2025/03/11 by Žan Grad, Grad, Žan · 2 citations
Chemistry · Mathematics · #Axial and Atropisomeric Chirality Synthesis #Molecular spectroscopy and chirality #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2503.08873
For a Lie groupoid G, the differential forms on its nerve comprise a double complex. A natural question is if this statement extends to forms with values in a representation V of G. In this paper, we research two types of covariant derivatives which commute with the simplicial differential, yielding two types of "curved" double complexes of forms with coefficients in V. The naive approach is to consider a linear connection ∇ on V, in which case d^∇ commutes with the simplicial differential if and only if ∇ satisfies a certain (restrictive) invariance condition. The heart of this paper focuses on another, more compelling approach: using a multiplicative Ehresmann connection for a bundle of ideals. In this case, we obtain a geometrically richer curved double complex, where the cochain map is given by the horizontal exterior covariant derivative D, which generalizes the well-known operator from the theory of principal bundles. Moreover, both differential operators d^∇ and D are researched in the infinitesimal setting of Lie algebroids, as well as their relationship with the van Est map. We conclude by using the operator D to study the curvature of an (infinitesimal) multiplicative Ehresmann connection.