Engineering Mathematics

What Algebraic D-modules Are | KEVOS® Mathematics

A D-module is a module over a ring of differential operators. This page explains the dictionary between differential equations and modules over the Weyl algebra.

Engineering Mathematics9 Aug 2026 · 20 min read

Simplicity of the Weyl Algebra | KEVOS® Mathematics

The Weyl algebra has no two-sided ideals except zero and itself. The proof brackets a minimal-degree element down to a non-zero constant.

The Weyl Algebra and the Birth of Quantum Mechanics | KEVOS® Mathematics

How the Heisenberg commutation relation of 1925 became the Weyl algebra, why it admits no finite matrices, and how Dirac's quantum algebra survives in modern notation.

The Weyl Algebra in Positive Characteristic | KEVOS® Mathematics

Over a field of characteristic p the two definitions of the Weyl algebra give different rings. One has nilpotents, the other is not simple, and both have finite-dimensional modu…

Engineering Mathematics9 Aug 2026 · 20 min read

Why the Weyl Algebra Has No Finite-dimensional Representations | KEVOS® Mathematics

No non-zero finite-dimensional vector space carries an action of the Weyl algebra in characteristic zero. Three independent proofs, and exactly where each one uses the character…

The Weyl Algebra Is Noetherian | KEVOS® Mathematics

Every left ideal of the Weyl algebra is finitely generated. The proof lifts Hilbert's basis theorem across the Bernstein filtration from the graded ring.

Why the Weyl Algebra Is a Domain | KEVOS® Mathematics

The Weyl algebra has no zero divisors. The proof is one line from additivity of degree, and it fails in positive characteristic for an instructive reason.

The Weyl Algebra by Generators and Relations | KEVOS® Mathematics

The Weyl algebra is the free algebra on 2n generators modulo the canonical commutation relations, a presentation that gives it a universal property.

The Weyl Algebra Defined as a Ring of Operators | KEVOS® Mathematics

The n-th Weyl algebra is the algebra of operators on a polynomial ring generated by multiplication by each variable and differentiation with respect to it.

Proving the Canonical Basis Is a Basis | KEVOS® Mathematics

How the monomials x-alpha d-beta are proved to be a K-basis of the Weyl algebra: straightening gives spanning, and a test polynomial gives independence.

Commutation Relations in the Weyl Algebra | KEVOS® Mathematics

The relations between the generators of the Weyl algebra, how they follow from the product rule, and the identities and structural consequences they generate.

Canonical Form of an Element of the Weyl Algebra | KEVOS® Mathematics

Every element of the Weyl algebra is uniquely a finite sum of scalar multiples of monomials with all variables written to the left of all derivatives.

The Weyl Algebra as the Ring of Differential Operators on Affine Space | KEVOS® Mathematics

Proof that the ring of differential operators of a polynomial ring in characteristic zero is exactly the Weyl algebra, with both lemmas in full.

Twisting a Weyl Module by an Automorphism | KEVOS® Mathematics

Twisting a module by a ring automorphism keeps the underlying group and changes only the action, preserving simplicity and torsion while often changing the isomorphism class.

When Are Two Twisted Modules Isomorphic? | KEVOS® Mathematics

Twisting the polynomial module by automorphisms of the Weyl algebra yields simple modules that are isomorphic only when the twisting automorphisms agree.

Systems of Differential Equations and Presentation Matrices | KEVOS® Mathematics

Several unknown functions give a matrix of operators, a quotient of a free module over the Weyl algebra, and a solution space that is again a space of homomorphisms.

Solutions as Module Homomorphisms | KEVOS® Mathematics

The solutions of a linear system in a chosen space are exactly the module homomorphisms from the system module into that space, a bijection that is linear and functorial.

Why the Polynomial Ring Is a Simple Weyl Module | KEVOS® Mathematics

Over a field of characteristic zero every non-zero polynomial generates the whole polynomial ring as a module over the Weyl algebra, so the module is simple.

Rings of Differential Operators: The Inductive Definition | KEVOS® Mathematics

The intrinsic definition of the ring of differential operators of a commutative algebra, by induction on order through iterated commutators.

The Polynomial Ring as a Module over the Weyl Algebra | KEVOS® Mathematics

The polynomial ring carries a canonical action of the Weyl algebra by multiplication and differentiation, and is isomorphic to the quotient by the left ideal of the partials.

Polynomial Maps and the Jacobian Matrix | KEVOS® Mathematics

How polynomial maps between affine spaces correspond to algebra homomorphisms of polynomial rings, and what the Jacobian determinant detects.

Engineering Mathematics9 Aug 2026 · 19 min read

The Noetherian Property in Short Exact Sequences | KEVOS® Mathematics

A module is Noetherian exactly when a submodule and the corresponding quotient both are, which makes the class closed under extensions and finite sums.

Noetherian Modules and the Ascending Chain Condition | KEVOS® Mathematics

A module is Noetherian when every submodule is finitely generated, a condition equivalent to the ascending chain condition and the maximal condition.

