Differential operators on commutative algebras

In [6], J. Dixmier posed six problems for the Weyl algebra A1 over a field K of characteristic zero. Problems 3, 5,and 6 were solved respectively by Joseph and Stein [7]; the author [1]; and Joseph [7]. Problems 1, 2, and 4 are still open. For an arbitrary algebra A, Dixmier's problem 6 is essentially aquestion: whether an inner derivation of the algebra A of the type ad f(a), a ∈ A, f(t) ∈ K[t], deg t(f(t)) > 1, has a nonzero eigenvalue. We prove that the answer is negative for many classes of algebras (e.g., rings of differential operators [Formula: see text] on smooth irreducible algebraic varieties, all prime factor algebras of the universal enveloping algebra [Formula: see text] of a completely solvable algebraic Lie algebra [Formula: see text]). This gives an affirmative answer (with a short proof) to an analogue of Dixmier's Problem 6 for certain algebras of small Gelfand–Kirillov dimension, e.g. the ring of differential operators [Formula: see text] on a smooth irreducible affine curve X, Usl(2), etc. (see [3] for details). In this paper an affirmative answer is given to an analogue of Dixmier's Problem 3 but for the ring [Formula: see text].

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CATEGORY OF VECTOR BUNDLES ON ALGEBRAIC CURVES AND INFINITE DIMENSIONAL GRASSMANNIANS

International Journal of Mathematics ◽

10.1142/s0129167x90000174 ◽

1990 ◽

Vol 01 (03) ◽

pp. 293-342 ◽

Cited By ~ 35

Author(s):

MOTOHICO MULASE

Keyword(s):

Vector Bundles ◽

Differential Operators ◽

Algebraic Curves ◽

Classical Problem ◽

Vector Spaces ◽

Line Bundles ◽

Contravariant Functor ◽

Arbitrary Vector ◽

Infinite Dimensional ◽

Commutative Algebras

Equivalence between the following categories is established: 1) A category of arbitrary vector bundles on algebraic curves defined over a field of arbitrary characteristic, and 2) a category of infinite dimensional vector spaces corresponding to certain points of Grassmannians together with their stabilizers. Our contravariant functor between these categories gives a full generalization of the well-known Krichever map, which assigns points of Grassmannians to the geometric data consisting of curves and line bundles. As an application, a solution to the classical problem of Wallenberg-Schur of classifying all commutative algebras consisting of ordinary differential operators is obtained. It is also shown that the KP flows produce all generic vector bundles on arbitrary algebraic curves of genus greater than one.

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Krall–Jacobi commutative algebras of partial differential operators

Journal de Mathématiques Pures et Appliquées ◽

10.1016/j.matpur.2011.03.001 ◽

2011 ◽

Vol 96 (5) ◽

pp. 446-461 ◽

Cited By ~ 18

Author(s):

Plamen Iliev

Keyword(s):

Differential Operators ◽

Partial Differential ◽

Commutative Algebras ◽

Partial Differential Operators

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Gaudin subalgebras and stable rational curves

Compositio Mathematica ◽

10.1112/s0010437x11005306 ◽

2011 ◽

Vol 147 (5) ◽

pp. 1463-1478 ◽

Cited By ~ 11

Author(s):

Leonardo Aguirre ◽

Giovanni Felder ◽

Alexander P. Veselov

Keyword(s):

Lie Algebra ◽

Differential Operators ◽

Dimensional Space ◽

Maximal Dimension ◽

Central Character ◽

Genus Zero ◽

Stable Curves ◽

First Order ◽

Commutative Algebras ◽

Marked Points

AbstractGaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno–Drinfeld Lie algebra $\Xmathfrak {t}_{\hspace *{.3pt}n}$. We show that Gaudin subalgebras form a variety isomorphic to the moduli space $\bar M_{0,n+1}$ of stable curves of genus zero with n+1 marked points. In particular, this gives an embedding of $\bar M_{0,n+1}$ in a Grassmannian of (n−1)-planes in an n(n−1)/2-dimensional space. We show that the sheaf of Gaudin subalgebras over $\bar M_{0,n+1}$ is isomorphic to a sheaf of twisted first-order differential operators. For each representation of the Kohno–Drinfeld Lie algebra with fixed central character, we obtain a sheaf of commutative algebras whose spectrum is a coisotropic subscheme of a twisted version of the logarithmic cotangent bundle of $\bar M_{0,n+1}$.

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