Classical Lie algebras

The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types $$ A_n $$, $$ B_n $$, $$ C_n $$ and $$ D_n $$, where for $$\mathfrak{gl}(n)$$ the general linear Lie algebra and $$ I_n $$ the $$ n \times n  $$ identity matrix:


 * $$ A_n := \mathfrak{sl}(n+1) = \{ x \in \mathfrak{gl}(n+1) : \text{tr}(x) = 0 \} $$, the special linear Lie algebra;
 * $$ B_n := \mathfrak{so}(2n+1) = \{ x \in \mathfrak{gl}(2n+1) : x + x^{T} = 0 \} $$, the odd-dimensional orthogonal Lie algebra;
 * $$ C_n := \mathfrak{sp}(2n) = \{ x \in \mathfrak{gl}(2n) : J_nx + x^{T}J_n = 0, J_n = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix} \} $$, the symplectic Lie algebra; and
 * $$ D_n := \mathfrak{so}(2n) = \{ x \in \mathfrak{gl}(2n) : x + x^{T} = 0 \} $$, the even-dimensional orthogonal Lie algebra.

Except for the low-dimensional cases $$ D_1 = \mathfrak{so}(2) $$ and $$ D_2 = \mathfrak{so}(4) $$, the classical Lie algebras are simple.

The Moyal algebra is an infinite-dimensional Lie algebra that contains all classical Lie algebras as subalgebras.