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Polynomial and horizontally polynomial functions on Lie groups.

Gioacchino Antonelli1, Enrico Le Donne2,3,4

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Annali Di Matematica Pura Ed Applicata
|October 5, 2022
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Summary
This summary is machine-generated.

We introduce S-polynomial functions on Lie groups, generalizing polynomial functions and affine maps. These functions are analytic and form finite-dimensional spaces, particularly equivalent to Leibman polynomials in nilpotent Lie groups.

Keywords:
Horizontally affine functionsLeibman PolynomialNilpotent Lie groupsPolynomial mapsPolynomial on groupsPrecisely monotone sets

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Area of Science:

  • Lie group theory
  • Differential geometry
  • Harmonic analysis

Background:

  • Generalizing polynomial functions on Lie groups and horizontally affine maps on Carnot groups.
  • Defining S-polynomial functions based on iterated derivatives of left-invariant vector fields.

Purpose of the Study:

  • Generalize polynomial functions on Lie groups.
  • Characterize S-polynomial functions and their properties.
  • Establish equivalences between different notions of polynomiality in nilpotent Lie groups.

Main Methods:

  • Defining S-polynomial functions using iterated Lie derivatives.
  • Proving analyticity of S-polynomial functions.
  • Demonstrating finite-dimensionality for S-polynomial spaces under specific conditions.
  • Comparing S-polynomial functions with Leibman polynomials in nilpotent Lie groups.

Main Results:

  • S-polynomial functions and distributions are analytic.
  • If the degree is uniform, S-polynomials form a finite-dimensional vector space.
  • In connected nilpotent Lie groups, S-polynomial functions are equivalent to Leibman polynomials.
  • Equivalence of Leibman polynomiality, exponential chart polynomiality, and vanishing mixed derivatives in nilpotent Lie groups.

Conclusions:

  • The study extends the concept of polynomial functions to a broader class of Lie groups.
  • Established key properties and equivalences for S-polynomial functions, particularly in the context of nilpotent Lie groups.
  • Provides a unified framework for understanding polynomial structures in Lie group theory.