Vtyjkyuk

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

Consider the Hopf algebra of representative functions $H:=mathcalR_mathbbR(mathcalG)$ associated to $mathcalG$ and recall that $Hsubseteq mathcalC^infty(mathcalG)$ 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 $mathcalP(H^circ)$ of primitive elements of its finite dual is isomorphic the Lie algebra of left-invariant derivations $^HmathsfDer_mathbbR(H,H)$ (i.e., derivations $delta:Hto H$ such that $Deltadelta=(Hotimes delta)Delta$).

I would expect to have an isomorphism $mathfrakgcong ^HmathsfDer_mathbbR(H,H)$.

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 $mathcalG$. Then $boldsymbolX(varphi):mathcalGto mathbbR$ given by $boldsymbolX(varphi)(g) = X_g(varphi)$ ($X_gin T_gmathcalG$) is a smooth function and hence we have an assignment $boldsymbolX:mathcalC^infty(mathcalG)to mathcalC^infty(mathcalG)$. One can verify that this induces a Lie algebra map
$$mathfrakgto ^HmathsfDer_mathbbR(H,H): Xmapsto boldsymbolX.$$
To provide a candidate inverse to this morphism, I consider a left-invariant derivation $delta$ and for every $g$ in $mathcalG$ the function $delta_g:mathcalR_mathbbR(mathcalG)to mathbbR, varphimapstodelta(varphi)(g)$. Now, I would expect to be able to extend such a function to the whole $mathcalC^infty(mathcalG)$, 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?