[Paper Review] An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators

V. V. Bavula|arXiv (Cornell University)|Nov 12, 2010

Advanced Differential Equations and Dynamical Systems20 references3 citations

TL;DR

This paper proves that every algebra endomorphism of the algebra of polynomial integro-differential operators $π_1$ over a field of characteristic zero is an automorphism, establishing an analogue of Dixmier's conjecture for this non-simple, non-Noetherian algebra. The proof proceeds through a nine-step structural analysis of endomorphisms, leveraging eigenvalue arguments, module isomorphisms, and inner automorphism realization via unit conjugation in $1+F^*$, where $F$ is the unique proper ideal of $π_1$. The key result confirms the conjecture's validity in this extended setting despite the algebra's lack of standard finiteness properties.

ABSTRACT

Let $A_1:=K\langle x, \frac{d}{dx} angle$ be the Weyl algebra and $\mI_1:= K\langle x, \frac{d}{dx}, \int angle$ be the algebra of polynomial integro-differential operators over a field $K$ of characteristic zero. The Conjecture/Problem of Dixmier (1968) [still open]: {\em is an algebra endomorphism of the Weyl algebra $A_1$ an automorphism?} The aim of the paper is to prove that {\em each algebra endomorphism of the algebra $\mI_1$ is an automorphism}. Notice that in contrast to the Weyl algebra $A_1$ the algebra $\mI_1$ is a non-simple, non-Noetherian algebra which is not a domain. Moreover, it contains infinite direct sums of nonzero left and right ideals.

Motivation & Objective

• To establish an analogue of Dixmier's 1968 conjecture—whether every algebra endomorphism of the Weyl algebra is an automorphism—for the algebra $\mathbb{I}_1$ of polynomial integro-differential operators.
• To address the challenge posed by $\mathbb{I}_1$'s non-simplicity, non-Noetherian structure, and failure to be a domain, which distinguish it from the Weyl algebra $A_1$.
• To demonstrate that despite the existence of non-automorphic endomorphisms in related algebras (e.g., $B_1$), the full algebra $\mathbb{I}_1$ satisfies the automorphism property.
• To show that each endomorphism of $\mathbb{I}_1$ arises as an inner automorphism via conjugation by a unit in $(1+F)^*$, where $F$ is the unique proper ideal of $\mathbb{I}_1$.
• To extend the conceptual framework of Dixmier-type problems to algebras with integration operators, thereby broadening the scope of the conjecture.

Proposed method

• Analyzing endomorphisms $\sigma$ of $\mathbb{I}_1$ via their action on generators $H = \partial x$, $\partial = \frac{d}{dx}$, and $\int$, reducing the problem to tracking $H' = \sigma(H)$, $\partial' = \sigma(\partial)$, $\int' = \sigma(\int)$.
• Using the unique proper ideal $F$ of $\mathbb{I}_1$ to induce a quotient map $\pi: \mathbb{I}_1 \to B_1 \simeq K[H][\partial, \partial^{-1}; \tau]$, where $\tau(H) = H+1$, and analyzing the induced map $\overline{\sigma}$ on $B_1$.
• Applying eigenvalue and eigenspace analysis to the action of $H'$ on $K[x]$, showing that the multiplicity of eigenvalues under $H' \cdot$ must match the rank of the twisted module ${}^\sigma K[x]$, leading to $n=1$ in the normalization step.
• Establishing that $\sigma$ induces an $\mathbb{I}_1$-module isomorphism ${}^\sigma K[x] \simeq K[x]$ by comparing dimensions of kernels of $H - i$ and using the structure of $K[x]$ as a direct sum of eigenspaces.
• Proving $\mu = 0$ by comparing dimensions of $\partial^\prime$-invariant subspaces and using the identity $\dim_K(\partial^\prime K[\partial^\prime] \ast \ker(H' - (i+1))) = i$ for all $i \in \mathbb{N}$.
• Constructing an $\mathbb{I}_1$-module isomorphism $u: K[x] \to {}^\sigma K[x]$ via $x^{[i]} \mapsto x^{\prime[i]}$, and showing $\sigma = \omega_u$, the inner automorphism induced by $u$, with $u \in (1+F)^*$.

Experimental results

Research questions

• RQ1 Does every algebra endomorphism of the algebra $\mathbb{I}_1$ of polynomial integro-differential operators over a field of characteristic zero extend to an automorphism?
• RQ2 How does the presence of a non-trivial ideal $F$ and the non-regularity of $\partial$ in $\mathbb{I}_1$ affect the structure of endomorphisms compared to the Weyl algebra $A_1$?
• RQ3 Can the eigenvalue structure of the operator $H' \cdot$ on $K[x]$ be used to constrain the possible forms of $\sigma(H)$, $\sigma(\partial)$, and $\sigma(\int)$?
• RQ4 Is it possible for an endomorphism of $\mathbb{I}_1$ to be non-inner, despite the algebra's non-simple and non-Noetherian nature?
• RQ5 What role does the algebraic torus action $\mathbb{T}^1$ play in normalizing endomorphisms, and how does it simplify the proof structure?

Key findings

• Every algebra endomorphism of $\mathbb{I}_1$ is an automorphism, confirming the analogue of Dixmier's conjecture for this algebra.
• The endomorphism $\sigma$ must satisfy $\sigma(H) = H + \mu + h$, $\sigma(\partial) = \partial + g$, and $\sigma(\int) = \int + f$ for some $h, f, g \in F$, after normalization via the torus action.
• The parameter $\mu$ must vanish, i.e., $\mu = 0$, due to dimension constraints on $\partial^\prime$-invariant subspaces and the eigenvalue multiplicity condition.
• The endomorphism $\sigma$ is realized as an inner automorphism $\omega_u$ via conjugation by a unit $u \in (1+F)^*$, where $F$ is the unique proper ideal of $\mathbb{I}_1$.
• The twisted $\mathbb{I}_1$-module ${}^\sigma K[x]$ is isomorphic to $K[x]$ as a module, which forces the normalization $n=1$ in the initial scaling step.
• The proof relies on the fact that $H^\prime \cdot$ acting on $K[x]$ has distinct eigenvalues $\{\frac{1}{n}(i+1) + \mu\}$ for $i \geq s$, which only allows $n=1$ when the eigenspace multiplicities match the rank of the module.