The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types , , and , where for the general linear Lie algebra and the identity matrix:

  • , the special linear Lie algebra;
  • , the odd-dimensional orthogonal Lie algebra;
  • , the symplectic Lie algebra; and
  • , the even-dimensional orthogonal Lie algebra.

Except for the low-dimensional cases and , 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

References

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