Algorithmic calculus for Lie determining systems

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The infinitesimal symmetries of differential equations (DEs) or other geometric objects provide key insight into their analytical structure, including construction of solutions and of mappings between DEs. This article is a contribution to the algorithmic treatment of symmetries of DEs and their applications. Infinitesimal symmetries obey a determining system L of linear homogeneous partial differential equations, with the property that its solution vector fields form a Lie algebra L. We exhibit several algorithms that work directly with the determining system without solving it. A procedure is given that can decide if a system specifies a Lie algebra L, if L is abelian and if a system L′ specifies an ideal in L. Algorithms are described that compute determining systems for transporter, Lie product and Killing orthogonal subspace. This gives a systematic calculus for Lie determining systems, enabling computation of the determining systems for normalisers, centralisers, centre, derived algebra, solvable radical and key series (derived series, lower/upper central series). Our methods thereby give algorithmic access to new geometrical invariants of the symmetry action.




Lisle, I. G., & Huang, S. L. T. (2017). Algorithmic calculus for Lie determining systems. Journal of Symbolic Computation, 79, 482–498.

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