Connes–Moscovici characteristic map is a Lie algebra morphism

Let H be a Hopf algebra with a modular pair in involution (δ,1). Let A be a (module) algebra over H equipped with a non-degenerated δ-invariant 1-trace τ. We show that Connes–Moscovici characteristic map is a morphism of graded Lie algebras. We also have a morphism Φ of Batalin–Vilkovisky algebras from the cotorsion product of H, , 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 is injective.