Related papers: Foundations of Differential Calculus for modules o…

Persistence modules are representations of products of totally ordered sets in the category of vector spaces. They appear naturally in the representation theory of algebras, but in recent years they have also found applications in other…

Given a small category C, a C-module M is a functor from C to the category of finite-dimensional vector spaces over a field k. Associated to M is its local structure, given as a functor from C to the category of bi-closed multi-flags over…

Let $A$ be a finite-dimensional, commutative algebra over $\mathbb{R}$ or $\mathbb{C}$. The notion of $A$-differentiable functions on $A$ is extended to the notion of $A$-differentiable functions on a finitely generated $A$-module $B$. Let…

We define a functor from the category of Lie conformal algebras to the category of differential Lie coalgebras, which associates to any Lie conformal algebra $L$ a differential Lie coalgebra $L^{\,0}$, defined as the maximal good…

Given a finite group scheme $\cG$ over an algebraically closed field $k$ of characteristic $\Char(k)=p>0$, we introduce new invariants for a $\cG$-module $M$ by associating certain morphisms $\deg^j_M : U_M \lra \Gr_d(M) \ \…

We consider the 3-category $2\mathfrak{C}at$ whose objects are 2-categories, 1-morphisms are lax functors, 2-morphisms are lax transformations and 3-morphisms are modifications. The aim is to show that it carries interesting…

Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or…

The problem of formulating a correct notion of Laplacian on compact quantum groups (CQGs) has long been recognized as both fundamental and nontrivial. Existing constructions typically rely on selecting a specific first-order differential…

We give an algebraic description of several modules and algebras related to the vector partition function, and we prove that they can be realized as the equivariant K-theory of some manifolds that have a nice combinatorial description. We…

We show that differential calculus (in its usual form, or in the general form of topological differential calculus) can be fully imdedded into a functor category (functors from a small category of anchord tangent algebras to anchored sets).…

The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H.…

Over-extended Kac-Moody algebras contain so-called gradient structures - a gl(d)-covariant level decomposition of the algebra contains strings of modules at different levels that can be interpreted as spatial gradients. We present an…

Let $\mathcal{A}$ denote a real, $n$-dimensional, unital, associative algebra.This paper provides an introductory exposition of calculus over $\mathcal{A}$. An $\mathcal{A}$-differentiable function is one for which the differential is…

For coalgebras $C$ over a field, we study when the categories ${}^C\Mm$ of left $C$-comodules and $\Mm^C$ of right $C$-comodules are symmetric categories, in the sense that there is a duality between the categories of finitely presented…

A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a…

Let M be a smooth manifold, and let O(M) be the poset of open subsets of M. Manifold calculus, due to Goodwillie and Weiss, is a calculus of functors suitable for studying contravariant functors (cofunctors) F: O(M)--> Top from O(M) to the…

Let ${\cal D}^k$ be the space of $k$-th order linear differential operators on ${\bf R}$: $A=a_k(x)\frac{d^k}{dx^k}+\cdots+a_0(x)$. We study a natural 1-parameter family of $\Diff(\bf R)$- (and $\Vect(\bf R)$)-modules on ${\cal D}^k$. (To…

We propose a sheaf-theoretic approach to the theory of differential calculi on quantum principal bundles over non-affine bases. After recalling the affine case we define differential calculi on sheaves of comodule algebras as sheaves of…

The main goals of this paper are: i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without…

We pose a new algebraic formalism for studying differential calculus in vector bundles. This is achieved by studying various functors of differential calculus over arbitrary graded commutative algebras (DCGCA) and applying this language to…