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;
• $C_{n}:={\mathfrak {sp}}(2n)=\{x\in {\mathfrak {gl}}(2n):J_{n}x+x^{T}J_{n}=0,J_{n}={\begin{pmatrix}0&I_{n}\\-I_{n}&0\end{pmatrix}}\}$, 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.

See also

• Simple Lie algebra
• Classical group