Lie algebras MOC

Lie algebra extension

Let be Lie algebras. An extension of by is a Lie algebra together with a short exact sequence lie

Hence the “covers” with kernel . Note that is necessarily an ideal, giving the quotient by the First isomorphism theorem. Two extensions of by are said to be equivalent iff there exists an isomorphism such that the following diagram commutes

Classification

Consider an extension .

1. Iff is abelian, one speaks of an abelian extension,
2. Iff is a central ideal, one speaks of a central extension.
3. Iff (Semidirect product of Lie algebras), one speaks of a split extension, equivalently is split epic.
4. Iff (Direct product of Lie algebras), one speaks of a trivial extension.

Proof of equivalence in 3.

proof

See also

• Group extension (the structure of that Zettel deliberately mirrors this one)

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