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 $\tau$. We show that Connes-Moscovici
characteristic map $\varphi_\tau:HC^*_{(\Character,1)}(H)\to HC^*_\lambda(A)$
is a morphism of graded Lie algebras. We also have a morphism $\Phi$ of
Batalin-Vilkovisky algebras from the cotorsion product of $H$,
$\text{Cotor}_H^*({\Bbbk},{\Bbbk})$, to the Hochschild cohomology of $A$,
$HH^*(A,A)$.