the last equality we just reordered terms; hence $J$ is two-sided. By construction it is clear that the quotient ${\mathcal{U}}({\mathcal{A}},{\mathfrak{g}}):={\mathcal{A}}\rtimes U({\mathfrak{g}})/J$ together with the obvious quotient morphisms $\imath_{\mathcal{A}}$ and $\imath_{\mathfrak{g}}$ possesses the universal property. Note that $\imath_{\mathcal{A}}$ is injective. A morphism $(\phi,\psi):({\mathcal{A}},{\mathfrak{g}})\to({\mathcal{A}}',{\mathfrak{g}}')$ induces a morphism of algebras ${\mathcal{U}}(\phi,\psi):{\mathcal{U}}({\mathcal{A}},{\mathfrak{g}})\to{\mathcal{U}}({\mathcal{A}}',{\mathfrak{g}}')$, and vice versa; this is an equivalence of categories.

One obtains by this construction the ordinary algebra of differential operators ${\mathbb{D}}(M)={\mathcal{U}}({\mathcal{F}}(M),{\mathfrak{X}}(M))$. It is also clear that ${\mathcal{U}}({\mathbb{C}},{\mathfrak{g}})={\mathcal{U}}({\mathfrak{g}})$ with trivial action of ${\mathfrak{g}}$. Just like the enveloping algebra, the universal algebra ${\mathcal{U}}({\mathcal{A}},{\mathfrak{g}})$ is filtered, with an associated graded object $\operatorname{gr}\,{\mathcal{U}}({\mathcal{A}},{\mathfrak{g}})$, which is a commutative graded ${\mathcal{A}}$-algebra. There is also a Poincaré–Birkhoff–Witt theorem for ${\mathcal{U}}({\mathcal{A}},{\mathfrak{g}})$ when ${\mathfrak{g}}$ is projective over ${\mathcal{A}}$ —which is the case in the geometrical examples. It claims that if $S_{\mathcal{A}}[{\mathfrak{g}}]$ is the symmetric ${\mathcal{A}}$-algebra on ${\mathfrak{g}}$, then the natural surjection $$S_{\mathcal{A}}[{\mathfrak{g}}] \to \operatorname{gr}\,{\mathcal{U}}({\mathcal{A}},{\mathfrak{g}}),
\label{eq:Casimir-future}$$ is an isomorphism of ${\mathcal{A}}$-algebras, rather like the $S[{\mathfrak{g}}]\simeq\operatorname{gr}\,
{\mathcal{U}}({\mathfrak{g}})$ effected, say, through the ‘symmetrization’ map $\sigma:S[{\mathfrak{g}}]
\to{\mathcal{U}}({\mathfrak{g}})$. In that case $\imath_{\mathfrak{g}}$ is of course injective.

Coming by Hopf algebra
======================

We begin here a journey from the geometrical to the Hopf world. Let us start by some well-known observations [@KawskiS]. A smooth manifold $M$ is 