Distributions over an algebra of truncated polynomial

We study integrable distributions over the $K$-algebra $\mathscr O_n$ of truncated polynomials, where $K$ is a field of characteristic $p>0$. We obtain an analogue of the theorem of Frobenius; we describe the equivalence classes of $TI$-distributions, i.e., of those distributions $\mathscr L$ with respect to which the algebra $\mathscr O_n$ has no nontrivial $\mathscr L$-invariant ideals; we show that over a perfect field any $TI$-distribution is equivalent to a general Lie algebra of Cartan type $W_s(\mathscr F)$; and we find all the forms of the Zassenhaus algebra, in the process making essential use of the theory of representations of the chromatic quiver $_\circ\overrightarrow{_\rightsquigarrow}_\circ$ of Kronecker.
Bibliography: 13 titles.