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

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

Except for the low-dimensional cases ${\displaystyle D_1=\mathfrak{𝔰}\mathfrak{𝔬}(2)}$ and ${\displaystyle D_2=\mathfrak{𝔰}\mathfrak{𝔬}(4)}$, the classical Lie algebras are simple.^[1][2]

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

• Simple Lie algebra
• Classical group

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