文摘
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 <em>L em> of linear homogeneous partial differential equations, with the property that its solution vector fields form a Lie algebra class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300025&_mathId=si1.gif&_user=111111111&_pii=S0747717116300025&_rdoc=1&_issn=07477171&md5=6b59be463dce3062309ca84687bca067" title="Click to view the MathML source">Lclass="mathContainer hidden">class="mathCode">. 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300025&_mathId=si1.gif&_user=111111111&_pii=S0747717116300025&_rdoc=1&_issn=07477171&md5=6b59be463dce3062309ca84687bca067" title="Click to view the MathML source">Lclass="mathContainer hidden">class="mathCode">, if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300025&_mathId=si1.gif&_user=111111111&_pii=S0747717116300025&_rdoc=1&_issn=07477171&md5=6b59be463dce3062309ca84687bca067" title="Click to view the MathML source">Lclass="mathContainer hidden">class="mathCode"> is abelian and if a system class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300025&_mathId=si2.gif&_user=111111111&_pii=S0747717116300025&_rdoc=1&_issn=07477171&md5=322c9f62c44eecf23ccfc2d2ca2dda78" title="Click to view the MathML source">L′class="mathContainer hidden">class="mathCode"> specifies an ideal in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300025&_mathId=si1.gif&_user=111111111&_pii=S0747717116300025&_rdoc=1&_issn=07477171&md5=6b59be463dce3062309ca84687bca067" title="Click to view the MathML source">Lclass="mathContainer hidden">class="mathCode">. 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.