lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -21,7 +21,7 @@ called \emph{Lie algebras}, and these will be the focus of these notes.
   Given a field \(K\), a Lie algebra over \(K\) is a \(K\)-vector space
   \(\mathfrak{g}\) endowed with an antisymmetric bilinear map \([\, ,] :
   \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}\) -- which we call its
-  \emph{Lie brackets} -- satisfying the Jacobi identity
+  \emph{Lie bracket} -- satisfying the Jacobi identity
   \[
     [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0
   \]
@@ -31,7 +31,7 @@ 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 sense that
+  bracket} in the sense that
   \[
     f([X, Y]) = [f(X), f(Y)]
   \]
@@ -48,7 +48,7 @@ associative algebras.
 
 \begin{example}\label{ex:inclusion-alg-in-lie-alg}
   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 -
+  over \(K\) with the Lie bracket 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
   \(\mathfrak{gl}(V)\). We may also regard the Lie algebra \(\mathfrak{gl}_n(K)
@@ -111,7 +111,7 @@ One specific instance of this last example is\dots
   the left translation by \(g\). The commutator of invariant fields is
   invariant, so the space \(\mathfrak{g} = \operatorname{Lie}(G)\) of all
   invariant vector fields has the structure of a Lie algebra over
-  \(\mathbb{R}\) with brackets given by the usual commutator of fields. Notice
+  \(\mathbb{R}\) with bracket given by the usual commutator of fields. Notice
   that an invariant field \(X\) is completely determined by \(X_1 \in T_1 G\).
   Hence there is a linear isomorphism \(\mathfrak{g} \isoto T_1 G\). In
   particular, \(\mathfrak{g}\) is finite-dimensional.
@@ -134,7 +134,7 @@ this last construction.
   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
-  structure of a Lie algebra over \(K\) with brackets given by the commutator
+  structure of a Lie algebra over \(K\) with bracket given by the commutator
   of derivations. Again, \(\operatorname{Lie}(G)\) is isomorphic to the Zariski
   tangent space \(T_1 G\), which is finite-dimensional.
 \end{example}
@@ -199,7 +199,7 @@ this last construction.
       X
     \right\},
   \]
-  with brackets given by the usual commutator of matrices -- where
+  with bracket given by the usual commutator of matrices -- where
   \(\operatorname{Id}_n\) denotes the \(n \times n\) identity matrices.
 \end{example}
 
@@ -309,7 +309,7 @@ is only natural to define\dots
 \begin{example}
   Let \(\mathfrak{g}_1\) and \(\mathfrak{g}_2\) be a Lie algebras over \(K\).
   Then the space \(\mathfrak{g}_1 \oplus \mathfrak{g}_2\) is a Lie algebra with
-  brackets
+  bracket
   \[
     [X_1 + X_2, Y_1 + Y_2] = [X_1, Y_1] + [X_2, Y_2],
   \]
@@ -368,7 +368,7 @@ There is also a natural analogue of quotients.
   any linear map \(\mathfrak{g} \to \mathfrak{h}\) between Abelian Lie algebras
   \(\mathfrak{g}\) and \(\mathfrak{h}\) is a homomorphism of Lie algebras. In
   particular, any linear isomorphism \(\mathfrak{g} \isoto K^n\) -- where
-  \(K^n\) is endowed with the trivial brackets \([v, w] = 0 \, \forall v, w \in
+  \(K^n\) is endowed with the trivial bracket \([v, w] = 0 \, \forall v, w \in
   K^n\) -- is an isomorphism of Lie algebras for Abelian \(\mathfrak{g}\).
 \end{note}
 
@@ -531,7 +531,7 @@ semisimple and reductive algebras by modding out by certain ideals, known as
 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
+\(A\) to the Lie algebra \(A\) with bracket given by commutators. We can also
 go the other direction by embedding a Lie algebra \(\mathfrak{g}\) in an
 associative algebra, known as \emph{the universal enveloping algebra of
 \(\mathfrak{g}\)}.