implies the non-degenaracy of its
+  Hence the non-degeneracy of \(B\) implies the non-degeneracy of its
   restriction.
 \end{proof}
 
@@ -1922,7 +1922,7 @@ appendix D of \cite{fulton-harris} and in \cite{humphreys}.
 We begin our analysis by remarking that in both \(\mathfrak{sl}_2(K)\) and
 \(\mathfrak{sl}_3(K)\), the roots were symmetric about the origin and spanned
 all of \(\mathfrak{h}^*\). This turns out to be a general fact, which is a
-consequence of the nondegeneracy of the restriction of the Killing form to the
+consequence of the non-degeneracy of the restriction of the Killing form to the
 Cartan subalgebra.
 
 \begin{proposition}\label{thm:weights-symmetric-span}
@@ -2131,14 +2131,14 @@ theorem~\ref{thm:sl3-existence-uniqueness}. Lo and behold\dots
 \end{definition}
 
 \begin{theorem}\label{thm:dominant-weight-theo}
-  For each dominant integeral \(\lambda \in P\) there exists precisely one
+  For each dominant integral \(\lambda \in P\) there exists precisely one
   irreducible finite-dimensional representation \(V\) of \(\mathfrak{g}\) whose
   highest weight is \(\lambda\).
 \end{theorem}
 
 Fix some dominant integral \(\lambda \in P\). The ``uniqueness'' part of the
 theorem follows at once from the argument used for \(\mathfrak{sl}_3(K)\). The
-``existance'' part is more nuanced. Our first instinct is, of course, to try to
+``existence'' part is more nuanced. Our first instinct is, of course, to try to
 generalize the proof used for \(\mathfrak{sl}_3(K)\). The issue is that our
 proof relied heavily on our knowledge of the roots of \(\mathfrak{sl}_3(K)\).
 Instead, we need a new strategy for the general setting. To that end, we
@@ -2172,7 +2172,7 @@ 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 vetor given by \(v^+ =
+  highest weight of \(M(\lambda)\), with highest weight vector given by \(v^+ =
   1 \otimes v^+ \in M(\lambda)\) as in definition~\ref{def:verma}.
 \end{proposition}
 
@@ -2233,8 +2233,8 @@ Moreover, we find\dots
   finitely many monomials \(F_{\alpha_1}^{k_1} F_{\alpha_2}^{k_2} \cdots
   F_{\alpha_n}^{k_n}\) such that \(\mu = \lambda + k_1 \cdot \alpha_1 + \cdots
   + k_n \cdot \alpha_n\). Since \(M(\lambda)_\mu\) is spanned by the images of
-  \(v^+\) under such monimials, we conclude \(\dim M(\lambda) < \infty\). In
-  particular, there is a single monimials \(F_{\alpha_1}^{k_1}
+  \(v^+\) under such monomials, we conclude \(\dim M(\lambda) < \infty\). In
+  particular, there is a single monomials \(F_{\alpha_1}^{k_1}
   F_{\alpha_2}^{k_2} \cdots F_{\alpha_n}^{k_n}\) such that \(\lambda = \lambda
   + k_1 \cdot \alpha_1 + \cdots + k_n \cdot \alpha_n\) -- which is, of course,
   the monomial where \(k_1 = \cdots = k_n = 0\). Hence \(\dim V_\lambda = 1\).
@@ -2292,9 +2292,9 @@ whose highest weight is \(\lambda\).
   \end{multline*}
 
   By applying the same procedure over and over again we can see that \(v_1 = X
-  v \in V\) for some \(X \in \mathcal{U}(\mathfrak{g})\). Furtheremore, if we
+  v \in V\) for some \(X \in \mathcal{U}(\mathfrak{g})\). Furthermore, if we
   reproduce all this for \(v_2 + \cdots + v_n = v - v_1 \in V\) we get that
-  \(v_2 \in V\). Now by applying the same procesude over and over we find
+  \(v_2 \in V\). Now by applying the same procedure over and over we find
   \(v_1, \ldots, v_n \in V\). Hence
   \[
     V = \bigoplus_\mu V_\mu = \bigoplus_\mu M(\lambda)_\mu \cap V