Central quotient of a Lie group

Definition

Let $G$ be a Lie group and let $Z$ be a closed subgroup of its center. The central quotient of $G$ by $Z$ is the quotient Lie group

G/Z,

whose multiplication is $(gZ)(hZ)=ghZ$ . Centrality makes $Z$ normal, and closedness gives the coset space its canonical smooth Lie-group structure.

Infinitesimal effect

If $\mathfrak z=\operatorname{Lie}(Z)$ , then

\operatorname{Lie}(G/Z)\cong \mathfrak g/\mathfrak z.

When $Z$ is discrete—for example, finite—the quotient map $G\to G/Z$ is a local diffeomorphism and the two groups have isomorphic Lie algebras. A finite central quotient can therefore change global topology and available representations without changing the infinitesimal symmetry algebra.

Representations

A representation $\rho:G\to GL(V)$ descends to $G/Z$ exactly when $Z\subseteq\ker\rho$ . Equivalently, representations of $G/Z$ are representations of $G$ on which every element of $Z$ acts trivially.

Example

The effective Standard Model internal symmetry group is often written

\bigl(U(1)\times SU(2)\times SU(3)\bigr)/\mathbb Z_6.

The central $\mathbb Z_6$ acts trivially in the one-generation Standard Model representation, and the quotient is isomorphic to $S(U(2)\times U(3))$ .

References

Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, second edition, Springer, 2015. DOI record. Relevant: quotient Lie groups and covering homomorphisms.
John C. Baez and John Huerta, “The Algebra of Grand Unified Theories,” Bulletin of the American Mathematical Society 47 (2010), 483–552. arXiv record. Relevant: §3.1.