Related papers: On a Commutative Ring of Two Variable Differential…

The free Baker-Akhiezer modules on rational varieties obtained from ${\mathbb C}P^{1}\times{\mathbb C}P^{n-1}$ by identification of two hypersurfaces are constructed. The corollary of this construction is the existence of embedding of…

In one variable, there exists a satisfactory classification of commutative rings of differential operators. In several variables, even the simplest generalizations seem to be unknown and in this report we give examples and pose questions…

We construct new examples of multidimensional commuting matrix differential operators and a multidimensional analog of the Kadomtsev--Petviashvili hierarchy.

In this paper we study one-point rank one commutative rings of difference operators. We find conditions on spectral data which specify such operators with periodic coefficients.

We give a natural generalization of the classification of commutative rings of ordinary differential operators, given in works of Krichever, Mumford, Mulase, and determine commutative rings of operators in a completed ring of partial…

A classification of commutative integral domains consisting of ordinary differential operators with matrix coefficients is established in terms of morphisms between algebraic curves.

Let R be a commutative ring and let n,m be two positive integers. Let be the polynomial ring in m x n commuting independent variables R. The symmetric group on n letters acts diagonally on A(n,m). We give generators and relations of the…

We show that the ring of multisymmetric functions over a commutative ring is isomorphic to the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices. As a consequence we give a…

Representations of polynomial covariance commutation relations by pairs of linear integral and differential operators are constructed in the space of infinitely continuously differentiable functions. Representations of polynomial covariance…

Several algebro-geometric properties of commutative rings of partial differential operators as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of…

We show that given two smooth affine varieties over $\mathbb{C}$ such that their rings of differential operators are Morita equivalent, then corresponding cotangent bundles are isomorphic as symplectic varieties.

We prove some algebraic results on the ring of matrix differential operators over a differential field in the generality of non-commutative principal ideal rings. These results are used in the theory of non-local Poisson structures.

For a rational function of several variables with nonnegative imaginary part on the upper poly-half-plane, the matrix representations are obtained.

In this paper we find the explicit formulas of two dimensional commuting ($2\times 2$)-matrix differential operators which were introduced by Nakayashiki. The common eigen functions and eigen values of these operators are parametrized by…

We study factorizations of rational matrix functions with simple poles on the Riemann sphere. For the quadratic case (two poles) we show, using multiplicative representations of such matrix functions, that a good coordinate system on this…

It is a long-established and heavily-used fact that the integral cohomology ring of the Deligne-Mumford moduli space of (complex) rational curves is the polynomial ring on the boundary divisors modulo the ideal generated by the obvious…

Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve with positive self-intersection. We prove that if there exists a non-constant meromorphic function on $F$, then the…

We study linear ordinary differential equations which are analytically parametrized on Hermitian symmetric spaces and invariant under the action of symplectic groups. They are generalizations of the classical Lam\'e equation. Our main…

Following the definition of quantum differential operators given by Lunts and Rosenberg in (Sel. math., New ser. 3 (1997) 335--359), we show that the ring of quantum differential operators on the affine line is the ring generated by x and…

Let $k$ be an algebraically closed field of characteristic 0 and let $A$ be a finitely generated $k$-algebra that is a domain whose Gelfand-Kirillov dimension is in $[2,3)$. We show that if $A$ has a nonzero locally nilpotent derivation…