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;
$B_{n}:={\mathfrak {o}}(2n+1)=\{x\in {\mathfrak {gl}}(2n+1):x+x^{T}=0\}$ , 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
$D_{n}:={\mathfrak {o}}(2n)=\{x\in {\mathfrak {gl}}(2n):x+x^{T}=0\}$ , 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.^[1]^[2]

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

References

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

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