Classical Lie algebras explained

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 ak{gl}(n) the general linear Lie algebra and I _{n} the n x n identity matrix:A _{n} : = ak{sl}(n+1) = \{ x \in ak{gl}(n+1) : tr (x) = 0 \} , the special linear Lie algebra;B _{n} : = ak{so}(2n+1) = \{ x \in ak{gl}(2n+1) : x + x ^{T} = 0 \} , the odd-dimensional orthogonal Lie algebra;C _{n} : = ak{sp}(2n) = \{ x \in ak{gl}(2n) : J _{nx} + x ^{T} J _{n} = 0 , J _{n} = \begin{pmatrix} 0 & I _{n} \ - I _{n} & 0 \end{pmatrix} \} , the symplectic Lie algebra; andD _{n} : = ak{so}(2n) = \{ x \in ak{gl}(2n) : x + x ^{T} = 0 \} , the even-dimensional orthogonal Lie algebra.

Except for the low-dimensional cases

D _{1} = ak{so}(2) and D _{2} = ak{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

Notes and References

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