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An element $x$ of a Hopf algebra $H$, is called a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$. The set of primitive elements of $H$ is denoted $P(H)$. It can be shown that:

"If $H$ is a $\mathbb{k}$-Hopf algebra of finite $\mathbb{k}$-dimension and $\mathbb{k}$ is a field of characteristic zero, then $P(H)=\{0\}$"

The proof of the above fact, amounts to showing -inductively- that, if there is $0\neq x \in P(H)$ then $x^{n}$ (for all positive integers $n$) are linearly independent, which is a contradiction to the finite dimensionality of $H$.

Now the question: If we drop the requirement of characteristic zero (for the field $\mathbb{k}$ of coefficients) what are some examples violating the above proposition? In other words, considering finite dimensional hopf algebras over some finite field, let's say the group Hopf algebra of the cyclic group of order $2$ over a field of characteristic $2$, are there -explicit- examples of primitive elements?

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  • $\begingroup$ this is similar to question math.stackexchange.com/questions/1924526/…, which however received no feedback. $\endgroup$ – Konstantinos Kanakoglou Sep 14 '16 at 20:27
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    $\begingroup$ An easy example is $k[x]/x^p$ where $k$ has characteristic $p$ and $x$ is primitive. This is the group scheme $\alpha_p$ whose functor of points sends a commutative $k$-algebra $A$ to the additive group of $a \in A$ such that $a^p = 0$. $\endgroup$ – Qiaochu Yuan Sep 14 '16 at 20:37
  • $\begingroup$ Thanks for mentioning this. It would be interesting if you find some time to add some details and turn your comment into an answer. However, I am mainly interested into some group hopf algebra example. $\endgroup$ – Konstantinos Kanakoglou Sep 14 '16 at 20:49
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    $\begingroup$ The primitive elements of a group Hopf algebra are always $0$. This is easy to check by computing the coproduct of $\sum_{g \in G} c_g g$ and comparing it with $\left(\sum_{g \in G} c_g g\right) \otimes 1 + 1 \otimes \left(\sum_{g \in G} c_g g\right)$. $\endgroup$ – darij grinberg Sep 14 '16 at 20:54
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    $\begingroup$ Yes. If at least one $h \neq 1$ satisfies $c_h \neq 0$, then $\Delta\left(\sum_{g \in G} c_g g\right)$ has a nonzero coefficient in front of $h \otimes h$, whereas $\left(\sum_{g \in G} c_g g\right) \otimes 1 + 1 \otimes \left(\sum_{g \in G} c_g g\right)$ does not. So the only elements that have a chance to be primitive are those of the form $c_1 1$. But those can only be primitive if $c_1 = 0$. $\endgroup$ – darij grinberg Sep 14 '16 at 21:08
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First notice that in any Hopf algebra, of any characteristic, the subalgebra generated by the primitive elements is also a Hopf algebra. Secondly, notice that the space of primitive elements is a lie algebra with respect to the product $$[x,y]=xy-yx.$$ This is an easy calculation. Thirdly, in characteristic $p$, if $x$ is primitive, so is $x^p$. Again, a direct calculation. The main point of this is the following: in characteristic $p$ you have the notion of restricted lie algebra. This is a lie algebra with a $p$-th power operation, see https://en.wikipedia.org/wiki/Restricted_Lie_algebra. Any finite dimensional restricted lie algebra has a restricted universal enveloping algebra, which is a finite dimensional Hopf algebra, in which the space of primitive elements is exactly the restricted lie algebra you started with. So up to taking sub-Hopf algebras, all your examples are restricted universal enveloping algebras.

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