Question

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!