# Differential Isomorphism and Equivalence of Algebraic Varieties

@article{Berest2003DifferentialIA, title={Differential Isomorphism and Equivalence of Algebraic Varieties}, author={Yuri Yu. Berest and George V. Wilson}, journal={arXiv: Algebraic Geometry}, year={2003} }

In this survey article we discuss the question: to what extent is an algebraic variety determined by its ring of differential operators? In the case of affine curves, this question leads to a variety of mathematical notions such as the Weyl algebra, Calogero-Moser spaces and the adelic Grassmannian. We give a fairly detailed overview of this material.

## 18 Citations

### On the isomorphism problem for the rings of differential operators on smooth affine varieties

- MathematicsJournal of Algebra
- 2022

### Diﬀerential operators on toric varieties and Fourier transform

- Mathematics
- 2008

. We show that Fourier transforms on the Weyl algebras have a geo-metric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of diﬀerential…

### Differential operators on toric varieties and Fourier transform

- Mathematics
- 2007

Abstract.We show that Fourier transforms on the Weyl algebras have a geometric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of differential…

### A Remark on Letzter-Makar-Limanov Invariants

- Mathematics
- 2003

We give a natural cohomological interpretation of Letzter-Makar-Limanov invariants for rings of differential operators on algebraic curves.

### $A_{\infty}$-modules and Calogero-Moser Spaces

- Mathematics
- 2004

We re-examine the bijective correspondence between the set of isomorphism classes of ideals of the first Weyl algebra and associated quiver varieties (Calogero-Moser spaces) \cite{BW1, BW2}. We give…

### On subgroups of the Dixmier group and Calogero-Moser spaces

- Mathematics
- 2011

We describe the structure of the automorphism groups of algebras
Morita equivalent to the first Weyl algebra $ A_1(k) $.
In particular, we give a geometric presentation for these groups in terms of…

### Cusps and D-modules

- Mathematics
- 2002

1.1. V-modules on singular varieties. Let Y denote a variety over a field and let Vy denote the full sheaf of differential operators on Y (in characteristic zero Vy is the familiar sheaf of…

### Quasi-invariants of complex reflection groups

- MathematicsCompositio Mathematica
- 2010

Abstract We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space of quasi-invariants of a given multiplicity is not, in…

### Trees, Amalgams and Calogero-Moser Spaces

- Mathematics
- 2010

We describe the structure of the automorphism groups of algebras Morita equivalent to the first Weyl algebra $ A_1 $. In particular, we give a geometric presentation for these groups in terms of…

### A∞-modules and Calogero-Moser spaces

- Mathematics
- 2007

The relations between these objects are well known and almost immediate. Thus, (1) is essentially the definition of (closed) points of HilbðCÞ. The bijection (1)! (2) is given by taking the quotient…

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