Classical Lie algebras

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

• [math]\displaystyle{ A_n := \mathfrak{sl}(n+1) = \{ x \in \mathfrak{gl}(n+1) : \text{tr}(x) = 0 \} }[/math], the special linear Lie algebra;
• [math]\displaystyle{ B_n := \mathfrak{so}(2n+1) = \{ x \in \mathfrak{gl}(2n+1) : x + x^{T} = 0 \} }[/math], the odd-dimensional orthogonal Lie algebra;
• [math]\displaystyle{ 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} \} }[/math], the symplectic Lie algebra; and
• [math]\displaystyle{ D_n := \mathfrak{so}(2n) = \{ x \in \mathfrak{gl}(2n) : x + x^{T} = 0 \} }[/math], the even-dimensional orthogonal Lie algebra.

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

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

See also

• Simple Lie algebra
• Classical group