TY - JOUR

T1 - Algorithmic calculus for Lie determining systems

AU - LISLE, Ian

AU - HUANG, Tracy

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - Algorithm

KW - Determining equations

KW - Differential elimination

KW - Lie algebra

KW - Lie symmetry

KW - Structure constants

UR - http://www.scopus.com/inward/record.url?scp=84963957949&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/algorithmic-calculus-lie-determining-systems

U2 - 10.1016/j.jsc.2016.03.002

DO - 10.1016/j.jsc.2016.03.002

M3 - Article

VL - 79

SP - 482

EP - 498

JO - Journal of Symbolic Computation

JF - Journal of Symbolic Computation

SN - 0747-7171

IS - 2

ER -