The LieAlgebrasOfVectorFields package provides a suite of commands and methods for the Lie equations which arise as the determining equations for symmetries of differential equations. The package attempts to extract information of Lie algebras of vector fields (LAVF) without having to solve its determining system. To be more specific, the package can extract information including both geometric and algebraic properties of a LAVF directly via its determining system.

The package deals with the determining system that can be in either un-solved or fully-solved, or any forms in between (i.e. partially solved).

The main object of this package, LAVF, represents a LAVF consisting of the determining system S and its formal vector field V where its components are (part of) dependent variables of S. Most methods are within the LAVF object including both algebraic & geometric, and covering both finite and infinite dimensional cases. See Overview of the LAVF object for more detail.

The primary job of this package is to provide constructor for the LAVF object, then many methods become accessible once a LAVF object is constructed.

The package also provides two other groups of commands for working with vector fields and linear homogeneous PDE. For more detail, see Overview of VectorField commands and Overview of Linear Homogeneous PDE commands.

The package utilizes several existing Maple packages such as DEtools and PDEtools. In particular, the command DEtools[rifsimp] is used heavily by the package.

Each command in the LieAlgebrasOfVectorFields package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

The long form, LieAlgebrasOfVectorFields:-command, is always available. The short form can be used after loading the package.

These are the exports of this package:

The following is a list of available methods for a LAVF object.

The following Maple builtins are extended to allow LAVF object: has, hastype, indets, type. See LAVF Object Overloaded Builtins for more detail.

The following is a list of available commands in a Distribution object.

The following Maple builtins functions are extended so that they work for a Distribution object: type, has, hastype, indets, convert. See Distribution Object Overloaded Builtins for more detail.

The following is a list of available methods for a VFPDO object.

A VFPDO object can also act as a differential operator. See VFPDO Object as Operator for more detail.

The following Maple builtins are extended to allow VFPDO object: has, hastype, indets, type. See VFPDO Object Overloaded Builtins for more detail.

The LieAlgebrasOfVectorFields package was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.