Let $A$ be a bialgebra over a field $k$. The space $A^* = \operatorname{Hom}(A, k)$ possesses an algebra structure, given by the convolution product $f*g = f \otimes g \circ \Delta$. Let $\gamma \in A^*$ be an idempotent, i.e. $\gamma * \gamma = \gamma$, i.e. $\gamma$ is a comultiplicative map from $A$ to $k$ (but not necessarily counital).
We could call $\delta \in \operatorname{End}_k(A)$ satisfying \begin{align*} \delta(ab) = \delta(a)b + \gamma(a_1) a_2 \delta(b) \end{align*} a "derivation up to $\gamma$" or something like that. Here I'm using a sumless Sweedler-type notation.
Now I want to put another condition $(P_\gamma)$, which is: $\delta$ should satisfy $$ \gamma(a_1) \delta(a_2) = \gamma(\delta(a_1)) \delta(a_2)\ . $$
With these conditions one computes:
Lemma. The usual commutator turns the space of derivations up to $\gamma$ satisfying condition $(P_\gamma)$ into a Lie-algebra
I have two (dependent) questions:
- Is this trivial? I've only checked in one example, namely the group Hopf algebra of the cyclic group of order 2. There is one non-trivial choice for $\gamma$, and then there's only one endomorphism satisfying $(P_\gamma)$, but it is not a "derivation up to $\gamma$".
I can't see why it should be trivial in general.
- If it's not trivial, then people must have thought about it before. Can anyone point me to a reference?
PS: Without writing out all my thoughts, the condition for $\gamma$ to be an idempotent could probably be replaced to $\gamma$ being central in $A^*$. But that leads to another structure.
Edit: I have computed for some other group algebras of cyclic groups, and just finished a computation in Sweedler's Hopf algebra. The space of endomorphisms satisfying both conditions was always zero (I have not tried each idempotent of the dual, though). This saddens me.
