First properties of Weyl algebras

For any ring $k$ we denote by $A_n(k)$ the Weyl algebra:

$$A_n(k)=k\langle \xi_1,\xi_2,\dots,\xi_n,\eta_1,\eta_2,\dots ,\eta_n \rangle / (\eta_j\xi_i-\xi_i\eta_j-\delta_{ij})$$

where $\delta_{ij}$ is the Kronecker's delta.

One of the good ways to compute multiplications of elements in $A_n(k)$ appears in [2, formula (11,4)]. For any variable $\eta,\xi$ with the canonical commutation relation $\eta\xi-\xi\eta=1$,and for any pair of ``normally ordered'' polynomials $f(\xi,\eta)=\sum f_{i,j}\xi^i \eta^j$ and $g(\xi,\eta)=\sum g_{i,j}\xi^i \eta^j$, one has the following formula

$$f(\xi,\eta)g(\xi,\eta) =\sum_{k=0}^\infty \frac{1}{k!}\partial_{\eta}^k f(\xi,\eta) * \partial_{\xi}^k g(\xi,\eta)$$

where $*$ stands for an `multiplication as though $\xi,\eta$ commutes'. This formula is valid and proved in the book cited above only if characteristic of the coefficient field is 0. If the characteristic $p$ of the coefficient field is positive, then we replace $\infty$ in the above formula by $p-1$ and obtain a valid formula in this case. The proof is almost the same. It is well suited for computer calculation using commutative polynomials and differentiations.

Dixmier conjectures that

Conjecture 1   Every $\mathbb{C}$-algebra endomorphism $\phi$ of $A_n(\mathbb{C})$ is invertible (that is, it is an automorphism).

Since $A_n(\mathbb{C})$ is simple (has no nontrivial both-sided ideal), we know that $\phi$ above is injective. The question therefore is the surjectivity of $\phi$.