I am reading the fascinating paper of Deligne on "le groupe fondamental de la droite projective moins trois points", and other stuffs related to anabelian geometry. This suggested the following question, which seems natural to me and that I haven't seen
answered in the literature (but this is perhaps a consequence of my still very low understanding of those matters).


Let $X$ be a proper and smooth, geometrically connected, variety over ${\mathbb Q}$. 
Let us assume that $X$ has a rational point $x$ over ${\mathbb Q}$. Let $\bar X = X \times\_{\mathbb Q} {\bar {\mathbb Q}}$. Then there is a natural action of $G\_{\mathbb Q}=Gal(\bar {\mathbb Q}/ {\mathbb Q})$ on the profinite fundamental group $\pi\_1(\bar X,x)$. As in Deligne, let us make $\pi\_1(\bar X,x)$ nilpotent of order $N$ (by quotienting it by $[\dots[[\pi\_1,\pi\_1],\pi\_1],\dots]$ where the number of $[$ is $N$), then take its pro-$l$-Sylow (for a given prime $l$), and quotient it by its torsion (which is a normal subgroup): we obtain a nilpotent torsion free pro-$l$ group $P\_{l,N}$ which inherits a continuous action of $G\_{\mathbb Q}$. Such a group has a finite dimensional Lie algebra over $Z\_l$, and let us call $V\_{l,N}$ this Lie algebra tensorized by $Q\_l$ (just for convenience, to have something over a field). Hence we have a finite dimensional Lie algebra $V\_{l,N}$ on which $G\_{{\mathbb Q}}$ acts continuously by Lie algebra automorphisms. In particular, forgetting the Lie algebra structure, $V\_{l,N}$ is a continuous Galois representation of $G\_{\mathbb Q}$.


(From here, I am really on uncharted territory for me, and I coulsd possibly have everything wrong). One of the first aim of Deligne's paper is to interpret $V\_{l,N}$ as the $l$-adic realization of a mixed motive. In Deligne's original paper,
a motive is just defined as a system of realizations, which means that what Deligne has to do, and does, is just to construct $l'$-adic analogs of $V\_{l,N}$ (for $l'$ an other prime - the construction is just by replacing $l$ by $l'$ from the beginning) and a Betty analog,
and a De Rham analog, and compatibilities between them. If I am not mistaken,
the underlying motive should be mixed, not pure, even if we have assumed $X$ proper and smooth (excepted if $N=1$ where $V\_{l,N}$ is just the $l$-adic homology of $X$ and hence
shoould be be pure). In particular, we do not expect $V\_{l,N}$ to be semi-simle as a Galois
representation in general. First question: is the above correct? Second question: if it is,



> 
> Do we know any case of $X$, $x$ as above where $V\_{l,N}$ is non semi-simple?
> 
> 
> 



Motives from the fundamental group made nilpotent
As you say, in general the representation should not be semi-simple even if $X$ is smooth projective. One can construct explicit examples as follows:


Let $X$ be any smooth projective curve of genus $g=2$ over $\mathbb{Q}$ with a rational point such that for some prime $p$, $X$ has a regular model over $\mathbb{Z}\_p$ with special fibre consisting of a union of two elliptic curves (one can do analogous things for larger $g$). Since the Jacobian of $X$ has good reduction at $p$ it follows that $H^1(X, \mathbb{Q}\_l)$ is unramified at $p$.


By a theorem of Takayuki Oda (Galois action on the nilpotent completion of the fundamental group of an algebraic curve. Advances in number theory (Kingston, ON, 1991), 213–232) it follows that since $X$ does not have good reduction at $p$, the Galois representation on $V\_{l,N}$ is ramified at $p$ for some $N$. Now the pronilponent completion has a filtration whose associated quotients are quotients of $H\_1(X,\mathbb{Q}\_l)^{\otimes m}$ for various $m$ and therefore unramified. It follows that $V\_{l,N}$ cannot be semisimple.


