Algorithmic calculus for Lie determining systems

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• 作者：Ian G. Lisle¹ ; ^{Ian.Lisle@canberra.edu.au" class="auth_mail" title="E-mail the corresponding author} ; S.-L. Tracy Huang ^{Tracy.Huang@uni.canberra.edu.au" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35-04 ; 35N10 ; 22-04 ; 53-04
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：March-April 2017
• 年：2017
• 卷：79
• 期：part_P2
• 页码：482-498
• 全文大小：477 K

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 L   of linear homogeneous partial differential equations, with the property that its solution vector fields form a Lie algebra 84687bca067" title="Click to view the MathML source">L$\mathcal{L}$. 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 84687bca067" title="Click to view the MathML source">L$\mathcal{L}$, if 84687bca067" title="Click to view the MathML source">L$\mathcal{L}$ is abelian and if a system L^′$L^{\prime }$ specifies an ideal in 84687bca067" title="Click to view the MathML source">L$\mathcal{L}$. 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.