A1 Journal article (refereed)
Polynomial and horizontally polynomial functions on Lie groups (2022)

Antonelli, G., & Le Donne, E. (2022). Polynomial and horizontally polynomial functions on Lie groups. Annali di Matematica Pura ed Applicata, 201(5), 2063-2100. https://doi.org/10.1007/s10231-022-01192-z

JYU authors or editors

Publication details

All authors or editors: Antonelli, Gioacchino; Le Donne, Enrico

Journal or series: Annali di Matematica Pura ed Applicata

ISSN: 0373-3114

eISSN: 1618-1891

Publication year: 2022

Publication date: 27/01/2022

Volume: 201

Issue number: 5

Pages range: 2063-2100

Publisher: Springer

Publication country: Germany

Publication language: English

DOI: https://doi.org/10.1007/s10231-022-01192-z

Publication open access: Openly available

Publication channel open access: Partially open access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/80056

Web address of parallel published publication (pre-print): https://arxiv.org/abs/2011.13665

Abstract

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.

Keywords: differential geometry; group theory; polynomials; harmonic analysis (mathematics)

Free keywords: nilpotent Lie groups; polynomial maps; Leibman Polynomial; polynomial on groups; horizontally affine functions; precisely monotone sets

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Ministry reporting: Yes

VIRTA submission year: 2022

JUFO rating: 1