Example 2

Let $g$ be a Lie algebra over $K$, $M$ a $K$-module and $\rho$ a homomorphism of $g$ into $\mathfrak{gl}(M)$. If $M$ is considered as a commutative Lie algebra, the Lie algebra of derivations of $M$ is $\mathfrak{gl}(M)$. We can therefore form the semi-direct product $h$ of $g$ by $M$ corresponding to $\rho$.

In particular, let $g = \mathfrak{gl}(M)$ and $\rho$ be the identity mapping of $\mathfrak{gl}(M)$. The semi-direct product of $g$ by $M$ is then denoted by $\mathfrak{af}(M)$ (or $\mathfrak{af}(n, K)$ if $M = K^n$). An element of $\mathfrak{af}(M)$ is an ordered pair $(m, u)$, where $m \in M, u \in \mathfrak{gl}(M)$; and the bracket is defined by

*When M is a finite-dimensional vector space over $\mathbf{R}$, $\mathfrak{af}(M)$ is canonically identified with the Lie algebra of the affine group of M.*

Let $t$ be a Lie algebra over K. A linear mapping $\theta$ of $t$ into $\mathfrak{af}(M)$ can be written $x \mapsto ((\zeta(x), \eta(x)))$, where $\zeta$ is a linear mapping of $t$ into $M$ and $\eta$ a linear mapping of $t$ into $gl(M)$. We examine the conditions that $\zeta$ and $\eta$ must satisfy for $\theta$ to be a homomorphism. For $x \in t, y \in t$, we must have

that is

Hence for $\theta$ to be a homomorphism of $t$ into $\mathfrak{af}(M)$, it is necessary and sufficient that $\eta$ be a homomorphism of $t$ into $gl(M)$ and that $\zeta$ satisfy the relation:

(13)

Let N be the K-module $M \times K$. We take $t$ to be the subalgebra of $gl(N)$ consisting of the $w \in gl(N)$ such that $w(N) \subset M$. For all $w \in t$, let $\eta(w) \in gl(M)$ be the restriction of $w$ to $M$ and let $\zeta(w) = w(0, 1) \in M$. For $w_1 \in t, w_2 \in t$,

Hence the mapping $w \mapsto (\zeta(w)), \eta(w))$ is a homomorphism $\theta$ of $t$ into $\mathfrak{af}(M)$. Clearly $\theta$ is bijective. Let $\phi = \theta^{-1}$. If $(m, u) \in \mathfrak{af}(M)$, $\phi(m, u)$ is the element $w$ of $t$ defined by

$\mathfrak{af}(M)$ is often identified with the subalgebra $t$ of $gl(N)$ under the isomorphism $\phi$.

When M is a finite-dimensional vector space over $\mathbf{R}$, the homomorphism $\phi$ of $\mathfrak{af}(M)$ into $gl(N)$ corresponds to a canonical homomorphism $\psi$ of the affine group A of M into the group $\mathbf{GL}(N)$; if $a \in A, \psi(a)$ is the unique element $g$ of $\mathbf{GL}(N)$ such that $g(m, 1) = (a(m), 1)$ for all $m \in M$. This homomorphism is injective and $\psi(A)$ is the set of automorphisms of N which leave invariant all the linear varieties of N parallel to M.