Integration of quadratic Lie algebroids to Riemannian Cartan-Lie 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 $\kappa$.

Accepted for publication in Letters in Mathematical Physics

References in corpus (5)

• Lectures on Integrability of Lie Brackets
• Lie algebroid morphisms, Poisson Sigma Models, and off-shell closed gauge symmetries
• Lie Algebroid Yang Mills with Matter Fields
• Riemannian metrics on differentiable stacks
• Cartan Connections on Lie Groupoids and their Integrability

Cited by in corpus (2)

• Lie algebroids, gauge theories, and compatible geometrical structures
• On the relation of Lie algebroids to constrained systems and their BV/BFV formulation