submitted version. Corollary 28 and Section 9 has been added. Section 9 computes the Batalin-Vilkovisky algebra on the rational cotor of an universal envelopping algebra of a lie algebra.Let H be a Hopf algebra with a modular pair in involution (\Character,1). Let A be a (module) algebra over H equipped with a non-degenerated \Character-invariant 1-trace τ. We show that Connes-Moscovici characteristic map \varphi_\tau:HC^*_{(\Character,1)}(H)\rightarrow HC^*_\lambda(A) is a morphism of graded Lie algebras. We also have a morphism Φ of Batalin-Vilkovisky algebras from the cotorsion product of H, CotorH∗(k,k), to the Hochschild cohomology of A, HH∗(A,A). Let K be both a Hopf algebra and a symmetric Frobenius algebra. Suppose that the square of its antipode is an inner automorphism by a group-like element. Then this morphism of Batalin-Vilkovisky algebras Φ:CotorK∨∗(F,F)≅ExtK(F,F)↪HH∗(K,K) is injective