Classical Lie algebras

The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types ${\displaystyle A_{n}}$, ${\displaystyle B_{n}}$, ${\displaystyle C_{n}}$ and ${\displaystyle D_{n}}$, where for ${\displaystyle {\mathfrak {gl}}(n)}$ the general linear Lie algebra and ${\displaystyle I_{n}}$ the ${\displaystyle n\times n}$ identity matrix:

• ${\displaystyle A_{n}:={\mathfrak {sl}}(n+1)=\{x\in {\mathfrak {gl}}(n+1):{\text{tr}}(x)=0\}}$, the special linear Lie algebra;
• ${\displaystyle B_{n}:={\mathfrak {o}}(2n+1)=\{x\in {\mathfrak {gl}}(2n+1):x+x^{T}=0\}}$, the odd orthogonal Lie algebra;
• ${\displaystyle C_{n}:={\mathfrak {sp}}(2n)=\{x\in {\mathfrak {gl}}(2n):J_{n}x+x^{T}J_{n}=0,J_{n}={\begin{pmatrix}0&I_{n}\\-I_{n}&0\end{pmatrix}}\}}$, the symplectic Lie algebra; and
• ${\displaystyle D_{n}:={\mathfrak {o}}(2n)=\{x\in {\mathfrak {gl}}(2n):x+x^{T}=0\}}$, the even orthogonal Lie algebra.

Except for the low-dimensional cases ${\displaystyle D_{1}={\mathfrak {so}}(2)}$ and ${\displaystyle D_{2}={\mathfrak {so}}(4)}$, the classical Lie algebras are simple.¹²

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