find tensors or differential forms which are invariant under the infinitesimal action of a set of matrices - Maple Programming Help

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Tensor[InvariantTensorsAtAPoint] - find tensors or differential forms which are invariant under the infinitesimal action of a set of matrices

Calling Sequences

InvariantTensorsAtAPoint(A, S, options)

Parameters

A        - a list of square matrices, with dimension equal to the dimension of the space on which the tensors S are defined

S        - a list of tensors or differential forms, each of the same index type

options  - the keyword argument output

Description

 • This command calculates the tensors in the span of the tensors in the list S which are invariant with respect to the infinitesimal action generated by the matrices in the list A. This is a pointwise calculation.
 • Let  be the coordinates in terms of which the tensors in the list S are defined. If and then If   , then If  and are tensors, then . Thus, the action of $P$ on a tensor  defined at a point coincides with the Lie derivative of (as a tensor with constant coefficients) with respect to the linear vector fieldthat is, . See Example 6 for examples of this action of matrices on tensors.
 • If and then InvariantTensorsAtAPoint(A, S) returns a basis for the vector space of all tensors  (constant) such that
 • If no invariant tensors exist, an empty list is returned.
 • With output = "list", a list of invariant tensors is returned. This is the default. With output = "general", a single tensor with arbitrary coefficients is returned. If the number of matrices in the list A is 1 and output = "action", then the action of the matrix in A on the tensors in S is returned.
 • In many cases, the list of tensors to be used by InvariantTensorsAtAPoint can be created with the commands GenerateTensors, GenerateSymmetricTensors, GenerateForms.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$$\mathrm{with}\left(\mathrm{GroupActions}\right):$

Example 1.

Define a list of matrices for the first argument of InvariantTensorsAtAPoint .

 > $A≔\left[\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,-1\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,1\right],\left[0,0\right]\right]\right)\right]$
 ${A}{:=}\left[\left[\begin{array}{rr}{1}& {0}\\ {0}& {-}{1}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {1}\\ {0}& {0}\end{array}\right]\right]$ (2.1)

Define a 2-dimensional space on which the tensors $S$ for the second argument of InvariantTensorsAtAPoint will be defined.

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.2)

We take for the space of all rank 2 covariant tensors on $M$.

 M > $S≔\mathrm{evalDG}\left(\left[\mathrm{dx}&t\mathrm{dx},\mathrm{dx}&t\mathrm{dy},\mathrm{dy}&t\mathrm{dx},\mathrm{dy}&t\mathrm{dy}\right]\right)$
 ${S}{:=}\left[{\mathrm{dx}}{}{\mathrm{dx}}{,}{\mathrm{dx}}{}{\mathrm{dy}}{,}{\mathrm{dy}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{dy}}\right]$ (2.3)
 M > $\mathrm{InvariantTensorsAtAPoint}\left(A,S\right)$
 $\left[{-}{\mathrm{dx}}{}{\mathrm{dy}}{+}{\mathrm{dx}}{}{\mathrm{dy}}\right]$ (2.4)

Example 2.

Here we consider a simple example where the matrices depend upon the coordinates of the manifold on which the tensors are defined.

 M > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.5)
 > $A≔\left[\mathrm{Matrix}\left(\left[\left[0,1,0\right],\left[-1,0,0\right],\left[0,0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0,1\right],\left[0,0,0\right],\left[-\frac{1}{{y}^{2}},0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0,0\right],\left[0,0,1\right],\left[0,-\frac{1}{{y}^{2}},0\right]\right]\right)\right]$
 ${A}{:=}\left[\left[\begin{array}{rrr}{0}& {1}& {0}\\ {-}{1}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{ccc}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {-}\frac{{1}}{{{y}}^{{2}}}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{ccc}{0}& {0}& {0}\\ {0}& {0}& {1}\\ {0}& {-}\frac{{1}}{{{y}}^{{2}}}& {0}\end{array}\right]\right]$ (2.6)

We take for the space of all symmetric rank-2 covariant tensors on $M$.

 M > $S≔\mathrm{evalDG}\left(\left[\mathrm{dx}&t\mathrm{dx},\mathrm{dx}&s\mathrm{dy},\mathrm{dx}&t\mathrm{dz},\mathrm{dy}&t\mathrm{dy},\mathrm{dy}&s\mathrm{dz},\mathrm{dz}&t\mathrm{dz}\right]\right)$
 ${S}{:=}\left[{\mathrm{dx}}{}{\mathrm{dx}}{,}\frac{{1}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dx}}{,}{\mathrm{dx}}{}{\mathrm{dz}}{,}{\mathrm{dy}}{}{\mathrm{dy}}{,}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dy}}{,}{\mathrm{dz}}{}{\mathrm{dz}}\right]$ (2.7)

We find that the $A$-invariant tensors vary with the coordinate.

 M > $\mathrm{InvariantTensorsAtAPoint}\left(A,S\right)$
 $\left[\frac{{\mathrm{dx}}{}{\mathrm{dx}}}{{{y}}^{{2}}}{+}\frac{{\mathrm{dy}}{}{\mathrm{dy}}}{{{y}}^{{2}}}{+}{\mathrm{dz}}{}{\mathrm{dz}}\right]$ (2.8)

Example 3.

The classical simple Lie algebras can be defined as matrix algebras which leave a tensor or a collection of tensors invariant. In this example we check that the matrices defining the real sympletic algebra leave invariant a non-degenerate 2-form.

We first use the commands SimpleLieAlgebraData and StandardRepresentation to obtain the matrices defining .

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("sp\left(4, R\right)",\mathrm{sp4R}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}\right]\right]$