Nonlinear Spencer operators on differentiable groupoids

TL;DR

This paper develops advanced nonlinear and linear Spencer complexes for differentiable Lie groupoids, extending diagonal calculus techniques to new contexts, which could impact the study of geometric structures and differential equations.

Contribution

It introduces the first comprehensive nonlinear and linear Spencer complexes for differentiable Lie groupoids, expanding the mathematical toolkit for geometric and algebraic analysis.

Findings

Constructed nonlinear Spencer complexes for Lie groupoids.

Extended diagonal calculus to the context of groupoid identities.

Provided a framework for analyzing differentiable groupoid structures.

Abstract

We construct the first, second and sophisticated non-linear and linear Spencer complexes for a differentiable Lie groupoid $G$. To do this, we extend the diagonal calculus, as applied by Malgrange to the groupoid $M\times M$, to the context of $I\times G$, where $I$ is the manifold of identities of $G$.