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Let $k$ be a field of characteristic $0$.

Let $H$ be a cocommutative connected filtered bialgebra over $k$. ("Connected" means that the counit, restricted to the $0$-th part of the filtration, is an isomorphism. Since $H$ is a connected filtered bialgebra, it is automatically a Hopf algebra.)

The convolution algebra $\mathcal L\left(H,H\right)$ of all $k$-linear maps $H\to H$ is local; its maximal ideal is the ideal $\mathfrak g\left(H,H\right)$ of all $k$-linear maps $H\to H$ sending $1$ to $0$. Denoting by $e$ the unity of the algebra $\mathcal L\left(H,H\right)$ (this $e$ is the map which sends every $x\in H$ to $\varepsilon\left(x\right)\cdot 1_H$), we can define an exponential mapping $\exp^{\ast}: \mathfrak g\left(H,H\right)\to e+\mathfrak g\left(H,H\right)$ and a logarithm mapping $\log^{\ast}: e+\mathfrak g\left(H,H\right)\to \mathfrak g\left(H,H\right)$ by the formulae

$\exp^{\ast}\left(f\right)=\sum\limits_{i\geq 0}\dfrac{f^{\ast i}}{i!}$

and

$\log^{\ast}\left(e+f\right)=\sum\limits_{i\geq 1}\dfrac{\left(-1\right)^{i-1}}{i}f^{\ast i}$,

where $f^{\ast i}$ denotes the $i$-th power of $f$ in the convolution algebra $\mathcal L\left(H,H\right)$. Both of these power series converge pointwise because $H$ is filtered. It is easy to see (and annoying to write up...) that $\exp^{\ast}\circ\log^{\ast}=\mathrm{id}$ and $\log^{\ast}\circ\exp^{\ast}=\mathrm{id}$.

So far we have been doing general stuff, not using the cocommutativity of $H$, and not using the fact that $H=H$ either (i. e., we could just as well have done all the above in $\mathcal L\left(C,A\right)$ with $C$ being a connected filtered $k$-coalgebra, and $A$ a $k$-algebra). But in the case of cocommutative $H$, we have something nice:

Theorem (Patras, Reutenauer, Garsia?). The map $\log^{\ast}\mathrm{id}$ is a projection of $H$ onto the subspace consisting of all primitive elements of $H$.

I can prove this using lots and lots of computations and an umbral-calculus trick (I can send the proof to anyone interested, but it's boredom guaranteed). The paper where I know this theorem from gives three references I cannot get any quick use of ([R2] [GR] [P1] as referenced in the Example on page 4; two of these are Solomon descent algebra literature, and the third is a book by Reutenauer that I'll one day read from one end to the other, but not now...). There is a shortcut through finite-dimensionality and dualization, but I don't like this kind of approaches as they use the fact of $k$ being a field (whereas the Theorem holds for $k$ being any commutative ring with $1$).

While I would be glad to hear a nicer proof of the Theorem, here is the

Actual question: For every $s\in\mathbb N$, we can define a mapping $\exp_s^{\ast}: \mathfrak g\left(H,H\right)\to e+\mathfrak g\left(H,H\right)$ by the formula

$\exp_s^{\ast}\left(f\right)=\sum\limits_{i=0}^s \dfrac{f^{\ast i}}{i!}$.

This is just the exponential series, stopped at $i=s$ and applied to $f$.

Now, I conjecture that $\log^{\ast}\left(\exp_s^{\ast}\left(\mathrm{id}-e\right)\right)$ is a projection of $H$ onto the subspace $C^{i+}$, where $\left(C^{\ell}\right)_{\ell\geq 0}$ is the coradical filtration of the coalgebra $H$ (note that $C^0 = H^0 = k\cdot 1$, and that $C^1 = k\cdot 1 + \left(\text{subspace of primitive elements of }H\right)$), and $S^+$ denotes $S\cap \mathrm{Ker}\varepsilon$ for every subspace $S$ of $H$.

The only evidence I have for this is that it's true for $s=0$ (obviously), for $s=1$ (this is the above Theorem) and for $s=\infty$ (not exactly a natural number, but a reasonable value to plug in the conjecture). Don't these data cry for a deeper meaning?...

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6 Answers

up vote 1 down vote accepted

Sorry, people. The conjecture is false. For a counterexample, take $H=k\left[x\right]$ (the usual polynomial algebra with shuffle comultiplication) and $s=2$. The image of $x^4$ under $\log^{\ast}\left(\exp^{\ast}_2\left(\mathrm{id}-e\right)\right)$ will be $-2x^4$, and this is not in the second term of the coradical filtration.

I was fooled by the fact that (in general, even if $H$ is not cocommutative!) the map $\log^{\ast}\left(\exp^{\ast}_s\left(\mathrm{id}-e\right)\right)$ restricts to the identity on $C^s$. This fact, though, is a triviality, since $C^s$ is a subcoalgebra of $C$ on which $\exp^{\ast}$ and $\exp^{\ast}_s$ are one and the same thing. (This argument doesn't always work over a ring, but this isn't hard to repair. Instead of restricting maps to $C^s$, consider maps modulo maps which vanish on $C^s$.)

