The Fundamental Theorems of Calculus and Zinbiel Algebras

Tuesday, 10-Sep-2024 

Abstract

In algebra, derivations and integrations generalize the differential operator and integral operator from calculus. Derivations are axiomatized by the Leibniz rule, while integrations are axiomatized by the Rota-Baxter rule, which an integral only version of integration by parts. On the other hand, Zinbiel algebras are a special kind of commutative non-unital associative algebras that capture the notion of riffle shuffle permutations. In this talk, I will explain how giving a Zinbiel algebra is equivalent to giving a derivation and integration which together satisfy the two fundamental theorems of calculus. This talk is based on my paper: https://arxiv.org/abs/2401.08223