Kernel of Ad and the center

Let $G$ be a Lie group with Lie algebra $\mathfrak g$, and let

be the adjoint representation , obtained by differentiating conjugation.

Lemma (Kernel of Ad).

• For any Lie group $G$, the center satisfies $Z(G)\subseteq \ker(\mathrm{Ad})$.
• If $G$ is connected, then $\ker(\mathrm{Ad})=Z(G)$.

More generally, for arbitrary (not necessarily connected) $G$, the kernel of $\mathrm{Ad}$ equals the centralizer of the identity component $G^\circ$.

Proof idea (connected case).
If $\mathrm{Ad}_g=\mathrm{id}$, then conjugation by $g$ has derivative equal to the identity at $e$, hence acts trivially on $\mathfrak g$. This forces conjugation by $g$ to fix a neighborhood of $e$ (via the exponential chart from the exponential map ), and since $G$ is generated by any neighborhood of $e$ when connected, $g$ commutes with all of $G$.

Context.
This lemma is used to relate faithfulness of the adjoint representation to the size of the center (compare Ad faithful iff center discrete ) and to connect $\ker(\mathrm{Ad})$ with the infinitesimal statement ker(ad) equals the Lie algebra center .