Algorithmic calculus for Lie determining systems. (English) Zbl 1361.35012
Summary: 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 \(\mathcal{L}\). 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 \(\mathcal{L}\), if \(\mathcal{L}\) is abelian and if a system \(L^\prime\) specifies an ideal in \(\mathcal{L}\). 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.
MSC:
| 35A30 | Geometric theory, characteristics, transformations in context of PDEs |
| 17B05 | Structure theory for Lie algebras and superalgebras |
| 68W30 | Symbolic computation and algebraic computation |
Keywords:
determining equations; Lie symmetry; Lie algebra; structure constants; differential elimination; algorithmReferences:
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