Central extension

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

1\rightarrow A\rightarrow E\rightarrow G\rightarrow 1

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

0\rightarrow \mathfrak{a}\rightarrow\mathfrak{e}\rightarrow\mathfrak{g}\rightarrow 0

such that \mathfrak{a} is in the center of \mathfrak{e}.

If the group G is a Lie group, then a central extension of G is a Lie group as well, and the Lie algebra of a central extension of G is a central extension of the Lie algebra of G. In the terminology of theoretical physics, the generators of E not included in G are called central charges. These generators are in the center of the Lie algebra of E; by Noether's theorem, generators of symmetry groups correspond to conserved quantities, referred to as charges.

In Lie group theory, central extensions arise in connection with algebraic topology. Suppose G is a connected Lie group that is not simply connected. Its universal cover G* is again a Lie group, in such a way that the projection

π: G* → G

is a group homomorphism, and surjective. Its kernel is (up to isomorphism) the fundamental group of G; this is known to be abelian (see H-space). This construction gives rise to central extensions.

A special case is that of the metaplectic groups. These stand in relation with the symplectic groups, in the same way that the spinor groups do with the special orthogonal groups. The case of SL2(R) involves a fundamental group that is infinite cyclic. Here the central extension involved is well known in modular form theory, in the case of forms of weight ½. A projective representation that corresponds is the Weil representation, constructed from the Fourier transform, in this case on the real line. Metaplectic groups also occur in quantum mechanics.

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