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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;
• ${\displaystyle B_{n}:={\mathfrak {so}}(2n+1)=\{x\in {\mathfrak {gl}}(2n+1):x+x^{T}=0\}}$, the odd-dimensional 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
• ${\displaystyle D_{n}:={\mathfrak {so}}(2n)=\{x\in {\mathfrak {gl}}(2n):x+x^{T}=0\}}$, the even-dimensional 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.

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

See also

• Simple Lie algebra
• Classical group

References

1. Antonino, Sciarrino; Paul, Sorba (2000-01-01). Dictionary on Lie algebras and superalgebras. Academic Press. ISBN 9780122653407. OCLC 468609320.
2. Sthanumoorthy, Neelacanta (18 April 2016). Introduction to finite and infinite dimensional lie (super)algebras. Amsterdam Elsevie. ISBN 9780128046753. OCLC 952065417.

• Lie algebras