Central Extensions of Quantum Symmetry Groups

A central extension retains the phase in a projective unitary representation as an additional group coordinate. The enlarged group acts by an ordinary representation, while its central phase subgroup acts by scalar operators. This construction explains how an exact symmetry action can coexist with a nontrivial multiplier. It does not assert that the original group’s multiplier can be removed.

Required background. Projective Representations and Multipliers supplies the cocycle and rephasing identities used below.

Build a group from a multiplier

Let $\omega:G\times G\to U(1)$ be a normalized unitary multiplier. On the set $U(1)\times G$, define

$$(z,g)(w,h)=(zw\,\omega(g,h),gh).$$

Call the resulting group $\widetilde G_\omega$. Associativity is exactly the cocycle identity. The identity is $(1,e)$, and an inverse is

$$(z,g)^{-1} =\bigl(z^{-1}\omega(g,g^{-1})^{-1},g^{-1}\bigr).$$

Normalization and the cocycle identity imply $\omega(g,g^{-1})=\omega(g^{-1},g)$, so this is both a left and a right inverse.

Projection $p(z,g)=g$ is a surjective group homomorphism. Its kernel consists of $(z,e)$, which commutes with every element. In compact notation,

$$1\longrightarrow U(1)\longrightarrow \widetilde G_\omega\overset{p}{\longrightarrow}G \longrightarrow1.$$

The word “central” refers to this kernel. The enlarged group itself need not be abelian, even when $G$ is.

The lifted representation has an exact multiplication law

If $U_gU_h=\omega(g,h)U_{gh}$, set

$$\widetilde U(z,g)=zU_g.$$

Then

$$\widetilde U(z,g)\widetilde U(w,h) =zw\,\omega(g,h)U_{gh} =\widetilde U\bigl((z,g)(w,h)\bigr).$$

The former scalar discrepancy is now part of the group product. The central subgroup acts through the fixed character $z\mapsto zI$. More general representations of an extension can carry other central characters, which must be specified when comparing physical sectors.

This is the structural content behind mass as a central charge in Galilean quantum mechanics and behind the Heisenberg group associated with canonical displacements. The physical symmetry group, its extension, and its chosen central character are separate data.

Splitting is stronger than having an extension

A splitting is a homomorphism $s:G\to\widetilde G_\omega$ with $p\circ s=\operatorname{id}_G$. Writing $s(g)=(b(g),g)$, the homomorphism condition is

$$b(g)b(h)\omega(g,h)=b(gh).$$

Thus a splitting exists precisely when a rephasing removes the multiplier. The map $g\mapsto(1,g)$ always gives a set-theoretic section, but it is a homomorphism only when $\omega=1$.

If $\omega'=(\beta(g)\beta(h)/\beta(gh))\omega$, the map $(z,g)\mapsto(z\beta(g),g)$ identifies $\widetilde G_{\omega'}$ with $\widetilde G_\omega$ while preserving kernel and projection. Equivalent multipliers therefore describe equivalent extensions, even though their coordinate multiplication formulas differ.

Canonical displacements supply a concrete example

For the symplectic vector space in the Weyl relations, the multiplier is $\omega(z,z')=e^{i\sigma(z,z')/(2\hbar)}$. One can use a real central coordinate $s$, with units of action, and the product

$$(s,z)(s',z') =\left(s+s'+\frac12\sigma(z,z'),\,z+z'\right).$$

The map $(s,z)\mapsto e^{is/\hbar}W(z)$ is an ordinary representation of this real central extension. The commutator of $(0,z)$ and $(0,z')$ is $(\sigma(z,z'),0)$.

This example shows why the central coordinate is meaningful even though it produces scalar operators in an irreducible representation. It records the noncommutativity of composing displacement operations. The detailed canonical representation and its uniqueness remain in the CCR chapter.

Distinguish topology and antiunitary phase actions

For a topological group, the preceding product gives a topological extension with the product topology if $\omega$ is continuous. More general continuous projective actions may require local sections: a global continuous multiplier need not be available in one product chart. A measurable section does not automatically trivialize the topology of the extension.

A covering group is a related but different construction. The cover $SU(2)\to SO(3)$ has discrete central kernel $\{\pm I\}$, rather than the entire phase circle. One must check which kernel elements act trivially or scalarly before descending a representation.

Finally, if antiunitary elements are included, the multiplication becomes

$$(z,g)(w,h) =\bigl(z\,w^{s(g)}\omega(g,h),gh\bigr),$$

using the twisted convention of the multiplier page. An antiunitary lift conjugates the phase subgroup by complex conjugation. That subgroup is then normal but generally not central. Calling every antiunitary extension a central extension would conceal this distinction.

Exercise: find the hidden noncommutativity

For an abelian group $G$ with a unitary multiplier, show that the commutator of the lifts $(1,g)$ and $(1,h)$ is the central phase $\omega(g,h)/\omega(h,g)$. Why does a value different from one obstruct splitting?

Solution

The two products have the same group coordinate $gh=hg$ and central coordinates $\omega(g,h)$ and $\omega(h,g)$. Their ratio is the displayed kernel element. Rephasing the lifts cancels from this ratio. A homomorphic section of an abelian group would have commuting images, so a nontrivial ratio rules out a splitting.

References

• V. Bargmann, “On Unitary Ray Representations of Continuous Groups”, Annals of Mathematics 59, 1–46, 1954 — projective representations and extensions.
• B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed., Graduate Texts in Mathematics 222, Springer, 2015 — covering groups and Lie-group representations.
• K. Landsman, Foundations of Quantum Theory: From Classical Concepts to Operator Algebras, Fundamental Theories of Physics 188, Springer, 2017, Chapter 5 — quantum symmetry groups and phase extensions.