Remark

(1) Let $E$ be a graded bigebra over $A$ and $N, N', N''$ three graded $A$-modules. Let $m$ be an $A$-bilinear mapping of $N \times N'$ into $N''$; for $u \in \mathrm{Homgr}_A(E, N)$ and $v \in \mathrm{Homgr}_A(E, N')$, let $u.v$ denote the graded homomorphism $m \circ (u \otimes v) \circ c$ of $E$ into $N''$. On the other hand, let $i(x)$ denote the (right or left) inner product by $x \in E$ in the $A$-modules $\mathrm{Homgr}_A(E, N)$, $\mathrm{Homgr}_A(E, N')$ and $\mathrm{Homgr}_A(E, N'')$. Then, if $x$ is of degree 1,

for all $u \in \mathrm{Homgr}_A(E, N)$ and $v \in \mathrm{Homgr}_A(E, N')$.

Under the same conditions, if $E$ is a skew graded bigebra and $u$ is homogeneous of degree $p$, then

The proofs are the same as in Proposition 9.

(2) The same argument as in the above proof proves, more generally, that for all $x \in E$, if $c(x) = \sum_j x'_j \otimes x''_j$, then, for all $u, v$ in $E^{*gr}$, the "Leibniz formula"

holds. In particular, for every primitive element of a graded bigebra $E$, that is such that $c(x) = x \otimes 1 + 1 \otimes x$, $i(x)$ is a derivation of $E^{*gr}$.