Geometric features of Vessiot-Guldberg Lie algebras of conformal and Killing vector fields on \(\mathbb{R}^2\)

Problem Statement

This work belongs to mathematical physics rather than machine learning. It studies finite-dimensional Lie algebras of vector fields on \(\mathbb{R}^2\), especially those whose elements are conformal or Killing vector fields for a chosen pseudo-Riemannian metric. The published paper was extended from my BSc thesis at the University of Warsaw.

The organizing principle is geometric preservation. A Killing vector field preserves a metric, while a conformal vector field preserves it up to local scaling. The paper asks which finite-dimensional Vessiot-Guldberg Lie algebras can arise under these geometric constraints.

The classification is local: it describes the possible algebraic structures and then studies associated geometric features, including invariant distributions and induced symplectic structures.

The results are illustrated through two examples with physical motivation: the Milne-Pinney equation and the projective Schrödinger equation on the Riemann sphere. In both cases, the differential equation is analyzed through the finite Lie algebra of vector fields that controls its dynamics.

Link to neural PDE solvers and neural operators

In current symmetry-aware PDE learning, this BSc work can be read as a symmetry-identification step. The paper is not about neural operators, but it uses the same mathematical objects: infinitesimal generators, their Lie algebraic closure, and the geometric structures they preserve.

In Lie Point Symmetry Data Augmentation for Neural PDE Solvers, Lie point symmetries are used as solution-preserving transformations for data augmentation. In Lie Point Symmetry and Physics Informed Networks, the infinitesimal generators induce additional PINN constraints so that symmetry-generated neighbouring solutions remain compatible with the PDE. In Lie Algebra Canonicalization: Equivariant Neural Operators under Arbitrary Lie Groups, the connection is closest to neural-operator modelling: Lie algebra generators are used to canonicalize inputs and obtain equivariant PDE evolution without designing a group-specific architecture for the full symmetry group.

The workflow is: first identify which vector-field algebra is present and what it preserves; then use that algebra as an inductive bias for learning solution maps, augmenting PDE data, regularizing PINNs, or constructing equivariant neural operators.

@article{lewandowski2017geometric,
  title = {Geometric features of {Vessiot--Guldberg} Lie algebras of conformal and Killing vector fields on {$\mathbb{R}^2$}},
  author = {Lewandowski, M. M. and de Lucas, J.},
  journal = {Banach Center Publications},
  volume = {113},
  pages = {243--262},
  year = {2017},
  doi = {10.4064/bc113-0-13},
  url = {https://doi.org/10.4064/bc113-0-13},
  eprint = {1708.09375},
  archivePrefix = {arXiv},
  primaryClass = {math-ph}
}