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 that may suggest a theory to be developed. In particular, we address the existence of a ``spectral variety'' generalizing the spectral curve of the one dimensional theory and the role of the differential resultant.
There is a relatively well-known description of the algebra of (higher order) left differential operators on commutative algebras. This note gives a construction of similar flavor for algebras of differential operators on not necessarily commutative algebras.
In this paper we establish a faithfulness result, in a homotopical sense, between a subcategory of the model category of augmented differential graded commutative algebras CDGA and a subcategory of the model category of augmented differential graded algebras DGA over the field of rational numbers $\mathbb{Q}$.
In this paper, we construct a canonical linear basis for free commutative integro-differential algebras by applying the method of Gröbner-Shirshov bases. We establish the Composition-Diamond Lemma for free commutative differential Rota-Baxter algebras of order $n$. We also obtain a weakly monomial order on these algebras, allowing us to obtain Gröbner-Shirshov bases for free commutative integro-differential algebras on a set. We finally generalize the concept of functional derivations to free differential algebras with arbitrary weight and generating sets from which to construct a canonical li…
In this survey, we outline two recent constructions of free commutative integro-differential algebras. They are based on the construction of free commutative Rota-Baxter algebras by mixable shuffles. The first is by evaluations. The second is by the method of Gröbner-Shirshov bases.
In this article, we show that for a partial skew group ring R*G, where R is a commutative ring, each non-zero ideal of R*G intersects R non-trivially if and only if R is a maximal commutative subring of R*G. As a consequence, we obtain necessary and sufficient conditions for simplicity; the partial skew group ring R*G is simple if and only if R is a G-simple and maximal commutative subring of R*G. We thereby generalize our previous results for skew group rings, to partial skew group rings which are not necessarily unital.
In this paper, the Composition-Diamond lemma for commutative algebras with multiple operators is established. As applications, the Gröbner-Shirshov bases and linear bases of free commutative Rota-Baxter algebra, free commutative $λ$-differential algebra and free commutative $λ$-differential Rota-Baxter algebra are given, respectively. Consequently, these three free algebras are constructed directly by commutative $Ω$-words.
We construct a cofibrantly generated model structure on the category of differential non-negatively graded quasi-coherent commutative $D_X$-algebras, where $D_X$ is the sheaf of differential operators of a smooth afine algebraic variety X. The paper contains an extensive appendix on D-modules, sheaves versus global sections, some more technical model categorical issues, as well as on relative Sullivan algebras. This article is the first of a series of works -located at the interface of homotopical algebra, algebraic geometry, and mathematical physics - on a derived D-geometric approach to the …
Let X be a singular affine normal variety with coordinate ring R and assume that there is an R-order admitting a stability structure such that the scheme of relevant semistable representations is smooth, then we construct a partial desingularization of X with classifiable remaining singularities. In dimension 3 this explains the omnipresence of conifold singularities in partial desingularizations of quotient singularities. In higher dimensions we have a small list of singularity types generalizing the role of the conifold singularity.
In this work, the cohomology theory for partial actions of co-commutative Hopf algebras over commutative algebras is formulated. This theory generalizes the cohomology theory for Hopf algebras introduced by Sweedler and the cohomology theory for partial group actions, introduced by Dokuchaev and Khrypchenko. Some nontrivial examples, not coming from groups are constructed. Given a partial action of a co-commutative Hopf algebra $H$ over a commutative algebra $A$, we prove that there exists a new Hopf algebra $\widetilde{A}$, over a commutative ring $E(A)$, upon which $H$ still acts partially a…
We use homological perturbation machinery specific for the algebra category [P. Real. Homological Perturbation Theory and Associativity. Homology, Homotopy and Applications vol. 2, n. 5 (2000) 51-88] to give an algorithm for computing the differential structure of a small 1--homological model for commutative differential graded algebras (briefly, CDGAs). The complexity of the procedure is studied and a computer package in Mathematica is described for determining such models.
