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

Abstract

Consider an anchored bundle $(E,\rho )$, i.e. a vector bundle $E\to M$ equipped with a bundle map $\rho :E\to TM$ 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 $(FR(E),\tilde{\nabla })$.

8+1 pages. This is an extended version of the original preprint arXiv:1603.04490, which was split into two parts for publication. The first, also slightly extended part, is available as a replacement under that number and has been published in Rev. Math. Phys. 31 (2019) 1950015

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• Lie Algebroid Yang Mills with Matter Fields