Classical Lie algebras

The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types An, Bn, Cn and Dn, where for 𝔤𝔩(n) the general linear Lie algebra and In the n×n identity matrix:

* An:=𝔰𝔩(n+1)={x∈𝔤𝔩(n+1):tr(x)=0}, the special linear Lie algebra;
* Bn:=𝔰𝔬(2n+1)={x∈𝔤𝔩(2n+1):x+xT=0}, the odd-dimensional orthogonal Lie algebra;
* Cn:=𝔰𝔭(2n)={x∈𝔤𝔩(2n):Jnx+xTJn=0,Jn=(0In−In0)}, the symplectic Lie algebra; and
* Dn:=𝔰𝔬(2n)={x∈𝔤𝔩(2n):x+xT=0}, the even-dimensional orthogonal Lie algebra.

Except for the low-dimensional cases D1=𝔰𝔬(2) and D2=𝔰𝔬(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.

## 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. https://www.worldcat.org/oclc/468609320.
2. ↑ Sthanumoorthy, Neelacanta (18 April 2016). Introduction to finite and infinite dimensional lie (super)algebras. Amsterdam Elsevie. ISBN 9780128046753. OCLC 952065417. https://www.worldcat.org/oclc/952065417.

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