This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action $α_G$ of a finite group $G$ on an algebra $S$ such that $S$ is an $α_G$-partial Galois extension of $S^{α_G}$ and a normal subgroup $H$ of $G$, we prove that $α_G$ induces a unital partial action $α_{G/H}$ of $G/H$ on the subalgebra of invariants $S^{α_H}$ of $S$ such that $S^{α_H}$ is an $α_{G/H}$-partial Galois extension of $S^{α_G}$. Second, assuming that $G$ is abelian, we construct a commutative inverse semigroup $T_{par}(G,R)$, whose elements are equivalence classes of $α_G$-part…
We show that a semi-commutative Galois extension of a unital associative algebra can be endowed with the structure of a graded q-differential algebra. We study the first and higher order noncommutative differential calculus of semi-commutative Galois extension induced by the graded q-differential algebra. As an example we consider the quaternions which can be viewed as the semi-commutative Galois extension of complex numbers.
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.
We study the relations between partial and global group cohomology with values in a commutative unital ring $\mathcal{A}$. In particular, for a unital partial action of a group $G$ on $\mathcal{A}$, such that $\mathcal{A}$ is a direct product of commutative indecomposable rings, we show that any partial $n$-cocycle of $G$ with values in $\mathcal{A}$ is globalizable.
The first part of the paper will describe a recent result of K. Retert in (\cite{Ret}) for $k[x_1,\ldots,x_n]$ and $k[[x_1,\ldots,x_n]]$. This result states that if $\mathfrak{D}$ is a set of commute $k$-derivations of $k[x,y]$ such that both $\partial_x \in \mathfrak{D}$ and the ring is $\mathfrak{D}$-simple, then there is $d \in \mathfrak{D}$ such that $k[x,y]$ is $\{\partial_x,d\}$-simple. As applications, we obtain relationships with known results of A. Nowicki on commutative bases of derivations.
We introduce a theory of cyclic Kummer extensions of commutative rings for partial Galois extensions of finite groups, extending some of the well-known results of the theory of Kummer extensions of commutative rings developed by A. Z. Borevich. In particular, we provide necessary and sufficient conditions to determine when a partial $n$-kummerian extension is equivalent to either a radical or a $I$-radical extension, for some subgroup $I$ of the cyclic group $C_n$.
We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables. We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various…
In this paper we undertake a basic study on connective commutative differential graded algebras (CDGA), more precisely, piecewise Noetherian CDGA, which is a DG-counter part of commutative Noetherian algebra. We establish basic results for example, Auslaner-Buchsbaum formula and Bass formula without any unnecessary assumptions. The key notion is the sup-projective (sppj) and inf-injective (ifij) resolutions introduced by the author, which are DG-versions of the projective and injective resolution for ordinary modules. These are different from DG-projective and DG-injective resolutions which is…
The aim of this article is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x;id,δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x;σ,δ] is a maximal commutative subring containing R and, in the case when σ=id, we show that it intersects every non-zero ideal of R[x;id,δ] non-trivially. Using this we show that if R is δ-simple and maximal commutative in R[x;id,δ], then R[x;id,δ] i…
Let $R$ be a commutative ring, $\mathcal A$ an $R$-algebra (not necessarily commutative) and $V$ an $R$-subspace or $R$-submodule of $\mathcal A$. By the radical of $V$ we mean the set of all elements $a\in \mathcal A$ such that $a^m\in V$ for all $m\gg 0$. We derive (and show) some necessary conditions satisfied by the elements in the radicals of the kernel of some (partial) differential operators, such as all differential operators of commutative algebras; the differential operators $P(D)$ of (noncommutative) $\mathcal A$ with certain conditions, where $P(\cdot)$ is a polynomial in $n$ commu…
A class of linear codes that extends classic Goppa codes to a non-commutative context is defined. An efficient decoding algorithm, based on the solution of a non-commutative key equation, is designed. We show how the parameters of these codes, when the alphabet is a finite field, may be adjusted to propose a McEliece-type cryptosystem.
