## Gradings for nilpotent Lie algebras

We present a constructive approach to torsion-free gradings of Lie algebras. Our main result is the computation of a maximal grading. Given a Lie algebra, using its maximal grading we enumerate all of its torsion-free gradings as well as its positive gradings. As applications, we classify gradings in low dimension, we consider the enumeration of Heintze groups, and we give methods to find bounds for non-vanishing \(\ell^{q,p}\) cohomology.

### Acknowledgements Acknowledgements

All of the authors were 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). E.H. was also supported by the Vilho, Yrjö and Kalle Väisälä Foundation, and by the SISSA project DIP_ECC_MATE_CoordAreaMate_0459 - Dipartimenti di Eccellenza 2018 - 2022 (CUP: G91|18000050006). V.K. was also supported by the Emil Aaltonen foundation. F.T. was also supported by the University of Bologna, funds for selected research topics, and by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 777822 GHAIA Geometric and Harmonic Analysis with Interdisciplinary Applications, and the Swiss National Foundation grant 200020_191978.