The Koszul homomorphism for a pair of Lie algebras in the theory of exotic characteristic classes of Lie algebroids

- Functoriality and homotopy properties of the exotic characteristic homomorphism for Lie algebroids.
- The Koszul homomorphism versus the universal exotic characteristic homomorphism.
- Monomorphicity of the universal exotic characteristic homomorphism for a pair of integrable Lie algebroids.
- Monomorphicity of the universal exotic characteristic homomorphism for non-integrable Lie algebroids of TC-foliations.

We examine fundamental properties of the universal exotic characteristic homomorphism in the category of Lie algebroids, introduced by the authors in Balcerzak and Kubarski (2012) [4]. The properties under study include: (a) functorial properties with respect to arbitrary morphisms of Lie algebroids, (b) homotopy properties, (c) relationships with the Koszul homomorphism for a pair of isotropy Lie algebras, (d) conditions under which the universal exotic characteristic homomorphism is a monomorphism.