Fundamental Theorems of Calculus and Zinbiel Algebras
Jean-Simon Pacaud Lemay
TL;DR
This work develops an algebraic framework for the Fundamental Theorems of Calculus by introducing FTC-pairs $(\mathsf{D}: A \to M, \mathsf{P}: M \to A)$ with $\mathsf{D} \circ \mathsf{P} = \mathrm{id}_M$ and $\mathsf{P}(\mathsf{D}(a)) = a - c_a$ for some $c_a \in \ker(\mathsf{D})$ satisfying $c_{ab} = c_a c_b$. It establishes a categorical equivalence between the FTC-pair category and the category of Zinbiel algebras, with augmented FTC-pairs corresponding to $k$-Zinbiel algebras, unifying derivations, integrations, and Zinbiel structures. The paper also connects FTC-pairs to integro-differential algebras and Rota-Baxter theory, and provides diverse constructions and examples (polynomials, smooth functions, Hurwitz series, shuffle algebras) demonstrating the breadth of the framework. It further suggests extensions to differential categories and noncommutative contexts (via dendriform algebras), offering a new perspective on antiderivatives and algebraic calculus. Overall, the results reframe Zinbiel algebras as module-generalized FTC-algebras and pave the way for algebraic and categorical treatments of integration and differentiation.
Abstract
Derivations are linear operators which satisfy the Leibniz rule, while integrations are linear operators which satisfy the Rota-Baxter rule. In this paper, we introduce the notion of an FTC-pair, which consists of an algebra and module with a derivation and integration between them which together satisfy analogues of the two Fundamental Theorems of Calculus. In the special case of when an algebra is seen as a module over itself, we show that this sort of FTC-pair is precisely the same thing as an integro-differential algebra. We provide various constructions of FTC-pairs, as well as numerous examples of FTC-pairs including some based on polynomials, smooth functions, Hurwitz series, and shuffle algebras. Moreover, it is well known that integrations are closely related to Zinbiel algebras. The main result of this paper is that the category of FTC-pairs is equivalent to the category of Zinbiel algebras.