How to discretize the differential forms on the interval

English

Bandiera, Ruggero[Università degli Studi di Roma "La Sapienza" > Department of Mathematics]

Schätz, Florian[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]

2016

29

No

[en] homotopical algebra ; A-infty algebras ; Magnus expansion

[en] We provide explicit quasi-isomorphisms between the following three algebraic structures associated to the unit interval: i) the commutative dg algebra of differential forms, ii) the non-commutative dg algebra of simplicial cochains and iii) the Whitney forms, equipped with a homotopy commutative and homotopy associative, i.e. C-∞, algebra structure. Our main interest lies in a natural `discretization' C-∞ quasi-isomorphism φ from differential forms to Whitney forms. We establish a uniqueness result that implies that φ coincides with the morphism from homotopy transfer, and obtain several explicit formulas for φ, all of which are related to the Magnus expansion. In particular, we recover combinatorial formulas for the Magnus expansion due to Mielnik and Plebanski.