# Connes-Moscovici characteristic map is a Lie algebra morphism

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1 LAREMA - Laboratoire Angevin de REcherche en MAthématiques

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 $\varphi \tau:HC^* {\Character,1}H ightarrow HC^* \lambdaA$ 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 $A$, $HH^*A,A$. 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 $\Phi:\text{Cotor} {K^\vee}^*\mathbb{F},\mathbb{F}\cong \text{Ext} {K}\mathbb{F},\mathbb{F} \hookrightarrow HH^*K,K$ is injective.

Keywords : Batalin-Vilkovisky algebra Hochschild cohomology cyclic cohomology Hopf algebra Frobenius algebra

Author: ** Luc Menichi - **

Source: https://hal.archives-ouvertes.fr/