Example. Differentiating a matrix representation [liu2016lie, Sections 2.1 and 2.4, pp. 9, 12] [fcap-001V]
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Let \(V\) be a finite-dimensional real vector space and let \[\rho :G\longrightarrow \operatorname {GL}(V)\] be a smooth representation. Its differential is a Lie-algebra representation \[d\rho _{e_G}:\mathfrak g\longrightarrow \mathfrak {gl}(V).\] Naturality of the exponential becomes \[\rho (\exp _G X)=\exp \bigl (d\rho _{e_G}(X)\bigr ),\] where the exponential on the right is the ordinary matrix exponential. For example, for the determinant homomorphism \(\det :\operatorname {GL}_n(\mathbb R)\to \mathbb R^\times \), \[d(\det )_I(A)=\operatorname {tr}(A).\]

Example. Differentiating a matrix representation [liu2016lie, Sections 2.1 and 2.4, pp. 9, 12] [fcap-001V]
✍️source

Reference. Notes for lie groups and representations [liu2016lie]

Differentiating smooth group homomorphisms [fcap-001Q]

Definition. The differential of a smooth homomorphism [liu2016lie, Section 2.1, p. 9] [fcap-001R]

Lemma. The differential preserves the Lie bracket [liu2016lie, Section 2.1, p. 9] [fcap-001S]

Theorem. The Lie functor [liu2016lie, Section 2.2, p. 10] [fcap-001T]

Theorem. Naturality of the exponential [liu2016lie, Section 2.4, pp. 12--13] [fcap-001U]

Proof.

Example. Differentiating a matrix representation [liu2016lie, Sections 2.1 and 2.4, pp. 9, 12] [fcap-001V]
✍️source

Remark. Why explicit smoothness changes the roadmap [liu2016lie, Sections 2.1--2.4 and 2.8, pp. 9--13, 19--20]; [isaev2018theory, Section 3.1.2, pp. 106--107] [fcap-001W]

Remark. Why explicit smoothness changes the roadmap [liu2016lie, Sections 2.1--2.4 and 2.8, pp. 9--13, 19--20]; [isaev2018theory, Section 3.1.2, pp. 106--107] [fcap-001W]