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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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