$$\bigl(a\rho(X)\bigr)v = a\bigl(\rho(X)\bigr)v; \qquad \rho(X)(av)
= a\,\rho(X)v + {\alpha}(X)a\,v.$$ If $V$ is moreover a ${\mathfrak{g}}$-module, the connection is flat (as the curvature defined in the obvious way vanishes).

A morphism $({\mathcal{A}},{\mathfrak{g}})\to({\mathcal{A}}',{\mathfrak{g}}')$ of Lie–Rinehart algebras is a pair $(\phi,\psi)$ of an algebra morphism $\phi:{\mathcal{A}}\to{\mathcal{A}}'$ and an ${\mathcal{A}}$-module morphism $\psi:{\mathfrak{g}}\to {\mathfrak{g}}'$, where the action of ${\mathcal{A}}$ on ${\mathfrak{g}}'$ is given by $aX':=\phi(a)X'$, intertwining the anchors: $$\phi\bigl({\alpha}(X)\,a\bigr) = {\alpha}'\bigl(\psi(X)\bigl)\,\phi(a).$$ An important example by Grabowski is as follows [@JanuszBifronte]. A linear operator $D:{\mathfrak{g}}\to{\mathfrak{g}}$ is called a *quasi-derivation* for ${\mathcal{A}}$, and we write $D\in\operatorname{Qder}_{\mathcal{A}}({\mathfrak{g}})$, if for each $a\in{\mathcal{A}}$ there exists $\widehat D(a)\in{\mathcal{A}}$ —necessarily unique— such that $[D,m_a]=m_{\widehat D(a)}$, where the bracket is the usual commutator. It is easily seen that $\widehat D\in\operatorname{Der}({\mathcal{A}})$. Then $({{\mathrm{id}}}_{\mathcal{A}},\operatorname{ad}):({\mathcal{A}},{\mathfrak{g}})\to\bigl({\mathcal{A}},\operatorname{Qder}_{\mathcal{A}}({\mathfrak{g}})\bigr)$, where $\operatorname{ad}:X\mapsto\operatorname{ad}_X$, is a morphism of Lie–Rinehart algebras.

Our first example of Hopf algebra comes now across: the (universal) enveloping algebra ${\mathcal{U}}({\mathfrak{g}})$ of the Lie algebra ${\mathfrak{g}}$. The enveloping algebra is Hopf because there is the diagonal algebra homomorphism $${\Delta}: {\mathcal{U}}({\mathfrak{g}}) \to {\mathcal{U}}({\mathfrak{g}}{\oplus}{\mathfrak{g}}) \simeq {\mathcal{U}}({\mathfrak{g}}) {\otimes}{\mathcal{U}}({\mathfrak{g}}) {\quad\mbox{by}\quad} X
\mapsto X {\oplus}X \mapsto X {\otimes}1 + 1 {\otimes}X, \; {\Delta}1 = 1 {\otimes}1,$$ for every $X\in {\mathfrak{g}}$.

When ${\mathfrak{g}}={\mathfrak{X}}(M)$, the enveloping algebra should not be confused with the algebra (with the usual 