[Paper Review] Automorphisms and Ideals of the Weyl Algebra

Yuri Berest, George Wilson|ArXiv.org|Feb 25, 2001

Advanced Differential Equations and Dynamical Systems11 references4 citations

TL;DR

This paper establishes a bijective correspondence between the space of isomorphism classes of non-zero right ideals in the first Weyl algebra $A_1$ and a disjoint union of spaces $\mathcal{C}_n$ of pairs of $n\times n$ matrices satisfying $[X,Y] - I$ has rank at most 1. It shows that the automorphism group $G$ of $A_1$ acts on this space via a structure analogous to the action on Hilbert schemes in the commutative case, revealing deep parallels between noncommutative ideals and commutative algebraic geometry despite the absence of finite-dimensional representations in $A_1$. The key contribution is the identification of $\mathcal{C}_n$ as coadjoint orbits under a central extension of the automorphism group, linking noncommutative algebra to integrable systems and infinite-dimensional algebraic groups.

ABSTRACT

Let $A_1$ be the (first) Weyl algebra, and let $G$ be its automorphism group. We study the natural action of $G$ on the space of isomorphism classes of right ideals of $A_1$ (equivalently, of finitely generated rank 1 torsion-free right $A_1$-modules). We show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety. Our results are strikingly similar to those for the commutative algebra of polynomials in two variables; however, we do not know of any general principle that would allow us to predict this in advance. As a key step in the proof, we obtain a new description of the bispectral involution of \cite{W1}. We also make some comments on the group $G$ from the viewpoint of Shafaravich's theory of infinite dimensional algebraic groups.

Motivation & Objective

• To understand the structure of isomorphism classes of right ideals in the first Weyl algebra $A_1$, which lacks finite-dimensional representations and thus defies classical reduction techniques.
• To describe the action of the automorphism group $G$ of $A_1$ on the space $\mathcal{R}$ of isomorphism classes of non-zero right ideals.
• To establish a geometric and dynamical correspondence between the noncommutative setting of $A_1$ and the well-understood commutative case of $\mathbb{C}[x,y]$, despite fundamental differences in representation theory.
• To explore the algebraic group structure of $G$ via Shafarevich's theory of infinite-dimensional algebraic groups, particularly in relation to the free associative algebra $\mathbb{C}\langle x,y\rangle$.
• To clarify the role of the adelic Grassmannian and the KP hierarchy in parametrizing ideals of $A_1$ through the composition $\omega = \alpha \beta$.

Proposed method

• The authors use the bijective map $\beta: \mathcal{C} \to \mathrm{Gr}^{\mathrm{ad}}$ from the KP hierarchy's adelic Grassmannian, constructed in [W2], to parametrize approximate representations of $A_1$.
• They combine this with the inverse map $\alpha: \mathrm{Gr}^{\mathrm{ad}} \to \mathcal{R}$ from Cannings and Holland's work to define $\omega = \alpha \beta$, establishing a bijection $\omega: \mathcal{C} \to \mathcal{R}$.
• The automorphism group $G$ of $A_1$ acts on $\mathcal{R}$, and this action is transferred via $\omega$ to $\mathcal{C}$, where it is described explicitly using the action of automorphisms $\Phi_p$ and $\Psi_q$ on matrix pairs.
• The action of $G$ on $\mathcal{C}_n$ is shown to preserve a holomorphic symplectic structure, suggesting that each $\mathcal{C}_n$ is a coadjoint orbit of a central extension of the automorphism group.
• The authors analyze the group $G$ using Shafarevich's framework for infinite-dimensional algebraic groups, distinguishing $G_1$ (for $A_1$) from $G_0$ (for $\mathbb{C}[x,y]$) via their Lie algebras: $A_1/\mathbb{C}$ vs. $A_0/\mathbb{C}$.
• They identify $G$ as a quotient of the group $\mathcal{G}$ of unimodular automorphisms of the free algebra $\mathbb{C}\langle x,y\rangle$, which acts algebraically on $\mathcal{C}_n$, offering a natural algebraic group structure for the action.

Experimental results

Research questions

• RQ1 Can the space of isomorphism classes of right ideals in the Weyl algebra $A_1$ be parametrized in a way analogous to the Hilbert scheme in the commutative case?
• RQ2 How does the automorphism group $G$ of $A_1$ act on the space of isomorphism classes of right ideals, and can this action be described explicitly?
• RQ3 Why do the spaces $\mathcal{C}_n$ of matrix pairs with $[X,Y] - I$ of rank at most 1 form finite-dimensional algebraic varieties stable under the $G$-action, despite $A_1$ having no finite-dimensional representations?
• RQ4 Is there a geometric or algebraic group-theoretic structure underlying the $G$-action on $\mathcal{C}_n$, and how does it relate to known structures in integrable systems or coadjoint orbits?
• RQ5 Can the automorphism group of $A_1$ be realized as an infinite-dimensional algebraic group acting algebraically on $\mathcal{C}_n$, and if so, via which algebraic group structure?

Key findings

• The space $\mathcal{R}$ of isomorphism classes of non-zero right ideals in $A_1$ is in bijection with the disjoint union $\mathcal{C} = \bigsqcup_{n \geq 0} \mathcal{C}_n$, where $\mathcal{C}_n$ consists of $G$-orbits of $n \times n$ matrix pairs $(X,Y)$ such that $[X,Y] - I$ has rank at most 1.
• The automorphism group $G$ of $A_1$ acts on $\mathcal{C}_n$ via the formula $\sigma \cdot (X,Y) = (\sigma^{-1}(X), \sigma^{-1}(Y))$, analogous to the action on commuting matrices in the commutative case.
• Each $\mathcal{C}_n$ carries a natural holomorphic symplectic structure preserved by the $G$-action, suggesting that $\mathcal{C}_n$ is a coadjoint orbit of a central extension of $G$.
• The Lie algebra of the automorphism group $G_1$ of $A_1$ is isomorphic to $A_1 / \mathbb{C}$, while that of $G_0$ (the automorphism group of $\mathbb{C}[x,y]$) is isomorphic to $A_0 / \mathbb{C}$, showing that $G_1$ and $G_0$ are not isomorphic as algebraic groups.
• The group $G$ can be realized as a quotient of the group $\mathcal{G}$ of unimodular automorphisms of the free algebra $\mathbb{C}\langle x,y\rangle$, and $\mathcal{G}$ acts algebraically on $\mathcal{C}_n$, providing a natural algebraic group structure for the action.
• The space $\mathcal{C}_n$ is conjectured to be isomorphic to a coadjoint orbit of a central extension of $\mathcal{G}$, as later confirmed by Ginzburg.