Classical Lie Algebra

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 $\mathfrak(n)$ the general linear Lie algebra and $I_n$ the $n \times n$ identity matrix:

* $A_n := \mathfrak(n+1) = \$, the ''special linear Lie algebra'';
* $B_n := \mathfrak(2n+1) = \$, the ''odd-dimensional orthogonal Lie algebra'';
* $C_n := \mathfrak(2n) = \$, the ''symplectic Lie algebra''; and
* $D_n := \mathfrak(2n) = \$, the ''even-dimensional orthogonal Lie algebra''.

Except for the low-dimensional cases $D_1 = \mathfrak(2)$ and $D_2 = \mathfrak(4)$, 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

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Algebra