Simplicity of partial and Schmidt differential operator rings

On the Differential Simplicity of Polynomial Rings

Commutative differentially simple rings have proved to be quite useful as a source of examples in non-commutative algebra. In this paper we use the theory of holomorphic foliations to construct new families of derivations with respect to which the polynomial ring over a field of characteristic zero is differentially simple.

Simplicity of Rings of Differential Operators in Prime Characteristic

LetW be a finite dimensional representation of a linearly reductive group G over a field k. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under G of the symmetric algebra of W has a simple ring of differential operators. In this paper, we show that this is true in prime characteristic. Indeed, if R is a graded subring of a ...

Differential Polynomial Rings of Triangular Matrix Rings

Operator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries

The operator-Schmidt decomposition is useful in quantum information theory for quantifying the nonlocality of bipartite unitary operations. We construct a family of unitary operators on C ⊗ C whose operatorSchmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on C ⊗ C with operatorSchmidt number S for every S ∈ {1, ..., 9}. This corollary...

Localization and Catenarity in Iterated Differential Operator Rings

Let k be a field, R an associative k-algebra with identity, a finite set of derivations of R, and R 1 1 · · · n n an iterated differential operator k-algebra over R such that j i ∈ R 1 1 · · · i−1 i−1 1 ≤ i < j ≤ n. If R is Noetherian -hypercentral, then every prime ideal P of A is classically localizable. The aim of this article is to show that under some additional hypotheses on the -prime id...