Structure theorem for non connected graded Hopf algebras Let $V^{\bullet}$ be a graded vector space. Assume that $V^{i}=0$ for $i<1$. Then the tensor algebra has the structure of a connected Hopf algebra and in particular its Lie algebra of primitive $P(T(V))$ is naturally ismorphic to the free Lie algebra. This result is a consequence of the structure theorem for connected Hopf algebras: given a connected Hopf algebra $H$ then
$$ H\cong U(P(H)), $$
where $U$ is the universal enveloping algebra. Here my questions
1) Is the connectess of $H$ a necessary requirement for the structure theorem? Do you know some counter examples?
2) Assume $V^{i}=0$ for $i<0$, then the tensor algebra $T(V)$ is again a graded Hopf algebra but it is no longer connected. Let $L(V)$ be the graded free Lie algebra on $V^{\bullet}$, are $P(T(V))$ and $L(V)$ isomorphic?

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*More general hypotheses are possible, but there are counterexamples. Take, for example, the group algebra of a nontrivial finite group, concentrated in degree $0$ (it's not clear to me whether you're talking about graded Hopf algebras here).  

*$T(V)$ is, for formal reasons, always the universal enveloping algebra of $L(V)$: this follows from the observation that adjoints compose. The remaining question is whether it's always true that the Lie algebra of primitive elements of a universal enveloping algebra (in this graded setting) recovers the original Lie algebra. I believe this is also true, at least over a field of characteristic zero, but it's less formal; I think you need a PBW-type result to handle the degree $0$ part. 
