Identity component of a Lie group

Definition

Let $G$ be a Lie group with identity $e$ . The identity component (or neutral component) of $G$ is the connected component

G^{\circ}=G_e

that contains $e$ . It is a connected, open-and-closed normal Lie subgroup of $G$ , and every connected component of $G$ is a left and right coset of $G^{\circ}$ .

Component group and Lie algebra

The quotient

\pi_0(G)=G/G^{\circ}

is a discrete group, called the component group of $G$ . It records the global disconnectedness that is invisible to infinitesimal Lie theory. Indeed, the inclusion $G^{\circ}\hookrightarrow G$ induces an isomorphism

\operatorname{Lie}(G^{\circ})\cong\operatorname{Lie}(G).

Thus a Lie algebra determines the local group structure but does not determine the component group.

Why it is normal and open

Conjugation by any $g\in G$ is a homeomorphism fixing $e$ , so it preserves the component containing $e$ ; hence $gG^{\circ}g^{-1}=G^{\circ}$ . A finite-dimensional manifold is locally connected, so its connected components are open. The other components are then the translates $gG^{\circ}$ .

Stabilizer warning

If a possibly disconnected Lie group acts on a space and $G_x$ is a stabilizer, three groups can differ:

(G_x)^{\circ},\qquad G_x\cap G^{\circ},\qquad G_x.

The first is the identity component of the stabilizer, the second is the full stabilizer inside the acting identity component, and the third may contain additional components. Neither of the first two should be substituted for the full stabilizer without checking connectedness. This distinction matters in homogeneous-space descriptions of real Grassmannians and frame spaces.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2013, Chapter 20. Publisher record.
J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Springer, 2000, Chapter 1. Publisher record.