# The algebra generated by derivations

This may be a boring question.

Suppose everything in the following is on a field $k$ (of characteristic $0$) without specification otherwise. Consider a commutative ring $A$ and the Lie algebra of its derivations $Der(A)$. Note that

1. $Der(A)$ is an $A$-module;
2. both $A$ and $Der(A)$ are subspaces of the algebra $End(A)$ of endomorphisms of $A$;
3. $(A,Der(A))$ form a Lie-Rinehart pair.

Then, we have the following algebras:

1. the symmetric algebra $S=Sym_A(Der(A))$, which is a graded $A$-algebra;
2. the subalgebra $R$ of $End(A)$ generated by $A$ and $Der(A)$, which is a filtered algebra with filtration by power of $Der(A)$ and its associated graded algebra is a commutative $A$-algebra;
3. the universal algebra $U$ of the Lie-Rinehart pair $(A,Der(A))$, which is a filtered algebra with filtration by power of $Der(A)$ and its associated graded algebra is a commutative $A$-algebra.

Then, we have a homomorphism of graded $A$-algebras: $$\phi\colon S\longrightarrow\mathrm{gr}R,$$ and a homomorphism of filtered algebras: $$\psi\colon U\longrightarrow R.$$

In the case $A$ is regular, hence $Der(A)$ has a basis, both $\phi$ and $\psi$ are isomorphisms and $R$ is moreover equal to the ring of differential operators. However, what about general case?

My questions: Are $\phi$ and $\psi$ isomorphisms in general? If not, are they injective? surjective? and does $\phi$ factors through $\mathrm{gr}(\psi)$?