An identity on algebras over a Hopf algebra

Abstract

Let A be a connected Hopf algebra which has an associative comultiplication ψ : A → A ⊗ A \psi :A \to A \otimes A . Let χ : A → A \chi :A \to A be the canonical conjugation on A. Let M be a graded algebra over the Hopf algebra A. If x , y ∈ M , ψ ( a ) = Σ a ′ ⊗ a x,y \in M,\psi (a) = \Sigma a’ \otimes a , then we have the identity \[ a x ⋅ y = Σ ( − 1 ) deg ⁡ x ⋅ deg ⁡ a a ′ ( x ⋅ χ ( a ) y ) . ax \cdot y = \Sigma {( - 1)^{\deg x \cdot \deg a}}a’(x \cdot \chi (a)y). \]