What are the primitive elements in a polynomial hopf algebra with primitive indeterminates?

Question

Is there a result that says that in any polynomial Hopf algebra $K[X_1, X_2, ...]$ over a field $K$ with indeterminates primitive, the primitive elements are precisely the linear homogeneous polynomials? Is so, what is a reference?

Answer 1

No, in general the claim is not true:

Consider a field $k$ of characteristic $p$ and take the polynomial hopf algebra $k[x]$ (in a single variable). Then $x$ is primitive and so is $x^p$: $$ \Delta(x^p)=1\otimes x^p+x^p\otimes 1 $$ (Remember that we are in characteristic $p$ and every binomial coefficient $\binom{p}{i}$ for $1\leq i\leq p-1 \ $ is divisible by $p$, thus zero).

However, in characteristic zero, $k[x]$ is generated by its primitive elements (which are in fact the homogeneous linear polynomials), in the sense that: $P(k[x])=kx$ and $k[x]\cong U\big(P(k[x])\big)$
(here $P(\cdot)$ denotes the set of primitives and $U(\cdot)$ the universal enveloping algebra of the Lie algebra of the primitives).