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## Reference for the fact that a coderivation of the (non reduced) tensor coalgebra is determined by its corestrictions.

If $V$ is a vector space, let us consider the tensor coalgebra $TV=\oplus_{k=0}^\infty V^{\otimes^k}$ with coproduct given by

$$\Delta (x_1\otimes \dots \otimes x_n):= \sum_{i=0}^{n}(x_1\otimes\dots\otimes x_{i-1})\bigotimes (x_i\otimes\dots\otimes x_{n}).$$

Its reduced version is $\overline{TV}=\oplus_{k=1}^\infty V^{\otimes^k}$ with coproduct $\bar{\Delta}$ given by replacing $1$ by $0$ in the previous formula. In other words: $$\Delta (x)=\bar{\Delta}(x)+1\otimes x + x\otimes 1.$$

It is well known that a coderivation of $\overline{TV}$, i.e $Q\in L(\overline{TV},\overline{TV})$ satisfying $(Q\otimes Id+Id\otimes Q)\circ \Delta=\Delta \circ Q$ is determined by its corestrictions, i.e. by $\Pi_V\circ Q$, where $\Pi_V\colon \overline{TV}\rightarrow V$ is the canonical projection on $V$.

What I am looking for is a reference for a proof of the same statement for $TV$, the non reduced tensor coalgebra.

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 arxiv.org/abs/math/0310337 – Fernando Muro Dec 28 2011 at 21:30 Thanks for the reference, but I think it does not address my question since in his Lemme 1.1.2.2 he only deals about coderivations of the reduced tensor coalgebra. – yael fregier Dec 28 2011 at 23:07