Group extension

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In mathematics, for Q a group, G is an extension of Q if there is an exact sequence

$1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1. \,\!$

This situation is sometimes described by saying that G is an extension of Q by N, or an extension of N by Q, depending on the author.

In other words: G is a group, N is a normal subgroup of G and the quotient group G/N is isomorphic to group Q. In contexts where the extension nomenclature is used, Q and N are known and the properties of G are to be determined.

One extension, the direct product, is immediately obvious. If one requires G and Q to be abelian groups, then the set of isomorphism classes of extensions of Q by a given (abelian) group N is in fact a group, which is isomorphic to $\operatorname{Ext}^1_{\mathbb Z}(Q,N)$ ; cf. the Ext functor. Several other general classes of extensions are known but no theory exists which treats all the possible extensions at one time. Group extension is usually described as a hard problem; it is termed the extension problem.

[edit] Central extension

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. The set of isomorphism classes of central extensions of G by A is in one-to-one correspondence with the cohomology group H²(G,A), where the action of G on A is trivial.

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.

In the case of finite perfect groups, there is a universal perfect central extension.

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 $\mathfrak{e}$ .

[edit] Lie groups

See also: Covering group

Central extensions of Lie groups are identical to covering spaces of Lie groups.

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.

Conversely, given a Lie group G, with non-trivial center Z, the quotient G/Z is a Lie group and G is a central extension of it.

The basic examples are:

the spin groups, which double cover the special orthogonal groups, which (in even dimension) double-cover the projective orthogonal group.
the metaplectic groups, which double cover the symplectic 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.