# Universal Cartan-Lie algebroid of an anchored bundle with connection and compatible geometries

## TL;DR

This paper constructs a universal Cartan-Lie algebroid extension of an anchored bundle with connection, ensuring compatibility with geometrical structures and invariance of tensor fields, generalizing Kapranov's work to include connections.

## Contribution

It adapts Kapranov's universal Lie algebroid construction to anchored bundles with arbitrary connections, establishing a unique compatible connection and invariance properties of tensors.

## Key findings

- Existence of a unique connection  on the universal Lie algebroid compatible with the original connection.
- Universal property: morphisms to Cartan-Lie algebroids factor uniquely through the universal extension.
- Tensor fields compatible with the anchored bundle are invariant under the associated Lie algebroid representation.

## Abstract

Consider an anchored bundle $(E,\rho)$, i.e. a vector bundle $E\to M$ equipped with a bundle map $\rho \colon E \to TM$ covering the identity. M.~Kapranov showed in the context of Lie-Rinehard algebras that there exists an extension of this anchored bundle to an infinite rank universal free Lie algebroid $FR(E)\supset E$. We adapt his construction to the case of an anchored bundle equipped with an arbitrary connection, $(E,\nabla)$, and show that it gives rise to a unique connection $\tilde \nabla$ on $FR(E)$ which is compatible with its Lie algebroid structure, thus turning $(FR(E), \tilde \nabla)$ into a Cartan-Lie algebroid. Moreover, this construction is universal: any connection-preserving vector bundle morphism from $(E,\nabla)$ to a Cartan-Lie Algebroid $(A,\bar \nabla)$ factors through a unique Cartan-Lie algebroid morphism from $(FR(E), \tilde \nabla)$ to $(A,\bar \nabla)$. Suppose that, in addition, $M$ is equipped with a geometrical structure defined by some tensor field $t$ which is compatible with $(E,\rho,\nabla)$ in the sense of being annihilated by a natural $E$-connection that one can associate to these data. For example, for a Riemannian base $(M,g)$ of an involutive anchored bundle $(E,\rho)$, this condition implies that $M$ carries a Riemannian foliation. %In general, the compatibility of a tensor $t$ with $(E,\rho,\nabla)$ implies its adequate invariance transversal to $\rho(E)$. It is shown that every $E$-compatible tensor field $t$ becomes invariant with respect to the Lie algebroid representation associated canonically to the Cartan-Lie algebroid $(FR(E), \tilde \nabla)$.