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 $\mathfrak{𝔤}\mathfrak{𝔩}(n)$ the general linear Lie algebra and $I_n$ the $n\times n$ identity matrix:

• $A_n:=\mathfrak{𝔰}\mathfrak{𝔩}(n+1)=\{x\in \mathfrak{𝔤}\mathfrak{𝔩}(n+1):\text{tr}(x)=0\}$, the special linear Lie algebra;
• $B_n:=\mathfrak{𝔰}\mathfrak{𝔬}(2n+1)=\{x\in \mathfrak{𝔤}\mathfrak{𝔩}(2n+1):x+x^T=0\}$, the odd-dimensional orthogonal Lie algebra;
• $C_n:=\mathfrak{𝔰}\mathfrak{𝔭}(2n)=\{x\in \mathfrak{𝔤}\mathfrak{𝔩}(2n):J_{n}x+x^T{J}_n=0,J_n=\left(\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right)\}$, the symplectic Lie algebra; and
• $D_n:=\mathfrak{𝔰}\mathfrak{𝔬}(2n)=\{x\in \mathfrak{𝔤}\mathfrak{𝔩}(2n):x+x^T=0\}$, the even-dimensional orthogonal Lie algebra.

Except for the low-dimensional cases $D_1=\mathfrak{𝔰}\mathfrak{𝔬}(2)$ and $D_2=\mathfrak{𝔰}\mathfrak{𝔬}(4)$, 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