## Distributions over an algebra of truncated polynomials.(English. Russian original)Zbl 0671.17009

Math. USSR, Sb. 64, No. 1, 187-205 (1989); translation from Mat. Sb., Nov. Ser. 136(178), No. 2(6), 187-205 (1988).
In classical differential geometry there is the well-known Frobenius theorem stating that a smooth constant dimensional distribution on a manifold is integrable if and only if it is involutive. In this work, some variant of this theorem concerning the distributions over the algebra of truncated polynomials $${\mathcal O}_ n$$ is obtained.
From the author’s summary: The author describes the equivalence classes of TI-distributions, i.e., of those distributions $${\mathcal L}$$ with respect to which the algebra $${\mathcal O}_ n$$ has no nontrivial $${\mathcal L}$$- invariant ideals; he shows that over a perfect field any TI-distribution is equivalent to a general Lie algebra of Cartan type $$W_ s({\mathcal F})$$, and he finds all the forms of the Zassenhaus algebra, in the process making essential use of the theory of representations of the chromatic quiver of Kronecker.
Reviewer: J.Kubarski

### MSC:

 17B50 Modular Lie (super)algebras 14K05 Algebraic theory of abelian varieties 16Gxx Representation theory of associative rings and algebras 17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
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