Differential Homological Algebra Research Articles

Killing Operator for the Kerr Metric

When D: E → F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: Jq(E) → F=F0 that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D1: F0 → F1. When D is involutive, that is when the corresponding system Rq = ker (Φ) is involutive, this procedure provides successive first order involutive operators D1, ..., Dn. Though D1 ο D = 0 implies ad (D) ο ad(D1) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D1) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When Rq is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.

Differential Correspondences and Control Theory

When a differential field K having n commuting derivations is given together with two finitely generated differential extensions L and M of K, an important problem in differential algebra is to exhibit a common differential extension N in order to define the new differential extensions L∩M and the smallest differential field (L,M ) ⊂ N containing both L and M. Such a result allows to generalize the use of complex numbers in classical algebra. Having now two finitely generated differential modules L and M over the non-commutative ring D = K [d1,...,dn] = K [d] of differential operators with coefficients in K, we may similarly look for a differential module N containing both L and M in order to define L∩M and L+M. This is exactly the situation met in linear or non-linear OD or PD control theory by selecting the inputs and the outputs among the control variables. However, in many recent books and papers, we have shown that controllability was a built-in property of a control system, not depending on the choice of inputs and outputs. The purpose of this paper is thus to revisit control theory by showing the specific importance of the two previous problems and the part plaid by N in both cases for the parametrization of the control system. An important tool will be the study of differential correspondences, a modern name for what was called Bäcklund problem during the last century, namely the elimination theory for groups of variables among systems of linear or nonlinear OD or PD equations. The main difficulty is to revisit differential homological algebra by using noncommutative localization as a way to generalize the symbolic calculus in the style of Heaviside and Mikusinski. Finally, when M is a D-module, this paper is using for the first time the fact that the system R = homK (M,K) is a D-module for the Spencer operator acting on sections, avoiding thus behaviours, trajectories and signal spaces in a purely formal way, contrary to a few recent works on this difficult subject.

Periodicity and cyclic homology. Para-$S$-modules and perturbation lemmas

In this paper, we introduce a paracyclic version of $S$-modules. These new objects are called para-$S$-modules. Paracyclic modules and parachain complexes give rise to para-$S$-modules much in the same way as cyclic modules and mixed complexes give rise to $S$-modules. More generally, para-$S$-modules provide us with a natural framework to get analogues for paracyclic modules and parachain complexes of various constructions and equivalence results for cyclic modules or mixed complexes. The datum of a para-$S$-module does not provide us with a chain complex, and so notions of homology and quasi-isomorphisms do not make sense. However, chain maps and chain homotopies continue to make sense, and so by equivalences we actually mean chain homotopy equivalences. We establish some generalizations for para-$S$-modules and parachain complexes of the basic perturbation lemma of differential homological algebra. These generalizations provide us with general recipes for converting deformation retracts of Hochschild chain complexes into deformation retracts of para-$S$-modules. By using ideas of Kassel this then allows us to get comparison results between the various para-$S$-modules associated with para-precyclic modules, and between them and Connes’ cyclic chain complex. These comparison results lead us to alternative descriptions of Connes’ periodicity operator. This has some applications in periodic cyclic homology. We also describe the counterparts of these results in cyclic cohomology. In particular, we obtain an explicit way to convert a periodic $(b, B)$-cocycle into a cohomologous periodic cyclic cocycle.

Differential Homological Algebra and General Relativity

In 1916, F.S. Macaulay developed specific localization techniques for dealing with “unmixed polynomial ideals” in commutative algebra, transforming them into what he called “inverse systems” of partial differential equations. In 1970, D.C. Spencer and coworkers studied the formal theory of such systems, using methods of homological algebra that were giving rise to “differential homological algebra”, replacing unmixed polynomial ideals by “pure differential modules”. The use of “differential extension modules” and “differential double duality” is essential for such a purpose. In particular, 0-pure differential modules are torsion-free and admit an “absolute parametrization” by means of arbitrary potential like functions. In 2012, we have been able to extend this result to arbitrary pure differential modules, introducing a “relative parametrization” where the potentials should satisfy compatible “differential constraints”. We recently noticed that General Relativity is just a way to parametrize the Cauchy stress equations by means of the formal adjoint of the Ricci operator in order to obtain a “minimum parametrization” by adding sufficiently many compatible differential constraints, exactly like the Lorenz condition in electromagnetism. In order to make this difficult paper rather self-contained, these unusual purely mathematical results are illustrated by many explicit examples, two of them dealing with contact transformations, and even strengthening the comments we recently provided on the mathematical foundations of General Relativity and Gauge Theory. They also bring additional doubts on the origin and existence of gravitational waves.

