The classical Lie algebras are finite-dimensional

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

s 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 In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...

: *

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.

References

{{reflist Algebra

See also

References