A projection from a vector space $V$ onto a subspace $U$ is characterized by two properties: that it restricts to the identity on $U$, and that it maps everything onto $U$. The first of these properties being satisfied, I got overly optimistic that the second one couldn't be that much false. Turned out that it is. (Also, I got overly optimistic by the fact that the conjecture holds for $s=2$ on the third graded component of the tensor Hopf algebra; but the fourth graded component gives a contradiction.)

I am left wondering whether there are some reasonable algebraically-defined projections onto $C^s$ for $s\neq 0,1,\infty$. We shouldn't necessarily look for variations on the Eulerian idempotent; there might also be something similar to the Dynkin idempotent (or the Klyachko one, but I haven't even started to learn about that).

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Indeed ! But, you need to pass properly to the limit. You can use $log_s$ (the $s$ first terms of the expansion) and then remark that, when you apply both sides to a given $f\in H$, the limit is obtained by the polynomial $log_s$ with $s$ sufficiently large.

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It's not the "passing to the limit" part that gave me trouble. It's the whole proof that $\Delta\circ\left(\log I\right) = \log\left(I\otimes I\right)\circ\Delta$ (or, more generally, that the $*$-logarithm of a coalgebra homomorphism from a connected filtered cocommutative coalgebra to a bialgebra is an $\left(\varepsilon,\varepsilon\right)$-coderivation). I don't think this is a direct consequence of the fact "that morphism of bi-algebras commute with convolution": –  darij grinberg Jul 21 '12 at 13:27
    
Ah, I see what you are suggesting! Since the convolution of $n$ coalgebra homomorphisms from a cocommutative coalgebra to bialgebra is also a coalgebra homomorphism, we know that $I^{\ast n}$ is a coalgebra homomorphism for any $n\in\mathbb N$. Thus, $\Delta \circ I^{\ast n} = \left(I^{\ast n}\otimes I^{\ast n}\right)\circ \Delta$ for any $n\in\mathbb N$. This rewrites as $\Delta \circ I^{\ast n} = \left(I\otimes I\right)^{\ast n}\circ \Delta$ for any $n\in\mathbb N$ (since $I^{\ast n}\otimes I^{\ast n}=\left(I\otimes I\right)^{\ast n}$). Thus, ... –  darij grinberg Jul 21 '12 at 13:40
    
... for any polynomial $P\in k\left[X\right]$, we have $\Delta\circ P\left(I\right) = P\left(I\otimes I\right)\circ \Delta$. This must also hold when $P$ is a formal power series (by obvious limiting arguments), and thus for $P=\log$, and so we get $\Delta\circ \log\left(I\right) = \log\left(I\otimes I\right)\circ \Delta$. Thanks for this argument; I don't know how I missed it! –  darij grinberg Jul 21 '12 at 13:42
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OK, The argument goes as follows : you do two or three commutative diagrams showing that morphism of bi-algebras commute with convolution (I can send you them) and end by

\begin{eqnarray} &&log(I\otimes I)=log((I\otimes e)*(e\otimes I))=\cr &&log(I\otimes e)+log(e\otimes I)=log(I)\otimes e +e\otimes log(I) \end{eqnarray}

This proves that log(I) maps INTO the set of primitives and to end the proof just remark that log(I)(f)=f when f is primitive because the series ends at the first term.

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Are you using $\Delta \circ \left(\log I\right) = \log\left(I\otimes I\right)\circ \Delta$ ? This was the messiest part of my proof. –  darij grinberg Jul 21 '12 at 13:11
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While I would be glad to hear a nicer proof of the Theorem,

Sorry for late answer. You can get a rather quick proof of this theorem by remarking that, in your situation, when H is cocommutative, the comultiplication is a morphism of bi-algebras.

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How exactly does this argument continue? –  darij grinberg Jul 21 '12 at 12:49
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OK, I did not apply the general fact you mention (which I do not use but can imagine true under certain conditions). I just used thet fact that, if $\phi : \mathcal{B}_1\rightarrow \mathcal{B}_2$ is a morphism of bialgebras, and $\phi\circ f_1=f_2\circ\phi\ ;\ \phi\circ g_1=g_2\circ\phi$ for $f_i,g_i\in End(\mathcal{B}_i)$, then $\phi\circ (f_1*g_1)=(f_2*g_2)\circ\phi$ and then, as $(I\otimes I)\circ\Delta=\Delta\circ I$, one has
$$ log_s(I\otimes I)\circ\Delta=\Delta\circ log_s(I) $$ Hope that it helps. Tell me.

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OK, you have got it. I am glad that it could help. If you happen to come to Paris, please let me know, we can discuss these matters (I am often around the IHP of IHES areas).

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