# Lie algebra of a compact Lie group

Let $$\mathcal{G}$$ be a compact (real) Lie group. We know that the Lie algebra $$\mathfrak{g}$$ of $$\mathcal{G}$$ is, by definition, the space of all left-invariant (smooth) vector fields over $$\mathcal{G}$$ with bracket given by the commutator. We also know that it is isomorphic, as a Lie algebra, with $$T_e\mathcal{G}$$ (the tangent space to $$\mathcal{G}$$ at the neutral element $$e$$).

Consider the Hopf algebra of representative functions $$H:=\mathcal{R}_{\mathbb{R}}(\mathcal{G})$$ associated to $$\mathcal{G}$$ and recall that $$H\subseteq \mathcal{C}^\infty(\mathcal{G})$$ is a dense subspace with respect to the supremum norm (Proposition I.3.12 of Brocker, Dieck, Representations of Compact Lie Groups together with Peter-Weyl Theorem). The Lie algebra $$\mathcal{P}(H^\circ)$$ of primitive elements of its finite dual is isomorphic the Lie algebra of left-invariant derivations $${^{H}\mathsf{Der}_{\mathbb{R}}(H,H)}$$ (i.e., derivations $$\delta:H\to H$$ such that $$\Delta\delta=(H\otimes \delta)\Delta$$).

I would expect to have an isomorphism $$\mathfrak{g}\cong {^{H}\mathsf{Der}_{\mathbb{R}}(H,H)}$$.

Q1. Is this true?

I didn't find it anywhere in the literature, whence I am trying to provide one by myself.

Let $$X$$ be a left-invariant vector field and let $$\varphi$$ be a smooth function on $$\mathcal{G}$$. Then $$\boldsymbol{X}(\varphi):\mathcal{G}\to \mathbb{R}$$ given by $$\boldsymbol{X}(\varphi)(g) = X_g(\varphi)$$ ($$X_g\in T_g\mathcal{G}$$) is a smooth function and hence we have an assignment $$\boldsymbol{X}:\mathcal{C}^\infty(\mathcal{G})\to \mathcal{C}^\infty(\mathcal{G})$$. One can verify that this induces a Lie algebra map $$\mathfrak{g}\to {^{H}\mathsf{Der}_{\mathbb{R}}(H,H)}: X\mapsto \boldsymbol{X}.$$ To provide a candidate inverse to this morphism, I consider a left-invariant derivation $$\delta$$ and for every $$g$$ in $$\mathcal{G}$$ the function $$\delta_g:\mathcal{R}_{\mathbb{R}}(\mathcal{G})\to \mathbb{R}, \varphi\mapsto\delta(\varphi)(g)$$. Now, I would expect to be able to extend such a function to the whole $$\mathcal{C}^\infty(\mathcal{G})$$, maybe resorting to the Continuous Linear Extension Theorem (see Theorem 5.7.6 in Foundations of Applied Mathematics, Volume I: Mathematical Analysis by Humpherys, Jarvis, Evans), but I didn't manage to.

Q2. Could somebody suggest a way to do this?

Q3. Is there some reference in which this is treated in some detail?