Example 1

Let $g$ be a Lie algebra over $K$ and $M$ a $g$-module. The $g$-module structure on $M$ and the trivial $g$-module structure on $K$ define a $g$-module structure on the $K$-module $N = \mathscr{L}(M, M; K)$ of bilinear forms on $M$. Then

for all $x \in g, m, m'$ in $M, \beta \in N$. If $\beta$ is a given element of $N$, the set of $x \in g$ such that $x_N \cdot \beta = 0$ is a subalgebra of $g$.

Let $M$ be a $K$-module and $\beta$ a bilinear form on $M$. By the above, the set of $x \in gl(M)$ such that

for all $m \in M$ and $m' \in M$ is a Lie subalgebra of $gl(M)$. Suppose that $K$ is a field, $M$ is finite-dimensional and $\beta$ is non-degenerate. Then every $x \in gl(M)$ admits a left adjoint $x^*$ (relative to $\beta$) which is everywhere defined and the subalgebra in question is the set of $x \in gl(M)$ such that $x^* = -x$. By this process we can construct two important examples of Lie algebras:

(a) Take $M = K^n$ and

We canonically identify $gl(K^n)$ with $M_n(K)$. Then the Lie algebra obtained is the Lie algebra of skew-symmetric matrices. (When $K = \mathbf{R}$, this algebra is the Lie algebra of the orthogonal group $O(n, \mathbf{R})$).

(b) Take $M = K^{2m}$ and

The matrix of $\beta$ with respect to the canonical basis of $K^{2m}$ is the matrix Let $U = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$ be the matrix with respect to the canonical basis of $K^{2m}$ of an element $u$ of $\mathrm{gl}(M)$ ($A, B, C, D$ lying in $\mathbf{M}_m(K)$). By formula (50) of Algebra, Chapter IX, § 1, no. 10, $u^*$ has with respect to the same basis the matrix The condition $u^* = -u$ is therefore equivalent to the conditions When $K = \mathbf{R}$, the Lie algebra obtained is the Lie algebra of the symplectic group $\mathbf{Sp}(2m, \mathbf{R})$.