The Central extension reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Central extension

In group theory, a central extension of a group G is an exact sequence of groups

such that A is in Z(E), the center of the group E.

Examples of central extensions can be constructed by taking any group G and any abelian group A, and setting E to be A×G. This kind of split example (a split extension in the sense of the extension problem, since G is present as a subgroup of E) isn't of particular interest. More serious examples are found in the theory of projective representations, in cases where the projective representation cannot be lifted to an ordinary linear representation.

Similarly, the central extension of a Lie algebra is an exact sequence

such that is in the center of .

If the group G is a Lie group, then central extension of G is a Lie group as well, and the Lie algebra of the central extension of G is the central extension of the Lie algebra of G.

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