Polynomial and horizontally polynomial functions on Lie groups
Antonelli, G., & Le Donne, E. (2022). Polynomial and horizontally polynomial functions on Lie groups. Annali di Matematica Pura ed Applicata, Early online. https://doi.org/10.1007/s10231-022-01192-z
Published inAnnali di Matematica Pura ed Applicata
DisciplineGeometrinen analyysi ja matemaattinen fysiikkaMatematiikkaAnalyysin ja dynamiikan tutkimuksen huippuyksikköGeometric Analysis and Mathematical PhysicsMathematicsAnalysis and Dynamics Research (Centre of Excellence)
© The Author(s) 2022
We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset S of the algebra g of left-invariant vector fields on a Lie group G and we assume that S Lie generates g. We say that a function f:G→R (or more generally a distribution on G) is S-polynomial if for all X∈S there exists k∈N such that the iterated derivative Xkf is zero in the sense of distributions. First, we show that all S-polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent k in the previous definition is independent on X∈S, they form a finite-dimensional vector space. Second, if G is connected and nilpotent, we show that S-polynomial functions are polynomial functions in the sense of Leibman. The same result may not be true for non-nilpotent groups. Finally, we show that in connected nilpotent Lie groups, being polynomial in the sense of Leibman, being a polynomial in exponential chart, and the vanishing of mixed derivatives of some fixed degree along directions of g are equivalent notions. ...
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Related funder(s)European Commission; Academy of Finland
Funding program(s)Academy Project, AoF; Research post as Academy Research Fellow, AoF
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
Additional information about fundingG.A. was partially supported by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’). E.L.D. was partially supported by the Academy of Finland (grant 288501 ‘Geometry of subRiemannian groups’ and by grant 322898 ‘Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory’) and by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’) ...
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