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Asked on math.stackexchange https://math.stackexchange.com/questions/2510606/what-are-the-primitive-elements-in-a-polynomial-hopf-algebra-with-primitive-inde but didn't get response (in fact got a negative vote without any comment), so trying here.

Is it true that in any polynomial Hopf algebra $K[X1,X2,...]$ over a field $K$ with indeterminates primitive, the primitive elements are precisely the linear homogeneous polynomials? (Perhaps with some additional assumptions like characteristic of $K$ is $0$?). If so, could someone kindly give me a reference? A paper I am reading says (without citation) that it is well-known.

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  • $\begingroup$ What paper are you reading? $\endgroup$
    – Yannic
    Commented Nov 9, 2017 at 13:15
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    $\begingroup$ Assuming you are in the graded situation and, yes, the characteristic is 0, this is standard. See the famous paper on Hopf algebras from the 1960's by Milnor and Moore. $\endgroup$
    – Nicholas Kuhn
    Commented Nov 9, 2017 at 15:53
  • $\begingroup$ Thank you for the reference. The paper I am reading is W. Schmitt Hopf algebra methods in graph theory (1995) $\endgroup$
    – John Doe
    Commented Nov 10, 2017 at 20:40
  • $\begingroup$ Schmitt does make a standing assumption that the characteristic of $K $ is zero in the paper you are reading. (Though it is somewhat hidden in Section 3.) $\endgroup$
    – darij grinberg
    Commented Dec 5, 2017 at 3:00
  • $\begingroup$ @darij He does make it clear earlier. But just wanted to know the precise result. My bad that I had been reading only the preprint, and should have looked at the published version which has more comments. $\endgroup$
    – John Doe
    Commented Dec 6, 2017 at 10:37

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No, in general the claim is not true:

To see why, 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 $x^p$: $$ \Delta(x^p)=1\otimes x^p+x^p\otimes 1 $$ (because in characteristic $p$: $\binom{p}{i}=0$, for $1\leq i\leq p-1 \ $).

However, in characteristic zero, $k[x]$ is generated by its primitive elements, which are in fact the homogeneous linear polynomials, in the sense that: $P(k[x])=kx$, as sets,and $k[x]\cong U\big(P(k[x])\big)\cong T(kx)$ as Hopf algebras.
($P(\cdot)$ denotes the Lie algebra of the primitives, $U(\cdot)$ stands for the universal enveloping algebra of $P(\cdot)$ and $T(.)$ the tensor or symmetric algebra of the one dimensional vector space $kx$).
Since you are also asking for some reference, the last statement (on characteristic zero) can be found cited explicitly (apart from the celebrated Milnor-Moore paper already cited at the comments to the OP) at the following sources:

  • Dascalescu's book, "Hopf algebras. an introduction", p.166, example 8
  • Montgomery's book, "Hopf algebras and their actions on rings", p.82, (see the discussion right after the example 5.6.8)
  • Abe's book, "Hopf algebras", p.61, example 2.6
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