Central Extensions of Lie Algebras

Extension of Lie Algebras

Lie Algebra Extension

An extension of a Lie algebra is another Lie algebra that is equipped with a surjective Lie algebra homomorphism to .
For non-trivial extensions, this homomorphism has a kernel , and one says that is an extension of by .
In particular, if happens to be abelian, i.e., its Lie bracket is trivial, then we call it an abelian extension.

Center of a Lie Algebra

The center of a Lie algebra is an abelian Lie subalgebra , consisting of all elements such that for all .

Central Extension

An abelian extension is called a central extension if the Lie bracket of vanishes as soon as already one of its arguments is in . That is, lies in the center of .

The Projective Representation

Projective Representation

Let be a Lie algebra over a field , be an -vector space. A projective representation loosens the requirement of a Lie algebra homomorphism, it allows to only satisfy

for some (typically central) bilinear map and identity .
Alternatively, it is a map .

For a projective representation, the usual Lie bracket relations are preserved up to a scalar, which obstructs them from being honest (linear) representations in .
To correct this, one can enlarge the Lie algebra by adding a central element governed by a 2-cocycle . This results in a central extension with Lie bracket: We obtained the following theorem:

Theorem

A projective representation of a Lie algebra can be lifted to a (linear) representation of its central extension.

Proof Define by and . Then Therefore, forms a linear representation.