The Lie Algebra of Vector Fields Package with Applications to Mappings of Differential Equations

  • Zahra MohammadiEmail author
  • Gregory J. Reid
  • S.-L. Tracy Huang
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 1125)


Lie symmetry groups of transformations (mappings) of differential equations leave them invariant, and are most conveniently studied through their Lie algebra of vector fields (essentially the linearization of the mappings around the identity transformation). Maple makes powerful and frequent use of such Lie algebras, mostly through routines that are dependent on Maple’s powerful exact integration routines, that essentially automate traditional hand-calculation strategies. However these routines are usually heuristic, and algorithmic approaches require a deeper integration of differential elimination (differential algebraic) approaches in applications to differential equations. This is the underlying motivation of the LieAlgebrasOfVectorFields (LAVF) package of Huang and Lisle. The LAVF package introduces a powerful algorithmic calculus for doing calculations with differential equations without the heuristics of integration to calculate efficiently many properties of such systems.

We use LAVF in the development of our MapDE package, which determines the existence of analytic invertible mappings of an input DE to target DE. Theory, algorithms, and examples of MapDE can be found in [5, 6]. Here we present a brief summary, through examples, of the application of LAVF to MapDE.


Symmetry Lie algebra Structure constants Differential algebra 


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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Zahra Mohammadi
    • 1
    Email author
  • Gregory J. Reid
    • 1
  • S.-L. Tracy Huang
    • 2
  1. 1.Department of Applied MathematicsUniversity of Western OntarioLondonCanada
  2. 2.Data61, CSIROCanberraAustralia

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