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Classical Lie algebras

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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 orthogonal Lie algebra;
  • , the symplectic Lie algebra; and
  • , the even 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.

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