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I am trying to understand a proof of a proposition dealing with primitive elements of the tensor bialgebra $(T(V),\mu ,\Delta')$. I want to show that for $x\in V^{\bigotimes n}$ $$x \text{ is primitive} \implies \gamma(x)=nx$$ where $$\gamma(v_1v_2...v_n)=[v_1,[v_2,...,[v_{n-1},v_n]...]]$$ and $[v,w]=vw-wv$.

The proof begins by defining the Euler operator $E:T(V)\to T(V)$ mapping $x\in V^{\bigotimes n}$ to $nx$.

Next, they say that one can show by induction that $\gamma \star id=E$. There are a two things that I don't understand, what is $\star$ in this context

What would $\gamma (1)$ be?

This is proposition 1.3.5 in algebraic operads (Loday & Vallette) by the way.

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The star is the convolution product of maps $TV\to TV$. It exists because TV is a bialgebra.

As for $\gamma(1)$: as the relation $\gamma\star\mathrm{id}=E$ is to hold, evaluating both sides on $1$ tells you that the only possibility is that $\gamma(1)=0$.

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  • $\begingroup$ Thank you, I had to read up on convolution but now I get it! $\endgroup$ Feb 6, 2017 at 9:57
  • $\begingroup$ I still can't show that $\gamma \star id=E$ using induction (according to the book this should be easy). If I understood the convolution right we have $$\gamma \star id=\mu \circ (\gamma \bigotimes id)\circ \Delta'$$ For $x\in K,x\in V$ it works out, but I fail to do the inductive step. The definition of $\Delta'(x),x\in V^{\bigotimes n}$ is quite complicated, it's a dubble sum over all $p+q=n$ and $\sigma\in sh(p,q)$, is it this definition that I should use or are there some other properties of the functions $\Delta',\mu, \gamma$ that makes the induction step easy? $\endgroup$ Feb 6, 2017 at 11:09

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