Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Commuting families in skew fields and quantization of Beauville's fibration

B. Enriquez, V. Rubtsov

Abstract

We construct commuting families in fraction fields of symmetric powers of algebras. The classical limit of this construction gives Poisson commuting families associated with linear systems. In the case of a K3 surface S, they correspond to lagrangian fibrations introduced by Beauville. When S is the canonical cone of an algebraic curve C, we construct commuting families of differential operators on symmetric powers of C, quantizing the Beauville systems.

Commuting families in skew fields and quantization of Beauville's fibration

Abstract

We construct commuting families in fraction fields of symmetric powers of algebras. The classical limit of this construction gives Poisson commuting families associated with linear systems. In the case of a K3 surface S, they correspond to lagrangian fibrations introduced by Beauville. When S is the canonical cone of an algebraic curve C, we construct commuting families of differential operators on symmetric powers of C, quantizing the Beauville systems.

Paper Structure

This paper contains 16 sections, 12 theorems, 111 equations.

Table of Contents

  1. Commuting families in skew fields
  2. Poisson commuting families on symmetric powers
  3. Poisson commuting families
  4. The Beauville fibration associated to a K3 surface
  5. An affine version of Beauville's fibration
  6. Beauville fibration in the case of the canonical cone of a curve
  7. Quantization of Beauville fibrations
  8. Quantization of fields
  9. Quantization of commuting families
  10. Quantization of Beauville systems
  11. The case of the canonical cone of an algebraic curve
  12. Algebras of rational differential operators
  13. Relation with $\operatorname{Cone}(C)$
  14. Commuting differential operators
  15. Explicit formulas in the rational case
  16. ...and 1 more sections

Key Result

Theorem 1.1

Assume that there exists a pair $(\phi,F)$ of a skew field $F$ and a ring injection $\phi : A^{\otimes n} \hookrightarrow F$. (We will identify elements of $A^{\otimes n}$ with their images by $\phi$.) Assume that $f_0,\ldots,f_n$ are linearly independent elements of $A$. Set for $i = 0,\ldots,n$. Then $\Delta_0 \neq 0$. The elements form a commutative family of elements of $K$: we have $H_i H_j

Theorems & Definitions (14)

  • Theorem 1.1
  • Corollary 1.1
  • Remark 1
  • Lemma 2.1
  • Theorem 2.1
  • Proposition 2.1
  • Remark 2
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • ...and 4 more