Classical Lie algebras

The classical Lie algebras are finite-dimensional <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebras</a> over a field which can be classified into four types 
 
 
 
 
 A
 
 n
 
 
 
 
 {\displaystyle A_{n}}
 
, 
 
 
 
 
 B
 
 n
 
 
 
 
 {\displaystyle B_{n}}
 
, 
 
 
 
 
 C
 
 n
 
 
 
 
 {\displaystyle C_{n}}
 
 and 
 
 
 
 
 D
 
 n
 
 
 
 
 {\displaystyle D_{n}}
 
, where for 
 
 
 
 
 
 g
 l
 
 
 (
 n
 )
 
 
 {\displaystyle {\mathfrak {gl}}(n)}
 
 the <a href="/facts/General_linear_group/B54gNILS">general linear Lie algebra</a> and 
 
 
 
 
 I
 
 n
 
 
 
 
 {\displaystyle I_{n}}
 
 the 
 
 
 
 n
 ×
 n
 
 
 {\displaystyle n\times n}
 
 <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>:

<ul><li>
 
 
 
 
 A
 
 n
 
 
 :=
 
 
 s
 l
 
 
 (
 n
 +
 1
 )
 =
 {
 x
 ∈
 
 
 g
 l
 
 
 (
 n
 +
 1
 )
 :
 
 tr
 
 (
 x
 )
 =
 0
 }
 
 
 {\displaystyle A_{n}:={\mathfrak {sl}}(n+1)=\{x\in {\mathfrak {gl}}(n+1):{\text{tr}}(x)=0\}}
 
, the special linear Lie algebra;</li>
<li>
 
 
 
 
 B
 
 n
 
 
 :=
 
 
 o
 
 
 (
 2
 n
 +
 1
 )
 =
 {
 x
 ∈
 
 
 g
 l
 
 
 (
 2
 n
 +
 1
 )
 :
 x
 +
 
 x
 
 T
 
 
 =
 0
 }
 
 
 {\displaystyle B_{n}:={\mathfrak {o}}(2n+1)=\{x\in {\mathfrak {gl}}(2n+1):x+x^{T}=0\}}
 
, the odd-dimensional orthogonal Lie algebra;</li>
<li>
 
 
 
 
 C
 
 n
 
 
 :=
 
 
 s
 p
 
 
 (
 2
 n
 )
 =
 {
 x
 ∈
 
 
 g
 l
 
 
 (
 2
 n
 )
 :
 
 J
 
 n
 
 
 x
 +
 
 x
 
 T
 
 
 
 J
 
 n
 
 
 =
 0
 ,
 
 J
 
 n
 
 
 =
 
 
 (
 
 
 
 0
 
 
 
 I
 
 n
 
 
 
 
 
 
 −
 
 I
 
 n
 
 
 
 
 0
 
 
 
 )
 
 
 }
 
 
 {\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</li>
<li>
 
 
 
 
 D
 
 n
 
 
 :=
 
 
 o
 
 
 (
 2
 n
 )
 =
 {
 x
 ∈
 
 
 g
 l
 
 
 (
 2
 n
 )
 :
 x
 +
 
 x
 
 T
 
 
 =
 0
 }
 
 
 {\displaystyle D_{n}:={\mathfrak {o}}(2n)=\{x\in {\mathfrak {gl}}(2n):x+x^{T}=0\}}
 
, the even-dimensional orthogonal Lie algebra.</li></ul>
Except for the low-dimensional cases 
 
 
 
 
 D
 
 1
 
 
 =
 
 
 s
 o
 
 
 (
 2
 )
 
 
 {\displaystyle D_{1}={\mathfrak {so}}(2)}
 
 and 
 
 
 
 
 D
 
 2
 
 
 =
 
 
 s
 o
 
 
 (
 4
 )
 
 
 {\displaystyle D_{2}={\mathfrak {so}}(4)}
 
, the classical Lie algebras are <a href="/facts/Simple_Lie_algebra/Hzmh6rYT">simple</a>.
The <a href="/facts/Moyal_bracket/EtUA7phv">Moyal algebra</a> is an infinite-dimensional Lie algebra that contains all classical Lie algebras as subalgebras.

Classical Lie algebras open-in-new

Classical Lie algebras