Rational Fibrations in Differential Homological Algebra

Rational Fibrations in Differential Homological Algebra

Rational fibrations in differential homological algebra

In this paper, a result of [6] is generalized as follows: Given a fibration F → E → p B F \to E\xrightarrow {p}B of simply connected spaces in which either, the fibre has finite dimensional rational cohomology, or, it has finite dimensional rational homotopy and ρ \rho induces a surjection in rational homotopy, we construct an explicit isomorphism, \[ φ : Ext C ∗ ( B , Q ) ⁡ ( Q , C ∗ ( B ; Q ) ) ⊗ ^ Ext C ∗ ( F ; Q ) ⁡ ( Q , C ∗ ( F , Q ) ) → ≅ Ext C ∗ ( E ; Q ) ⁡ ( Q , C ∗ ( E ; Q ) ) . \begin {array}{*{20}{c}} {\varphi :\operatorname {Ext}_{{C^\ast }(B,{\mathbf {Q}})}({\mathbf {Q}},{C^\ast }(B;{\mathbf {Q}}))\hat \otimes \operatorname {Ext}_{{C^\ast }(F;{\mathbf {Q}})}({\mathbf {Q}},{C^\ast }(F,{\mathbf {Q}}))} \\ {\xrightarrow { \cong }\operatorname {Ext}_{{C^\ast }(E;{\mathbf {Q}})}(Q,{C^\ast }(E;{\mathbf {Q}})).} \\ \end {array} \] This is deduced from its "algebraic translation," a more general result in the environment of graded differential homological algebra.

Perturbation theory in differential homological algebra II

Perturbation theory in differential homological algebra II

Perturbation theory in differential homological algebra I

Perturbation theory in differential homological algebra I

Commutative rings, algebraic topology, graded lie algebras and the work of Jan-Erik Roos

The classic book, Homological Algebra, by Cartan and Eilenberg begins in the following way: ‘During the last decade the methods of algebraic topology have invaded extensive- ly the domain of pure algebra . . . The invasion has occurred on three fronts through the construction of cohomology theories for groups, Lie algebras and associative algebras . . . We present here a single cohomology (and also a homology) theory which embodies all three.’ A few lines later the authors mention Hilbert’s syzygy theorem as an application; this is one instance of the usefulness of these methods when applied to the study of commutative rings. A central role in Homological Algebra is played by the functors Tor and Ext. Ap- plied to a ring homomorphism R + k (k a field) they give a graded k-vector space Tor$(k, k) whose graded dual Ei = Ext,*(k, k) is, equipped with the Yoneda product, a graded algebra. With the introduction of differential homological algebra by Eilenberg and Moore [14] the functor Tor was extended to differential graded algebras (DGA’s). They showed, in particular, that for the cochain algebra C* of a pointed l-connected CW complex of finite type: Torc*(k,

Differential homological algebra and homogeneous spaces

Differential homological algebra and homogeneous spaces

Derived categories and motives

Ab stract. We relate derived categories of modules over rational DGA’s to categories of comodules over associated Hopf algebras, and we explain how this implies the equivalence of definitions of mixed Tate motives proposed by Bloch and Deligne. We also describe an approach to integral mixed Tate motives in terms of the derived categories of modules over certain E∞ algebras. We first explain some new differential homological algebra — alias rational homotopy theory — over a field of characteristic zero and then use it to show the equivalence of two proposed definitions of mixed Tate motives [8, 9, 5] in algebraic geometry. One of these has been proven to admit Hodge