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
\(\mathfrak{g}\)}.