Modular Classes of Loday Algebroids

We introduce the concept of Loday algebroids, a generalization of Courant algebroids. We define the naive cohomology and modular class of a Loday algebroid, and we show that the modular class of the double of a Lie bialgebroid vanishes. For Courant algebroids, we describe the relation between the naive and standard cohomologies and we conjecture that they are isomorphic when the Courant algebroid is transitive. 1 Naive Cohomology Given a Courant algebroid (E, ρ, J·, ·K, 〈·, ·〉), let Γ(∧ ker ρ) denote the space of smooth sections of the (possibly singular) vector bundle ∧ ker ρ (i.e. smooth sections α of ∧E such that α|m ∈ ∧ k ker ρ for each m ∈ M). The extension of the pseudo-metric 〈·, ·〉 to ∧E naturally induces an isomorphism Ξ : ∧E → ∧E. Since, by definition, 〈Df, e〉 = 1 2 ρ(e)f , the sections of ∧ ker ρ are characterized as the elements ε ∈ Γ(∧E) such that ı̆Dfε = 0, ∀f ∈ C(M). Here ı̆Df = Ξ −1 ◦iDf ◦Ξ, where iDf : Γ(∧ E) → Γ(∧E) is the usual contraction of exterior forms with the section Df ∈ Γ(E). Define an operator d̆ : Γ(∧ ker ρ) → Γ(∧E) by (d̆α)(e0, · · · , ek) = k ∑ i=0 (−1)ρ(ei)α(e0, · · · , êi, · · · , ek)