# Leibniz rule and Alexander-Whitney coproduct

Is there anything more than a superficial similarity between the following?

• The Alexander-Whitney coproduct $\Delta$ on the tensor algebra $\bigotimes^\bullet V$ of a vector space $V$ is defined by the equalities $$\Delta(1) = 1,\quad \Delta(v) = v\otimes 1 + 1\otimes v$$ and compatibility with the product $\nabla = {-}\otimes{-}\,$.

• A derivation $D$ on a (unital associative) algebra $(A,\nabla,\eta)$ is defined as a linear map $A\to A$ satisfying the Leibniz rule $$D\circ\nabla =\nabla\circ (D\otimes\mathrm{id} +\mathrm{id}\otimes D);$$ dually, a coderivation $D$ on a coalgebra $(C,\Delta,\epsilon)$ is a linear map $C\to C$ satisfying the co-Leibniz rule $$\Delta\circ D = (D\otimes\mathrm{id} +\mathrm{id}\otimes D)\circ\Delta\,.$$

• A subspace $I\subseteq C$ of a coalgebra $(C,\Delta,\epsilon)$ is a coideal if $$\epsilon(I) = 0,\quad \Delta(I)\subseteq I\otimes C + C\otimes I.$$

• At least for derivations, as far as I can recall, what happens is that the coproduct you chose gives you an iterativity rule. If you replace it with something else, you get a different notion of derivation, which no longer follows the Leibniz rule. I vaguely recall them being studied in context of group schemes, but I don't have time to check details now. I imagine it is similar with the coideal. Nov 9 '17 at 5:08
• @tomasz I don’t know what an iterativity rule is—could you give some references? (Also, yes, the question about derivations basically is “what does the tensor coalgebra structure have to do with this?”, but so far I don’t see what.) Nov 9 '17 at 16:36
• Again, I don't remember the details, I've heard about it a few times in the past but never did any work related to it. You can try looking up Hasse-Schmidt derivations for connections between Hopf algebras and derivations. Nov 12 '17 at 3:57