The Dixmier problem for skew PBW extensions and rings

In this paper we discuss for skew $PBW$ extensions the famous Dixmier problem formulated by Jacques Dixmier in 1968. The skew $PBW$ extensions are noncommutative rings of polynomial type and covers several algebras and rings arising in mathematical physics and noncommutative algebraic geometry. For this purpose, we introduce the Dixmier algebras and we will study the Dixmier problem for algebras over commutative rings, in particular, for $\mathbb{Z}$-algebras, i.e., for arbitrary rings. The results are focused on the investigation of the Dixmier problem for matrix algebras, product of algebras, tensor product of algebras and also on the Dixmier question for the following particular key skew $PBW$ extension: Let $K$ be a field of characteristic zero and let $\mathcal{CSD}n(K)$ be the $K$-algebra generated by $n\geq 2$ elements $x_1,\dots,x_n$ subject to relations $$x_jx_i=x_ix_j+d{ij}, \ for \ all \ 1\leq i<j\leq n, \ with \ d_{ij}\in K-{0}$$. We prove that the algebra $\mathcal{CSD}_n(K)$ is central and simple. In the last section we present a matrix-computational approach to the problem formulated by Jacques Dixmier and also we compute some concrete nontrivial examples of automorphisms of the first Weyl algebra $A_1(K)$ and $\mathcal{CSD}_n(K)$ using the MAPLE library SPBWE developed for the first author. We compute the inverses of these automorphisms, and for $A_1(K)$, its factorization through some elementary automorphisms. For $n$ odd, we found some endomorphisms of $\mathcal{CSD}_n(K)$ that are not automorphisms. We conjecture that $\mathcal{CSD}_n(K)$ is Dixmier when $n$ is even.

💡 Research Summary

The paper investigates the celebrated Dixmier problem—whether every endomorphism of a given algebra is automatically an automorphism—in the broad setting of skew PBW (Poincaré‑Birkhoff‑Witt) extensions. After a concise historical overview linking the Dixmier conjecture to the Jacobian, Kernel, and Zariski cancellation problems, the authors introduce the notion of a “Dixmier algebra” (an algebra for which every endomorphism is an automorphism) and a “Dixmier ring” (the analogous property for rings, extended to matrix algebras). They establish elementary closure properties: direct sums, products, and tensor products of Dixmier algebras remain Dixmier, and if a matrix algebra (M_n(A)) is Dixmier then the base algebra (A) is Dixmier as well.

The core of the work focuses on skew PBW extensions, a class that encompasses Weyl algebras, quantum Weyl algebras, and many other non‑commutative polynomial‑type rings arising in mathematical physics and non‑commutative geometry. The authors prove a structural transfer theorem: if a skew PBW extension over a commutative base ring is a Dixmier algebra, then the base ring itself must be Dixmier. This result ties the Dixmier property of sophisticated non‑commutative algebras back to the underlying commutative coefficients.

A central example is the algebra (\mathcal{CSD}_n(K)) defined over a characteristic‑zero field (K) by generators (x_1,\dots,x_n) and relations \