Characteristic Classes of Lie Algebroid Morphisms

We extend R. Fernandes’ construction of the secondary characteristic classes of a Lie algebroid to the case of a base-preserving morphism between two Lie algebroids. Like in the case of a Lie algebroid, the simplest characteristic class of our construction coincides with the modular class of the morphism. In [4] R. Fernandes has constructed a sequence of secondary characteristic classes of a Lie algebroid whose first element coincides with the modular class. In this note we extend Fernandes’ construction and use the general definition of D. Lehmann [9] in order to produce secondary characteristic classes of a basepreserving morphism of two Lie algebroids. In particular, like in [4], we get a sequence of secondary characteristic classes whose first element coincides with the modular class of the morphism [5, 6]. We assume that the reader is familiar with Lie algebroids and Lie-algebroid connections and will consult [4, 10, 9, 11] whenever needed. The framework of the paper is the C-category. We mention that other constructions of secondary characteristic classes of Lie algebroids may also be found in the literature e.g., [1, 8]. 1 Selected topics on A-Connections Let (A, ♯A, [ , ]A) be a Lie algebroid and V a vector bundle with the same base manifold M (m = dimM). By an A-connection we shall understand an Acovariant derivative ∇ : ΓA× ΓV → ΓV (Γ denotes the space of cross sections of a vector bundle), written as (a, v) 7→ ∇av, which is R-bilinear and has the properties (1.1) ∇fav = f∇av, ∇a(fv) = f∇av + ♯Aa(f)v (f ∈ C (M)). Accordingly, the value ∇av(x) depends only on a(x) and on v|Ux where Ux is a neighborhood of x ∈ M . In order to write down the local expression of ∇, 2000 Mathematics Subject Classification: 53D17, 57R20.