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


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
Full Text: DOI