diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -170,38 +170,32 @@ this last construction.
The Lie algebra of the affine algebraic group
\[
\operatorname{Sp}_{2 n}(K)
- = \left \{ M \in \operatorname{SL}_{2 n}(K) :
- M
- \begin{pmatrix}
- 0 & \operatorname{Id}_n \\
- - \operatorname{Id}_n & 0
- \end{pmatrix}
- M^{-1} =
- \begin{pmatrix}
- 0 & \operatorname{Id}_n \\
- - \operatorname{Id}_n & 0
- \end{pmatrix}
- \right \}
+ = \{
+ g \in \operatorname{GL}_{2 n}(K) :
+ \omega(g \cdot v, g \cdot w) = \omega(v, w) \, \forall v, w \in K^{2n}
+ \}
\]
is canonically isomorphic to the Lie algebra
\[
\mathfrak{sp}_{2 n}(K) =
\left\{
- X \in \mathfrak{gl}_{2 n}(K) : X^\top
\begin{pmatrix}
- 0 & \operatorname{Id}_n \\
- - \operatorname{Id}_n & 0
+ X & Y \\
+ Z & -X^\top
\end{pmatrix}
- = -
- \begin{pmatrix}
- 0 & \operatorname{Id}_n \\
- - \operatorname{Id}_n & 0
- \end{pmatrix}
- X
+ : X, Y, Z \in \mathfrak{gl}_n(K), Y = Y^\top, Z = Z^\top
\right\},
\]
with bracket given by the usual commutator of matrices -- where
- \(\operatorname{Id}_n\) denotes the \(n \times n\) identity matrices.
+ \[
+ \omega(
+ (v_1, \ldots, v_n, \dot v_1, \ldots, \dot v_n),
+ (w_1, \ldots, w_n, \dot w_1, \ldots, \dot w_n)
+ )
+ = v_1 \dot w_1 + \cdots + v_n \dot w_n
+ - \dot v_1 w_1 - \cdots - \dot v_n w_n
+ \]
+ is, of course, the standard symplectic form of \(K^{2n}\).
\end{example}
It is important to point out that the construction of the Lie algebra