Why is it interesting?

Central extensions are a standard trick to convert projective representations of some group into true representations of another group.

This is necessary, because when we only consider the "naive" normal representations of a group like the Lorentz group, we miss an important representation (the spin $\frac{1}{2}$) representation). Thus, we can either use a less restrictive definition of a representation, i.e. use projective representations instead of true representations, or we could simply work with true representations of the central extension of the given group.

For example, the projective representations of $SO(3,1)$ correspond to regular representations of $SL(2,\mathbb{C})$.

"Central extensions play an important role in quantum mechanics: one of the earlier encounters is by means of Wigner’s theorem which states that a symmetry of a quantum mechanical system determines a (anti-) unitary transformation of the Hilbert space, which is unique up to a phase factor $e^{iϑ}$. As an immediate consequence of this phase factor, one deduces that given a quantum mechanical symmetry group $G$ there exists an extension $G_0$ of $G$ by $U(1)$ (the phase factors) which acts as a group of unitary transformations on the Hilbert space. In most cases physicists have been succesful in hiding these central extensions by using larger symmetry groups" http://math.univ-lille1.fr/~gmt/PaperFolder/CentralExtensions.pdf

Layman

Student

• See page 178 in Moonshine beyond the Monster by Terry Gannon

Researcher

Examples

Galilean group -> Bargmann group

The classical Galilean group needs to be extended by the introduction of a central charge, called mass, and this yields the Bargmann group. (This is shown very nicely in QUANTIZATION ON A LIE GROUP: HIGHER-ORDER POLARIZATIONS by V. Aldaya, J. Guerrero and G. Marmo).

SO(3) -> SU(2)

The standard spatial rotation group $SO(3)$ needs to be extended by $\mathbb{Z}_2$, which yields $SU(2)$, because otherwise we are not able to describe spin $\frac{1}{2}$ particles.

Mickelsson-Faddeev algebra

The algebra of fermionic non-Abelian charge densitites needs to be extended to the Mickelsson-Faddeev algebra (See this answer)

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