Example 3

The set $M_n(K)$ of square matrices of order $n$ can be considered as a Lie algebra over K canonically isomorphic to gl(n, K). Let $(E_{ij})$ be the canonical basis of $\mathbf{M}_n(K)$ (Algebra, Chapter II, § 10, no. 3). It follows easily that:

(5)

The Lie subalgebra of $\mathbf{M}_n(K)$ consisting of the triangular matrices (resp. triangular matrices of zero trace, resp. triangular matrices of zero diagonal) is denoted by $t(n, K)$ (resp. st(n, K), resp. n(n, K)) (Algebra, Chapter II, § 10; no. 7).

*Example 3. Let V be an infinitely differentiable real manifold. The differential operators with infinitely differentiable real coefficients constitute an associative algebra over $\mathbf{R}$ and hence, by Example 1, a Lie algebra $\Delta$ over $\mathbf{R}$. The bracket of two infinitely differentiable vector fields on V is an infinitely differentiable vector field and hence the infinitely differentiable vector fields on V constitute a Lie subalgebra $\mathfrak{f}$ of $\Delta$. If V is a real Lie group, the left invariant vector fields constitute a Lie subalgebra $g$ of $\mathfrak{f}$ called the Lie algebra of V. The vector space $g$ is identified with the tangent space to V at $e$ (the identity element of V). Let V' be another real Lie group, $e'$ its identity element and $g'$ its Lie algebra. Every analytic homomorphism of V into V' defines a linear mapping of the tangent space to V at $e$ into the tangent space to V' at $e'$; this mapping is a homomorphism of the Lie algebra $g$ into the Lie algebra $g'$. If V is the linear group of a finite-dimensional real vector space E there exists a canonical isomorphism of $gl(E)$ onto the Lie algebra $g$ of V, under which $g$ is identified with $gl(E)$.*