Editing Group extension (section)

===Lie groups===

In [[Lie group]] theory, central extensions arise in connection with [[algebraic topology]]. Roughly speaking, central extensions of Lie groups by discrete groups are the same as [[covering group]]s. More precisely, a [[connected space|connected]] [[covering space]] {{math|''G''<sup>∗</sup>}} of a connected Lie group {{math|''G''}} is naturally a central extension of {{math|''G''}}, in such a way that the projection

:<math>\pi\colon G^* \to G</math>

is a group homomorphism, and surjective. (The group structure on {{math|''G''<sup>∗</sup>}} depends on the choice of an identity element mapping to the identity in {{math|''G''}}.) For example, when {{math|''G''<sup>∗</sup>}} is the [[universal cover]] of {{math|''G''}}, the kernel of ''π'' is the [[fundamental group]] of {{math|''G''}}, which is known to be abelian (see [[H-space]]). Conversely, given a Lie group {{math|''G''}} and a discrete central subgroup {{math|''Z''}}, the quotient {{math|''G''/''Z''}} is a Lie group and {{math|''G''}} is a covering space of it.

More generally, when the groups {{math|''A''}}, {{math|''E''}} and {{math|''G''}} occurring in a central extension are Lie groups, and the maps between them are homomorphisms of Lie groups, then if the Lie algebra of {{math|''G''}} is {{math|'''g'''}}, that of {{math|''A''}} is {{math|'''a'''}}, and that of {{math|''E''}} is {{math|'''e'''}}, then {{math|'''e'''}} is a [[Lie algebra extension#Central extension|central Lie algebra extension]] of {{math|'''g'''}} by {{math|'''a'''}}.   In the terminology of [[theoretical physics]], generators of {{math|'''a'''}} are called [[central charge]]s.  These generators are in the center of {{math|'''e'''}}; by [[Noether's theorem]], generators of symmetry groups correspond to conserved quantities, referred to as [[charge (physics)|charges]].

The basic examples of central extensions as covering groups are:
* the [[spin group]]s, which double cover the [[special orthogonal group]]s, which (in even dimension) doubly cover the [[projective orthogonal group]].
* the [[metaplectic group]]s, which double cover the [[symplectic group]]s.
The case of {{math|[[SL2(R)|SL<sub>2</sub>('''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 {{math|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]].