# 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 $\varphi_\tau:HC^*_{(\Character,1)}(H)\rightarrow HC^*_\lambda(A)$ 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 $A$, $HH^*(A,A)$. 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 :
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-00454042
Contributor : Luc Menichi <>
Submitted on : Thursday, June 17, 2010 - 11:25:04 AM
Last modification on : Monday, March 9, 2020 - 6:15:57 PM
Long-term archiving on: : Thursday, September 23, 2010 - 5:51:22 PM

### Files

Ext_Hochschild.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-00454042, version 3
• ARXIV : 1002.1771

### Citation

Luc Menichi. Connes-Moscovici characteristic map is a Lie algebra morphism. 2010. ⟨hal-00454042v3⟩

Record views