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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  \[    a x ⋅  y = Σ  ( −  1 ) deg ⁡  x ⋅  deg ⁡  a a ′ ( x ⋅  χ  ( a ) y ) . ax \cdot y = \Sigma {( - 1)^{\deg x \cdot \deg a}}a’(x \cdot \chi (a)y).     \]

An identity on algebras over a Hopf algebra