Connes–Moscovici characteristic map is a Lie algebra morphism

Abstract

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 φ τ : H C ( δ , 1 ) ⁎ ( H ) → H C λ ⁎ ( A ) is a morphism of graded Lie algebras. We also have a morphism Φ of Batalin–Vilkovisky algebras from the cotorsion product of H, Cotor H ⁎ ( k , k ) , to the Hochschild cohomology of A, H H ⁎ ( 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 Φ : Cotor K ∨ ⁎ ( F , F ) ≅ Ext K ( F , F ) ↪ H H ⁎ ( K , K ) is injective.