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Let $N$ be a unipotent algebraic group over a field $k$ of characteristic $p>0$. Assume that $N$ is split (i.e. it admits a filtration whose quotients are isomorphic to the additive group). In particular, it is smooth and connected (since the additive group is smooth and connected and these two properties are stable under passing to extensions).

Then which of the following subgroups of $N$ are split?

  • the (scheme-theoretic) center of $N$ (or its identity component)
  • the maximal smooth central $k$-subgroup of $N$ (or its identity component)
  • the cc$kp$-kernel of $N$ (i.e. the maximal smooth connected central $p$-torsion $k$-subgroup of $N$)
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  • $\begingroup$ Your definition of "split" only makes sense for smooth connected unipotent groups, but there is a more general notion of unipotent group scheme (see section 1 of Expose XVII of SGA3). So probably you are assuming $N$ is smooth and connected, but its scheme-theoretic center is often neither smooth nor connected, so what exactly do you mean by $Z_N$? It will help to clarify what is your motivation. Would it be sufficient for your purposes (whatever they may be) to show that if $N$ is split and non-trivial then it admits $\mathbf{G}_{\rm{a}}$ as a central $k$-subgroup? (That does always hold.) $\endgroup$ – nfdc23 Aug 15 '17 at 23:35
  • $\begingroup$ Or maybe you just want that (when $N$ is split smooth connected unipotent and non-trivial) that there is a non-trivial split central smooth connected $k$-subgroup that is stable under all $k_s$-automorphisms of $N_{k_s}$? $\endgroup$ – nfdc23 Aug 16 '17 at 0:12
  • $\begingroup$ I think the definition of "split" makes sense for any unipotent group scheme, but indeed a split unipotent group is automatically smooth and connected (since these two properties hold true for the additive group and are stable under passing to extensions). As for my motivation it is indeed the following: any split unipotent group admits a (finite) series of characteristic split subgroups whose quotients are direct products of copies of the additive group. $\endgroup$ – Arkandias Aug 16 '17 at 8:06
  • $\begingroup$ (I can prove the result I need using the split part of the cckp-kernel, from Tits' Lecture on algebraic groups, but I came to think about this simple question on the center of split unipotent groups which I find interesting on its own.) $\endgroup$ – Arkandias Aug 16 '17 at 8:13
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    $\begingroup$ In Example K.1.1 of ams.org/open-math-notes/omn-view-listing?listingId=110663 you'll find an example of a 2-dimensional non-commutative extension $U$ of $\mathbf{G}_a$ by $\mathbf{G}_a$ over $\mathbf{F}_p$ such that the scheme-theoretic center $Z_U$ is equal to $\mathbf{G}_a \times \alpha_p$. (This non-commutative smooth connected $U$ is rather disorienting since its adjoint representation is trivial.) So the identity component of the scheme-theoretic center can fail to be smooth and hence not be "split" according to the definition you are using in the unipotent setting. $\endgroup$ – nfdc23 Aug 16 '17 at 19:18

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