# “Higher” Tangent spaces in char-p geometry - definition?

Hi, everyone!

I have some construction that requires exact definition.

I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more generally - from infinite Witt vectors or for example from $U_m(\Bbbk)$ - I'll write here the simplest case). There is a simple classification of such homorphisms: $$\lambda (\in \Bbbk) \mapsto \prod_{i=1}^{n} \exp(\sum_{j=0}^{m_i}a_{ij}\lambda^{p^j}N_i),$$ where $N_i$'s are commuting nil-triangular matrices with $N_i^p=0$.

$\sum_{i=0}^n a_{i0}N_i$ will be the tangent vector in unity of our one-parameter subgroup. But we see that $W_j=\sum_{i=0}^na_{ij}N_i$ are also important vectors (they are from the coefficient at $\lambda^{p^j}$ and play the role of "trimming" vectors). The similar situation will be in homomorphisms from infinite Witt vectors and from $U_m$

I seems that in char-p geometry must be "higher" tangent spaces for algebraic subvarieties (they come from $p^j$'s powers which are going to zero for any differentiation) - but I have no idea how can they be correctly constructed. Motivation for such spaces comes from algebraic groups in char-p.

So, my question will be about any possibilities for definition of such things.

Thanks!

-
I do not feel competent enough to give a proper answer, but it seems that the key-word you are looking for is "p-Lie algebra". For an algebraic group over a field of char. p>0, you have (at least) three infinitesimal invariants, of increasing strength : the Lie algebra, the p-Lie algebra, and the formal group. This hierarchy collapses in char. 0 where the Lie algebra determines the formal group. The p-Lie algebra can be exponentiated "up to height 1". For p-Lie algebras (also called restricted Lie algebras), see Borel, Linear Algebraic Groups, I.3.1, and Pseudo-reductive groups, App. A7 – Simon Pepin Lehalleur Oct 4 '12 at 12:41
@SimonPL, thanks for this references! However they seem to be unrelated to my question but still very interesting! – zroslav Oct 6 '12 at 22:06
Sorry, I also don't feel like I really understand the question, but so-called Hasse-Schmidt derivations give a rather natural extension of something that plays the role of the tangent space. These also have a sensible relationship to jet spaces, which can be thought of as higher "tangent spaces" in a sense. See here arxiv.org/pdf/math/0407113.pdf – Jon Beardsley Aug 30 '15 at 1:40