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 Throem). 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?

OT: I already asked it on MSE, but maybe it could be that this question is more suitable for MO.

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 Throem). 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?

OT: I already asked it on MSE, but maybe it could be that this question is more suitable for MO.