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Let $k$ be a commutative ring with $1$, and let $V$ be a $k$-module. Let $TV$ be the $k$-module $\bigoplus\limits_{n\in\mathbb N}V^{\otimes n}$, where all tensor products are over $k$.

We define a $k$-linear map $\mathrm{shf}:\left(TV\right)\otimes\left(TV\right)\to TV$ by

$\mathrm{shf}\left(\left(a_1\otimes a_2\otimes ...\otimes a_i\right)\otimes\left(a_{i+1}\otimes a_{i+2}\otimes ...\otimes a_n\right)\right)$

$= \sum\limits_{\sigma\in\mathrm{Sh}\left(i,n-i\right)} a_{\sigma^{-1}\left(1\right)} \otimes a_{\sigma^{-1}\left(2\right)} \otimes ... \otimes a_{\sigma^{-1}\left(n\right)}$

for every $n\in \mathbb N$ and $a_1,a_2,...,a_n\in V$. Here, $\mathrm{Sh}\left(i,n-i\right)$ denotes the set of all $\left(i,n-i\right)$-shuffles, i. e. of all permutations $\sigma\in S_n$ satisfying $\sigma\left(1\right) < \sigma\left(2\right) < ... < \sigma\left(i\right)$ and $\sigma\left(i+1\right) < \sigma\left(i+2\right) < ... < \sigma\left(n\right)$.

We define a $k$-linear map $\eta:k\to TV$ by $\eta\left(1\right)=1\in k=V^{\otimes 0}\subseteq TV$.

We define a $k$-linear map $\Delta:TV\to \left(TV\right)\otimes\left(TV\right)$ by

$\Delta\left(a_1\otimes a_2\otimes ...\otimes a_n\right) = \sum\limits_{i=0}^n \left(a_1\otimes a_2\otimes ...\otimes a_i\right)\otimes\left(a_{i+1}\otimes a_{i+2}\otimes ...\otimes a_n\right)$

for every $n\in \mathbb N$ and $a_1,a_2,...,a_n\in V$.

We define a $k$-linear map $\varepsilon:TV\to k$ by

$\varepsilon\left(x\right)=x$ for every $x\in V^{\otimes 0}=k$, and

$\varepsilon\left(x\right)=0$ for every $x\in V^{\otimes n}$ for every $n\geq 1$.

Then the claim is:

1. The $k$-module $TV$ becomes a Hopf algebra with multiplication $\mathrm{shf}$, unit map $\eta$, comultiplication $\Delta$ and counit $\varepsilon$. It even becomes a graded Hopf algebra with $n$-th graded component $V^{\otimes n}$.

2. The antipode $S$ of this Hopf algebra satisfies

$S\left(v_1\otimes v_2\otimes ...\otimes v_n\right) = \left(-1\right)^n v_n\otimes v_{n-1}\otimes ...\otimes v_1$

for every $n\in \mathbb N$ and any $v_1,v_2,...,v_n\in V$.

I call this Hopf algebra the shuffle Hopf algebra, although I am not sure whether this is the standard notion. What I know is that the algebra part of it is called the shuffle algebra (note that it is commutative), while the coalgebra part of it is called the tensor coalgebra or deconcatenation coalgebra.

Question: Is there a slick, or at least a not-too-long proof (I'm speaking of <10 pages in detail) for the statements 1 and 2? The best I can come up with is this here:

For 1, we WLOG assume that $V$ is a finite free $k$-module (because all we have to prove are some identities involving finitely many elements of $V$; now we can see these elements as images of a map from a finite free $k$-module $W$, and by functoriality it is thus enough to prove these identities in $W$). Then, we have $V^{\ast\ast}\cong V$, and we notice that the graded dual of our above graded Hopf algebra (we don't know that it is a graded Hopf algebra yet, but at least it has the right signature) is the tensor Hopf algebra of $V^{\ast}$, for which Hopf algebraicity is much easier to show. (Note that this only works with the graded dual, not with the standard dual, because $TW$ is free but not finite free.)

For 2, we prove that $v_1\otimes v_2\otimes ...\otimes v_n\mapsto \left(-1\right)^n v_n\otimes v_{n-1}\otimes ...\otimes v_1$ is indeed a $\ast$-inverse of $\mathrm{id}$ by checking the appropriate equalities combinatorially (i. e., showing that positive and negative terms cancel out).

These things are ultimately not really difficult, but extremely annoying to write up. Somehow it seems to me that there are simpler proofs, but I am unable to find any proof of this at all in literature (except of the "obviously" kind of proof).

One reason why I am thinking that there are simpler proofs is that the similar statements for the tensor Hopf algebra (this is another Hopf algebra with underlying $k$-module $TV$; it has the same counit and unit map as the shuffle Hopf algebra, but the multiplication is the standard tensor algebra multiplication, and the comultiplication is the so-called shuffle comultiplication) are significantly easier to prove. In particular, 2 holds verbatim for the tensor Hopf algebra, but the proof is almost trivial (since $v_1\otimes v_2\otimes ...\otimes v_n$ equals $v_1\cdot v_2\cdot ...\cdot v_n$ in the tensor Hopf algebra).

What would Grothendieck do? Is there a good functorial interpretation, i. e., is the algebraic group induced by the shuffle Hopf algebra (since it is commutative) of any significance?

2 the section about the embedding was nonsense.; edited tags; edited tags

One reason why I am thinking that there are simpler proof proofs is that the similar statements for the tensor Hopf algebra (this is another Hopf algebra with underlying $k$-module $TV$; it has the same counit and unit map as the shuffle Hopf algebra, but the multiplication is the standard tensor algebra multiplication, and the comultiplication is the so-called shuffle comultiplication) are significantly easier to prove. In particular, 2 holds verbatim for the tensor Hopf algebra, but the proof is almost trivial (since $v_1\otimes v_2\otimes ...\otimes v_n$ equals $v_1\cdot v_2\cdot ...\cdot v_n$ in the tensor Hopf algebra).

As far as I understand, we can identify the $k$-algebra $\left(TV,\mathrm{shf},\eta\right)$ with the symmetric part of the tensor algebra $TV$ if $k$ is a field of characteristic $0$ (or, more generally,

What would Grothendieck do? Is there a $\mathbb Q$-algebra)good functorial interpretation, but I am not sure about what this does to the coalgebra structurei. e., and even if it respects is the coalgebra structure, this would only give us a proof of algebraic group induced by the above statements for characteristic $0$, and we would have to do some additional work to generalize it shuffle Hopf algebra (although that since it is the easy part).commutative) of any significance?

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