Integration of quadratic Lie algebroids to Riemannian Cartan-Lie groupoids

TL;DR

This paper establishes the conditions under which positive quadratic Lie algebroids, equipped with compatible Riemannian metrics, can be integrated into Riemannian Cartan-Lie groupoids with bi-invariant metrics.

Contribution

It provides a characterization of when quadratic Lie algebroids with ad-invariant metrics can be integrated into Riemannian Cartan-Lie groupoids, extending the theory of Lie algebroid integration.

Findings

Derived necessary and sufficient conditions for integration.

Defined Riemannian Cartan-Lie groupoids with bi-invariant metrics.

Connected metric properties of algebroids and groupoids.

Abstract

Cartan-Lie algebroids, i.e. Lie algebroids equipped with a compatible connection, permit the definition of an adjoint representation, on the fiber as well as on the tangent of the base. We call (positive) quadratic Lie algebroids, Cartan-Lie algebroids with ad-invariant (Riemannian) metrics on their fibers and base $\kappa$ and $g$, respectively. We determine the necessary and sufficient conditions for a positive quadratic Lie algebroid to integrate to a Riemmanian Cartan-Lie groupoid. Here we mean a Cartan-Lie groupoid $\mathcal{G}$ equipped with a bi-invariant and inversion invariant metric $\eta$ on $T\mathcal{G}$ such that it induces by submersion the metric $g$ on its base and its restriction to the $t$-fibers coincides with $\kappa$.