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Invariant differential operators in positive characteristic
Stockholm University, Faculty of Science, Department of Mathematics. New York University Abu Dhabi, United Arab Emirates.
Number of Authors: 2
2018 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 499, p. 281-297Article in journal (Refereed) Published
Abstract [en]

In 1928, at the IMC, Veblen posed the problem: classify invariant differential operators between spaces of natural objects (in modern terms: either tensor fields, or jets) over a real manifold of any dimension. The problem was solved by Rudakov for unary operators (no nonscalar operators except the exterior differential); by Grozman for binary operators. In dimension one, Grozman discovered an indecomposable selfdual operator of order 3 that does not exist in higher dimensions. We solve Veblen's problem in the 1-dimensional case over any field of positive characteristic. Unary invariant operators are known: these are the exterior differential and analogs of the Berezin integral. We construct new binary operators from these analogs and discovered two more (up to dualizations) types of new indecomposable operators of however high order: analogs of the Grozman operator and a completely new type of operators. Gordan's transvectants, aka Cohen-Rankin brackets, always invariant with respect to the simple 3-dimensional Lie algebra, are also invariant, in characteristic 2, with respect to the whole Lie algebra of vector fields on the line when the height of the indeterminate is equal to 2.

Place, publisher, year, edition, pages
2018. Vol. 499, p. 281-297
Keyword [en]
Veblen's problem, Invariant differential operator, Positive characteristic
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-154702DOI: 10.1016/j.jalgebra.2017.11.048ISI: 000425578400011OAI: oai:DiVA.org:su-154702DiVA, id: diva2:1197259
Available from: 2018-04-12 Created: 2018-04-12 Last updated: 2018-04-12Bibliographically approved

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