Algorithmic calculus for Lie determining systems. (March 2017)

• Record Type: Journal Article
• Authors: Lisle, Ian G.; Huang, S.-L. Tracy
• Is Part Of: Journal of symbolic computation. Volume 79(2017)Part 2
• Page Start: 482
• Page End: 498
• Publication Date: 2017-03
• Subjects: 35-04 -- 35N10 -- 22-04 -- 53-04; Determining equations -- Lie symmetry -- Lie algebra -- Structure constants -- Differential elimination -- Algorithm; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2016.03.002
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: 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.