Engineering Mathematics9 Aug 2026 · 18 min read

Multi-index Notation for Differential Operators | KEVOS® Mathematics

Multi-index notation compresses monomials and iterated partial derivatives into a single exponent vector, and it is the working language of the Weyl algebra.

Engineering Mathematics9 Aug 2026 · 27 min read

Locally Nilpotent Derivations | KEVOS® Mathematics

A derivation is locally nilpotent when some power of it kills every element. In characteristic zero such derivations exponentiate to automorphisms, and a slice makes them partia…

The Jacobian Conjecture | KEVOS® Mathematics

The conjecture that a polynomial self-map of affine space with Jacobian determinant one has a polynomial inverse, what is proved and what is open.

Engineering Mathematics9 Aug 2026 · 20 min read

Holonomic Modules Have Finite Length | KEVOS® Mathematics

Multiplicity is additive and positive, so a holonomic module is artinian as well as noetherian, has a composition series, and its length never exceeds its multiplicity.

Holonomic Modules: Definition and First Examples | KEVOS® Mathematics

A finitely generated module over the n-th Weyl algebra is holonomic when it is zero or has dimension exactly n, the least value Bernstein's inequality allows.

Holomorphic Functions as a Module over the Weyl Algebra | KEVOS® Mathematics

The holomorphic functions on an open subset of the complex plane form a Weyl module that is neither simple nor torsion, as the function exp(exp z) shows.

The Hilbert Function and the Hilbert Polynomial | KEVOS® Mathematics

The Hilbert function of a finitely generated graded module counts dimensions degree by degree, and for large arguments it agrees exactly with a polynomial.

Engineering Mathematics9 Aug 2026 · 26 min read

The Dixmier Conjecture and Its Link to the Jacobian Conjecture | KEVOS® Mathematics

Dixmier asked whether every endomorphism of the Weyl algebra is an automorphism. A yes would prove the Jacobian conjecture; the two are now known equivalent.

Direct Limits of Modules | KEVOS® Mathematics

The direct limit of a directed family of modules: its construction as a quotient of a disjoint union, its universal property, and why the limit is an exact operation.

Engineering Mathematics9 Aug 2026 · 27 min read

Computing a Direct Limit: Worked Constructions | KEVOS® Mathematics

How to identify a direct limit in practice: a recognition criterion, then germs of holomorphic functions, localisation and the microfunction system computed in full.

Differential Operators on an Affine Variety | KEVOS® Mathematics

What the ring of differential operators looks like on an affine variety, why smoothness makes it well behaved, and how the cusp breaks the naive picture.

Order of a Differential Operator and the Order Filtration | KEVOS® Mathematics

The order of a differential operator, the increasing filtration it defines, and why composition adds orders while commutators lose one.

Derivations of the Polynomial Ring | KEVOS® Mathematics

Every K-linear derivation of a polynomial ring is a polynomial vector field: it is determined by its values on the variables and equals the sum of those values times the partials.

Engineering Mathematics9 Aug 2026 · 26 min read

Derivations of a Commutative Ring | KEVOS® Mathematics

Derivations of a commutative K-algebra: the Leibniz rule, the module and Lie algebra structure, and why they are the differential operators of order one.

The Degree of a Differential Operator | KEVOS® Mathematics

The degree of an element of the Weyl algebra is the top total degree in its canonical form. It is additive on products and drops by two on commutators.

The D-module Attached to a Differential Equation | KEVOS® Mathematics

How a linear system of differential equations with polynomial coefficients becomes a cyclic module over the Weyl algebra: quotient by the ideal of its consequences.

Cyclic Weyl Modules and Their Presentations | KEVOS® Mathematics

A module generated by one element is a quotient of the Weyl algebra by a left ideal; this page shows how to read off that ideal and when it exists.

The Bernstein-Sato Polynomial | KEVOS® Mathematics

The Bernstein-Sato polynomial of p is the monic generator of the ideal of all b(s) admitting an operator D(s) with b(s) p^s = D(s) p^(s+1).

Bernstein's Inequality | KEVOS® Mathematics

Bernstein's inequality bounds the dimension of every non-zero finitely generated module over the n-th Weyl algebra between n and 2n.

Automorphisms of the Weyl Algebra | KEVOS® Mathematics

Every endomorphism of the Weyl algebra is injective because the algebra is simple, so the Dixmier conjecture is a surjectivity question, and it implies the Jacobian conjecture.

The Associated Graded Algebra | KEVOS® Mathematics

How a filtration of a ring produces a graded algebra of symbols, and why the Weyl algebra's Bernstein filtration yields a polynomial ring in 2n variables.

A Roadmap Through Algebraic D-module Theory | KEVOS® Mathematics

How the elementary theory of D-modules is organised: ring theory first, then invariants, then operations, and finally applications, with the dependencies made explicit.