TY - JOUR

T1 - Invariant differential operators in positive characteristic

AU - Bouarroudj, Sofiane

AU - Leites, Dimitry

PY - 2018/4/1

Y1 - 2018/4/1

N2 - 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.

AB - 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.

KW - Invariant differential operator

KW - Positive characteristic

KW - Veblen's problem

UR - http://www.scopus.com/inward/record.url?scp=85039771800&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85039771800&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2017.11.048

DO - 10.1016/j.jalgebra.2017.11.048

M3 - Article

AN - SCOPUS:85039771800

VL - 499

SP - 281

EP - 297

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -