Algorithms of SageMath designed to construct optimal sets of subalgebras of Lie algebra of symmetries for 3D plasticity equations with Huber - von Mises criterion

For Lie algebra of point symmetries g, admitted by the system of differential equations for a quasi-static state of a perfectly rigid-plastic solid with Huber – von Mises yield criterion, we classify subalgebras up to dimension three. The group of inner automorphisms Int(g), generated by the adjoint algebra ad(g), is the principal tool to divide all subalgebras of g into classes of equivalence. Sometimes external discrete automorphisms of g are used to reduce the number of such classes and finally to form a set of representatives of each class. Such set is called the optimal system of subalgebras. To realize the classification, a set of algorithms was designed and implemented in CAS SageMath.

Although the classification process was not completely automated, its semi-automatic application was what allowed the result to be obtained. The algorithms and the files that are the result of the classification can be viewed in corresponding folders: Algorithms and Results. Files: CompiladoAlgoritmos_2022, AnalisisSubalgebras, AutomorfismoRotacion, Automorfismos, AutomorfismosAE, CambioVariableNuevo, CondicionSubalgebra, ContenenciaSubalgebras, ImprimirTexto, ReduccionGaussiana, varios with extension .ipynb are the modules that store the algorithms, which are executed from the SageMath notebook CompiledAlgorithms_2022.ipynb.

File tables_subalgebras_2D.pdf contains the classification results for two-dimensional subalgebras. File tables_subalgebras_3D.pdf includes all non-similar classes of three-dimensional subalgebras.

针对满足许布-冯·米塞斯（Huber – von Mises）屈服准则的理想刚塑性固体准静态状态的微分方程组所容许的点对称李代数𝔤，我们对其三维及以下的子代数完成分类。由伴随代数ad(𝔤)生成的内自同构群Int(𝔤)，是将𝔤的全体子代数划分为等价类的核心工具。有时我们会借助𝔤的外离散自同构来缩减此类等价类的数量，最终得到每个等价类的代表元集合，该集合被称为子代数最优系统。为完成该分类任务，我们设计了一套算法，并在计算机代数系统（Computer Algebra System, CAS）SageMath中予以实现。尽管整个分类流程并未实现完全自动化，但依托半自动化的实施路径，我们最终成功获取了分类结果。相关算法及分类结果文件可在对应文件夹“Algorithms”与“Results”中查阅。其中扩展名为.ipynb的算法模块文件包括CompiladoAlgoritmos_2022、AnalisisSubalgebras、AutomorfismoRotacion、Automorfismos、AutomorfismosAE、CambioVariableNuevo、CondicionSubalgebra、ContenenciaSubalgebras、ImprimirTexto、ReduccionGaussiana及varios，所有算法均通过SageMath笔记本CompiledAlgorithms_2022.ipynb启动运行。文件tables_subalgebras_2D.pdf收录了二维子代数的分类结果，tables_subalgebras_3D.pdf则涵盖了三维子代数的全部非相似等价类。