Properties of the Exotic Characteristic Homomorphism for a Pair of Lie Algebroids, Relationship with the Koszul Homomorphism for a Pair of Lie algebras

TL;DR

This paper explores the properties of an exotic characteristic homomorphism for pairs of Lie algebroids, establishing its functoriality, homotopy invariance, and conditions for injectivity, with connections to the Koszul homomorphism for Lie algebras.

Contribution

It introduces a universal homomorphism for pairs of Lie algebroids and investigates its properties, including injectivity in special cases, linking to the Koszul homomorphism.

Findings

Proves the Rigidity Theorem for the homomorphism.

Establishes a factorization through a universal homomorphism.

Shows injectivity in specific cases related to Lie algebra pairs.

Abstract

We examine functorial and homotopy properties of the exotic characteristic homomorphism in the category of Lie algebroids which was lastly obtained by the authors in [4]. This homomorphism depends on a triple (A,B,$\nabla$) where B $\subset$ A are regular Lie algebroids, both over the same regular foliated manifold (M,F), and $\nabla$ is a flat L-connection in A, where L is an arbitrary Lie algebroid over M. The Rigidity Theorem (i.e. the independence from the choice of homotopic Lie subalgebroids of B) is obtained. The exotic characteristic homomorphism is factorized by one (called universal) obtained for a pair of regular Lie algebroids. We raise the issue of injectivity of the universal homomorphism and establish injectivity for special cases. Here the Koszul homomorphism for pairs of isotropy Lie algebras plays a major role.