Connes-Moscovici characteristic map is a Lie algebra morphism

Abstract

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 $φ_τ:HC^*_{(\Character,1)}(H)\rightarrow HC^*_λ(A)$ is a morphism of graded Lie algebras. We also have a morphism $\Phi$ of Batalin-Vilkovisky algebras from the cotorsion product of $H$, ${\text{Cotor}}_H^{\ast }(\mathbb{k},\mathbb{k})$, to the Hochschild cohomology of $A$, $H{H}^{\ast }(A,A)$. 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 $\Phi :{\text{Cotor}}_{K^{\vee }}^{\ast }(\mathbb{F},\mathbb{F})\cong {\text{Ext}}_K(\mathbb{F},\mathbb{F})\hookrightarrow H{H}^{\ast }(K,K)$ is injective.

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